The Vertical Linear Fractional Initialization Problem

NASA Technical Reports Server (NTRS)

Lorenzo, Carl F.; Hartley, Tom T.

1999-01-01

This paper presents a solution to the initialization problem for a system of linear fractional-order differential equations. The scalar problem is considered first, and solutions are obtained both generally and for a specific initialization. Next the vector fractional order differential equation is considered. In this case, the solution is obtained in the form of matrix F-functions. Some control implications of the vector case are discussed. The suggested method of problem solution is shown via an example.

Numerical solution of distributed order fractional differential equations

NASA Astrophysics Data System (ADS)

Katsikadelis, John T.

2014-02-01

In this paper a method for the numerical solution of distributed order FDEs (fractional differential equations) of a general form is presented. The method applies to both linear and nonlinear equations. The Caputo type fractional derivative is employed. The distributed order FDE is approximated with a multi-term FDE, which is then solved by adjusting appropriately the numerical method developed for multi-term FDEs by Katsikadelis. Several example equations are solved and the response of mechanical systems described by such equations is studied. The convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results.

Symmetry and singularity properties of second-order ordinary differential equations of Lie's type III

NASA Astrophysics Data System (ADS)

Andriopoulos, K.; Leach, P. G. L.

2007-04-01

We extend the work of Abraham-Shrauner [B. Abraham-Shrauner, Hidden symmetries and linearization of the modified Painleve-Ince equation, J. Math. Phys. 34 (1993) 4809-4816] on the linearization of the modified Painleve-Ince equation to a wider class of nonlinear second-order ordinary differential equations invariant under the symmetries of time translation and self-similarity. In the process we demonstrate a remarkable connection with the parameters obtained in the singularity analysis of this class of equations.

On the boundedness and integration of non-oscillatory solutions of certain linear differential equations of second order.

PubMed

Tunç, Cemil; Tunç, Osman

2016-01-01

In this paper, certain system of linear homogeneous differential equations of second-order is considered. By using integral inequalities, some new criteria for bounded and [Formula: see text]-solutions, upper bounds for values of improper integrals of the solutions and their derivatives are established to the considered system. The obtained results in this paper are considered as extension to the results obtained by Kroopnick (2014) [1]. An example is given to illustrate the obtained results.

Lines of Eigenvectors and Solutions to Systems of Linear Differential Equations

ERIC Educational Resources Information Center

Rasmussen, Chris; Keynes, Michael

2003-01-01

The purpose of this paper is to describe an instructional sequence where students invent a method for locating lines of eigenvectors and corresponding solutions to systems of two first order linear ordinary differential equations with constant coefficients. The significance of this paper is two-fold. First, it represents an innovative alternative…

Some efficient methods for obtaining infinite series solutions of n-th order linear ordinary differential equations

NASA Technical Reports Server (NTRS)

Allen, G.

1972-01-01

The use of the theta-operator method and generalized hypergeometric functions in obtaining solutions to nth-order linear ordinary differential equations is explained. For completeness, the analysis of the differential equation to determine whether the point of expansion is an ordinary point or a regular singular point is included. The superiority of the two methods shown over the standard method is demonstrated by using all three of the methods to work out several examples. Also included is a compendium of formulae and properties of the theta operator and generalized hypergeometric functions which is complete enough to make the report self-contained.

On solutions of the fifth-order dispersive equations with porous medium type non-linearity

NASA Astrophysics Data System (ADS)

Kocak, Huseyin; Pinar, Zehra

2018-07-01

In this work, we focus on obtaining the exact solutions of the fifth-order semi-linear and non-linear dispersive partial differential equations, which have the second-order diffusion-like (porous-type) non-linearity. The proposed equations were not studied in the literature in the sense of the exact solutions. We reveal solutions of the proposed equations using the classical Riccati equations method. The obtained exact solutions, which can play a key role to simulate non-linear waves in the medium with dispersion and diffusion, are illustrated and discussed in details.

FAST TRACK COMMUNICATION Quasi self-adjoint nonlinear wave equations

NASA Astrophysics Data System (ADS)

Ibragimov, N. H.; Torrisi, M.; Tracinà, R.

2010-11-01

In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation.

Semicommuting and Commuting Operators for the Heun Family

NASA Astrophysics Data System (ADS)

Batic, D.; Mills, D.; Nowakowski, M.

2018-04-01

We derive the most general families of first- and second-order differential operators semicommuting with the Heun class differential operators. Among these families, we classify all the families that commute with the Heun class. In particular, we find that a certain generalized Heun equation commutes with the Heun differential operator, which allows constructing a general solution of a complicated fourth-order linear differential equation with variable coefficients whose solution cannot be obtained using Maple 16.

The numerical solution of linear multi-term fractional differential equations: systems of equations

NASA Astrophysics Data System (ADS)

Edwards, John T.; Ford, Neville J.; Simpson, A. Charles

2002-11-01

In this paper, we show how the numerical approximation of the solution of a linear multi-term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity. We begin by showing how our method applies to a simple class of problems and we give a convergence result. We solve the Bagley Torvik equation as an example. We show how the method can be applied to a general linear multi-term equation and give two further examples.

Solving ay'' + by' + cy = 0 with a Simple Product Rule Approach

ERIC Educational Resources Information Center

Tolle, John

2011-01-01

When elementary ordinary differential equations (ODEs) of first and second order are included in the calculus curriculum, second-order linear constant coefficient ODEs are typically solved by a method more appropriate to differential equations courses. This method involves the characteristic equation and its roots, complex-valued solutions, and…

Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models.

PubMed

Shah, A A; Xing, W W; Triantafyllidis, V

2017-04-01

In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach.

Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models

PubMed Central

Xing, W. W.; Triantafyllidis, V.

2017-01-01

In this paper, we develop reduced-order models for dynamic, parameter-dependent, linear and nonlinear partial differential equations using proper orthogonal decomposition (POD). The main challenges are to accurately and efficiently approximate the POD bases for new parameter values and, in the case of nonlinear problems, to efficiently handle the nonlinear terms. We use a Bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values. Computational efficiency is ensured by using manifold learning to perform the emulation in a low-dimensional space. The accuracy of the method is demonstrated on a linear and a nonlinear example, with comparisons with a global basis approach. PMID:28484327

Improved pedagogy for linear differential equations by reconsidering how we measure the size of solutions

NASA Astrophysics Data System (ADS)

Tisdell, Christopher C.

2017-11-01

For over 50 years, the learning of teaching of a priori bounds on solutions to linear differential equations has involved a Euclidean approach to measuring the size of a solution. While the Euclidean approach to a priori bounds on solutions is somewhat manageable in the learning and teaching of the proofs involving second-order, linear problems with constant co-efficients, we believe it is not pedagogically optimal. Moreover, the Euclidean method becomes pedagogically unwieldy in the proofs involving higher-order cases. The purpose of this work is to propose a simpler pedagogical approach to establish a priori bounds on solutions by considering a different way of measuring the size of a solution to linear problems, which we refer to as the Uber size. The Uber form enables a simplification of pedagogy from the literature and the ideas are accessible to learners who have an understanding of the Fundamental Theorem of Calculus and the exponential function, both usually seen in a first course in calculus. We believe that this work will be of mathematical and pedagogical interest to those who are learning and teaching in the area of differential equations or in any of the numerous disciplines where linear differential equations are used.

Application of a local linearization technique for the solution of a system of stiff differential equations associated with the simulation of a magnetic bearing assembly

NASA Technical Reports Server (NTRS)

Kibler, K. S.; Mcdaniel, G. A.

1981-01-01

A digital local linearization technique was used to solve a system of stiff differential equations which simulate a magnetic bearing assembly. The results prove the technique to be accurate, stable, and efficient when compared to a general purpose variable order Adams method with a stiff option.

Linearly first- and second-order, unconditionally energy stable schemes for the phase field crystal model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yang, Xiaofeng, E-mail: xfyang@math.sc.edu; Han, Daozhi, E-mail: djhan@iu.edu

2017-02-01

In this paper, we develop a series of linear, unconditionally energy stable numerical schemes for solving the classical phase field crystal model. The temporal discretizations are based on the first order Euler method, the second order backward differentiation formulas (BDF2) and the second order Crank–Nicolson method, respectively. The schemes lead to linear elliptic equations to be solved at each time step, and the induced linear systems are symmetric positive definite. We prove that all three schemes are unconditionally energy stable rigorously. Various classical numerical experiments in 2D and 3D are performed to validate the accuracy and efficiency of the proposedmore » schemes.« less

The Use of Graphs in Specific Situations of the Initial Conditions of Linear Differential Equations

ERIC Educational Resources Information Center

Buendía, Gabriela; Cordero, Francisco

2013-01-01

In this article, we present a discussion on the role of graphs and its significance in the relation between the number of initial conditions and the order of a linear differential equation, which is known as the initial value problem. We propose to make a functional framework for the use of graphs that intends to broaden the explanations of the…

Third-Order Elliptic Lowpass Filter for Multi-Standard Baseband Chain Using Highly Linear Digitally Programmable OTA

NASA Astrophysics Data System (ADS)

Elamien, Mohamed B.; Mahmoud, Soliman A.

2018-03-01

In this paper, a third-order elliptic lowpass filter is designed using highly linear digital programmable balanced OTA. The filter exhibits a cutoff frequency tuning range from 2.2 MHz to 7.1 MHz, thus, it covers W-CDMA, UMTS, and DVB-H standards. The programmability concept in the filter is achieved by using digitally programmable operational transconductors amplifier (DPOTA). The DPOTA employs three linearization techniques which are the source degeneration, double differential pair and the adaptive biasing. Two current division networks (CDNs) are used to control the value of the transconductance. For the DPOTA, the third-order harmonic distortion (HD3) remains below -65 dB up to 0.4 V differential input voltage at 1.2 V supply voltage. The DPOTA and the filter are designed and simulated in 90 nm CMOS technology with LTspice simulator.

Keep Your Distance! Using Second-Order Ordinary Differential Equations to Model Traffic Flow

ERIC Educational Resources Information Center

McCartney, Mark

2004-01-01

A simple mathematical model for how vehicles follow each other along a stretch of road is presented. The resulting linear second-order differential equation with constant coefficients is solved and interpreted. The model can be used as an application of solution techniques taught at first-year undergraduate level and as a motivator to encourage…

New Operational Matrices for Solving Fractional Differential Equations on the Half-Line

PubMed Central

2015-01-01

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques. PMID:25996369

New operational matrices for solving fractional differential equations on the half-line.

PubMed

Bhrawy, Ali H; Taha, Taha M; Alzahrani, Ebraheem O; Alzahrani, Ebrahim O; Baleanu, Dumitru; Alzahrani, Abdulrahim A

2015-01-01

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Choi, Cheong R.

The structural changes of kinetic Alfvén solitary waves (KASWs) due to higher-order terms are investigated. While the first-order differential equation for KASWs provides the dispersion relation for kinetic Alfvén waves, the second-order differential equation describes the structural changes of the solitary waves due to higher-order nonlinearity. The reductive perturbation method is used to obtain the second-order and third-order partial differential equations; then, Kodama and Taniuti's technique [J. Phys. Soc. Jpn. 45, 298 (1978)] is applied in order to remove the secularities in the third-order differential equations and derive a linear second-order inhomogeneous differential equation. The solution to this new second-ordermore » equation indicates that, as the amplitude increases, the hump-type Korteweg-de Vries solution is concentrated more around the center position of the soliton and that dip-type structures form near the two edges of the soliton. This result has a close relationship with the interpretation of the complex KASW structures observed in space with satellites.« less

Modification of 2-D Time-Domain Shallow Water Wave Equation using Asymptotic Expansion Method

NASA Astrophysics Data System (ADS)

Khairuman, Teuku; Nasruddin, MN; Tulus; Ramli, Marwan

2018-01-01

Generally, research on the tsunami wave propagation model can be conducted by using a linear model of shallow water theory, where a non-linear side on high order is ignored. In line with research on the investigation of the tsunami waves, the Boussinesq equation model underwent a change aimed to obtain an improved quality of the dispersion relation and non-linearity by increasing the order to be higher. To solve non-linear sides at high order is used a asymptotic expansion method. This method can be used to solve non linear partial differential equations. In the present work, we found that this method needs much computational time and memory with the increase of the number of elements.

Matrix form of Legendre polynomials for solving linear integro-differential equations of high order

NASA Astrophysics Data System (ADS)

Kammuji, M.; Eshkuvatov, Z. K.; Yunus, Arif A. M.

2017-04-01

This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods.

Time-optimal Aircraft Pursuit-evasion with a Weapon Envelope Constraint

NASA Technical Reports Server (NTRS)

Menon, P. K. A.

1990-01-01

The optimal pursuit-evasion problem between two aircraft including a realistic weapon envelope is analyzed using differential game theory. Six order nonlinear point mass vehicle models are employed and the inclusion of an arbitrary weapon envelope geometry is allowed. The performance index is a linear combination of flight time and the square of the vehicle acceleration. Closed form solution to this high-order differential game is then obtained using feedback linearization. The solution is in the form of a feedback guidance law together with a quartic polynomial for time-to-go. Due to its modest computational requirements, this nonlinear guidance law is useful for on-board real-time implementation.

An invariant asymptotic formula for solutions of second-order linear ODE's

NASA Technical Reports Server (NTRS)

Gingold, H.

1988-01-01

An invariant-matrix technique for the approximate solution of second-order ordinary differential equations (ODEs) of form y-double-prime = phi(x)y is developed analytically and demonstrated. A set of linear transformations for the companion matrix differential system is proposed; the diagonalization procedure employed in the final stage of the asymptotic decomposition is explained; and a scalar formulation of solutions for the ODEs is obtained. Several typical ODEs are analyzed, and it is shown that the Liouville-Green or WKB approximation is a special case of the present formula, which provides an approximation which is valid for the entire interval (0, infinity).

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

NASA Astrophysics Data System (ADS)

Abdelaziz, Y.; Maillard, J.-M.

2017-05-01

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first illustrate covariance properties on order-two linear differential operators associated with identities relating the same {}_2F1 hypergeometric function with different rational pullbacks. These rational transformations are solutions of a differentially algebraic equation that already emerged in a paper by Casale on the Galoisian envelopes. We provide two new and more general results of the previous covariance by rational functions: a new Heun function example and a higher genus {}_2F1 hypergeometric function example. We then focus on identities relating the same {}_2F1 hypergeometric function with two different algebraic pullback transformations: such remarkable identities correspond to modular forms, the algebraic transformations being solution of another differentially algebraic Schwarzian equation that also emerged in Casale’s paper. Further, we show that the first differentially algebraic equation can be seen as a subcase of the last Schwarzian differential condition, the restriction corresponding to a factorization condition of some associated order-two linear differential operator. Finally, we also explore generalizations of these results, for instance, to {}_3F2 , hypergeometric functions, and show that one just reduces to the previous {}_2F1 cases through a Clausen identity. The question of the reduction of these Schwarzian conditions to modular correspondences remains an open question. In a _2F1 hypergeometric framework the Schwarzian condition encapsulates all the modular forms and modular equations of the theory of elliptic curves, but these two conditions are actually richer than elliptic curves or {}_2F1 hypergeometric functions, as can be seen on the Heun and higher genus example. This work is a strong incentive to develop more differentially algebraic symmetry analysis in physics.

A three operator split-step method covering a larger set of non-linear partial differential equations

NASA Astrophysics Data System (ADS)

Zia, Haider

2017-06-01

This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized derivative non-linear Schrödinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies to is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material.

Nonclassical point of view of the Brownian motion generation via fractional deterministic model

NASA Astrophysics Data System (ADS)

Gilardi-Velázquez, H. E.; Campos-Cantón, E.

In this paper, we present a dynamical system based on the Langevin equation without stochastic term and using fractional derivatives that exhibit properties of Brownian motion, i.e. a deterministic model to generate Brownian motion is proposed. The stochastic process is replaced by considering an additional degree of freedom in the second-order Langevin equation. Thus, it is transformed into a system of three first-order linear differential equations, additionally α-fractional derivative are considered which allow us to obtain better statistical properties. Switching surfaces are established as a part of fluctuating acceleration. The final system of three α-order linear differential equations does not contain a stochastic term, so the system generates motion in a deterministic way. Nevertheless, from the time series analysis, we found that the behavior of the system exhibits statistics properties of Brownian motion, such as, a linear growth in time of mean square displacement, a Gaussian distribution. Furthermore, we use the detrended fluctuation analysis to prove the Brownian character of this motion.

High-order Newton-penalty algorithms

NASA Astrophysics Data System (ADS)

Dussault, Jean-Pierre

2005-10-01

Recent efforts in differentiable non-linear programming have been focused on interior point methods, akin to penalty and barrier algorithms. In this paper, we address the classical equality constrained program solved using the simple quadratic loss penalty function/algorithm. The suggestion to use extrapolations to track the differentiable trajectory associated with penalized subproblems goes back to the classic monograph of Fiacco & McCormick. This idea was further developed by Gould who obtained a two-steps quadratically convergent algorithm using prediction steps and Newton correction. Dussault interpreted the prediction step as a combined extrapolation with respect to the penalty parameter and the residual of the first order optimality conditions. Extrapolation with respect to the residual coincides with a Newton step.We explore here higher-order extrapolations, thus higher-order Newton-like methods. We first consider high-order variants of the Newton-Raphson method applied to non-linear systems of equations. Next, we obtain improved asymptotic convergence results for the quadratic loss penalty algorithm by using high-order extrapolation steps.

Legendre-tau approximation for functional differential equations. Part 2: The linear quadratic optimal control problem

NASA Technical Reports Server (NTRS)

Ito, K.; Teglas, R.

1984-01-01

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Legendre-tau approximation for functional differential equations. II - The linear quadratic optimal control problem

NASA Technical Reports Server (NTRS)

Ito, Kazufumi; Teglas, Russell

1987-01-01

The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed-loop eigenvalues of the feedback system. A comparison with existing methods (based on averaging and spline approximations) is made.

Chaotic Oscillations of Second Order Linear Hyperbolic Equations with Nonlinear Boundary Conditions: A Factorizable but Noncommutative Case

NASA Astrophysics Data System (ADS)

Li, Liangliang; Huang, Yu; Chen, Goong; Huang, Tingwen

If a second order linear hyperbolic partial differential equation in one-space dimension can be factorized as a product of two first order operators and if the two first order operators commute, with one boundary condition being the van der Pol type and the other being linear, one can establish the occurrence of chaos when the parameters enter a certain regime [Chen et al., 2014]. However, if the commutativity of the two first order operators fails to hold, then the treatment in [Chen et al., 2014] no longer works and significant new challenges arise in determining nonlinear boundary conditions that engenders chaos. In this paper, we show that by incorporating a linear memory effect, a nonlinear van der Pol boundary condition can cause chaotic oscillations when the parameter enters a certain regime. Numerical simulations illustrating chaotic oscillations are also presented.

Design of an all-optical fractional-order differentiator with terahertz bandwidth based on a fiber Bragg grating in transmission.

PubMed

Liu, Xin; Shu, Xuewen

2017-08-20

All-optical fractional-order temporal differentiators with bandwidths reaching terahertz (THz) values are demonstrated with transmissive fiber Bragg gratings. Since the designed fractional-order differentiator is a minimum phase function, the reflective phase of the designed function can be chosen arbitrarily. As examples, we first design several 0.5th-order differentiators with bandwidths reaching the THz range for comparison. The reflective phases of the 0.5th-order differentiators are chosen to be linear phase, quadratic phase, cubic phase, and biquadratic phase, respectively. We find that both the maximum coupling coefficient and the spatial resolution of the designed grating increase when the reflective phase varies from quadratic function to cubic function to biquadratic function. Furthermore, when the reflective phase is chosen to be a quadratic function, the obtained grating coupling coefficient and period are more likely to be achieved in practice. Then we design fractional-order differentiators with different orders when the reflective phase is chosen to be a quadratic function. We see that when the designed order of the differentiator increases, the obtained maximum coupling coefficient also increases while the oscillation of the coupling coefficient decreases. Finally, we give the numerical performance of the designed 0.5th-order differentiator by showing its temporal response and calculating its cross-correlation coefficient.

An Efficient Spectral Method for Ordinary Differential Equations with Rational Function Coefficients

NASA Technical Reports Server (NTRS)

Coutsias, Evangelos A.; Torres, David; Hagstrom, Thomas

1994-01-01

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple three-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e. matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation N, is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.

Numerical Solution of Systems of Loaded Ordinary Differential Equations with Multipoint Conditions

NASA Astrophysics Data System (ADS)

Assanova, A. T.; Imanchiyev, A. E.; Kadirbayeva, Zh. M.

2018-04-01

A system of loaded ordinary differential equations with multipoint conditions is considered. The problem under study is reduced to an equivalent boundary value problem for a system of ordinary differential equations with parameters. A system of linear algebraic equations for the parameters is constructed using the matrices of the loaded terms and the multipoint condition. The conditions for the unique solvability and well-posedness of the original problem are established in terms of the matrix made up of the coefficients of the system of linear algebraic equations. The coefficients and the righthand side of the constructed system are determined by solving Cauchy problems for linear ordinary differential equations. The solutions of the system are found in terms of the values of the desired function at the initial points of subintervals. The parametrization method is numerically implemented using the fourth-order accurate Runge-Kutta method as applied to the Cauchy problems for ordinary differential equations. The performance of the constructed numerical algorithms is illustrated by examples.

Lie group classification of first-order delay ordinary differential equations

NASA Astrophysics Data System (ADS)

Dorodnitsyn, Vladimir A.; Kozlov, Roman; Meleshko, Sergey V.; Winternitz, Pavel

2018-05-01

A group classification of first-order delay ordinary differential equations (DODEs) accompanied by an equation for the delay parameter (delay relation) is presented. A subset of such systems (delay ordinary differential systems or DODSs), which consists of linear DODEs and solution-independent delay relations, have infinite-dimensional symmetry algebras—as do nonlinear ones that are linearizable by an invertible transformation of variables. Genuinely nonlinear DODSs have symmetry algebras of dimension n, . It is shown how exact analytical solutions of invariant DODSs can be obtained using symmetry reduction.

System of polarization correlometry of polycrystalline layers of urine in the differentiation stage of diabetes

NASA Astrophysics Data System (ADS)

Ushenko, Yu. O.; Pashkovskaya, N. V.; Marchuk, Y. F.; Dubolazov, O. V.; Savich, V. O.

2015-08-01

The work consists of investigation results of diagnostic efficiency of a new azimuthally stable Muellermatrix method of analysis of laser autofluorescence coordinate distributions of biological liquid layers. A new model of generalized optical anisotropy of biological tissues protein networks is proposed in order to define the processes of laser autofluorescence. The influence of complex mechanisms of both phase anisotropy (linear birefringence and optical activity) and linear (circular) dichroism is taken into account. The interconnections between the azimuthally stable Mueller-matrix elements characterizing laser autofluorescence and different mechanisms of optical anisotropy are determined. The statistic analysis of coordinate distributions of such Mueller-matrix rotation invariants is proposed. Thereupon the quantitative criteria (statistic moments of the 1st to the 4th order) of differentiation of human urine polycrystalline layers for the sake of diagnosing and differentiating cholelithiasis with underlying chronic cholecystitis (group 1) and diabetes mellitus of degree II (group 2) are estimated.

A neuro approach to solve fuzzy Riccati differential equations

NASA Astrophysics Data System (ADS)

Shahrir, Mohammad Shazri; Kumaresan, N.; Kamali, M. Z. M.; Ratnavelu, Kurunathan

2015-10-01

There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.

A neuro approach to solve fuzzy Riccati differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shahrir, Mohammad Shazri, E-mail: mshazri@gmail.com; Telekom Malaysia, R&D TM Innovation Centre, LingkaranTeknokrat Timur, 63000 Cyberjaya, Selangor; Kumaresan, N., E-mail: drnk2008@gmail.com

There are many applications of optimal control theory especially in the area of control systems in engineering. In this paper, fuzzy quadratic Riccati differential equation is estimated using neural networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). The solution can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that NN approach shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over RK4.

On shifted Jacobi spectral method for high-order multi-point boundary value problems

NASA Astrophysics Data System (ADS)

Doha, E. H.; Bhrawy, A. H.; Hafez, R. M.

2012-10-01

This paper reports a spectral tau method for numerically solving multi-point boundary value problems (BVPs) of linear high-order ordinary differential equations. The construction of the shifted Jacobi tau approximation is based on conventional differentiation. This use of differentiation allows the imposition of the governing equation at the whole set of grid points and the straight forward implementation of multiple boundary conditions. Extension of the tau method for high-order multi-point BVPs with variable coefficients is treated using the shifted Jacobi Gauss-Lobatto quadrature. Shifted Jacobi collocation method is developed for solving nonlinear high-order multi-point BVPs. The performance of the proposed methods is investigated by considering several examples. Accurate results and high convergence rates are achieved.

Algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations with the use of parallel computations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Moryakov, A. V., E-mail: sailor@orc.ru

2016-12-15

An algorithm for solving the linear Cauchy problem for large systems of ordinary differential equations is presented. The algorithm for systems of first-order differential equations is implemented in the EDELWEISS code with the possibility of parallel computations on supercomputers employing the MPI (Message Passing Interface) standard for the data exchange between parallel processes. The solution is represented by a series of orthogonal polynomials on the interval [0, 1]. The algorithm is characterized by simplicity and the possibility to solve nonlinear problems with a correction of the operator in accordance with the solution obtained in the previous iterative process.

Determining Dissolved Oxygen Levels

ERIC Educational Resources Information Center

Boucher, Randy

2010-01-01

This project was used in a mathematical modeling and introduction to differential equations course for first-year college students. The students worked in two-person groups and were given three weeks to complete the project. Students were given this project three weeks into the course, after basic first order linear differential equation and…

Oscillation criteria for half-linear dynamic equations on time scales

NASA Astrophysics Data System (ADS)

Hassan, Taher S.

2008-09-01

This paper is concerned with oscillation of the second-order half-linear dynamic equation(r(t)(x[Delta])[gamma])[Delta]+p(t)x[gamma](t)=0, on a time scale where [gamma] is the quotient of odd positive integers, r(t) and p(t) are positive rd-continuous functions on . Our results solve a problem posed by [R.P. Agarwal, D. O'Regan, S.H. Saker, Philos-type oscillation criteria for second-order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085-1104; S.H. Saker, Oscillation criteria of second order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005) 375-387] and our results in the special cases when and involve and improve some oscillation results for second-order differential and difference equations; and when , and , etc., our oscillation results are essentially newE Some examples illustrating the importance of our results are also included.

Finite-time H∞ filtering for non-linear stochastic systems

NASA Astrophysics Data System (ADS)

Hou, Mingzhe; Deng, Zongquan; Duan, Guangren

2016-09-01

This paper describes the robust H∞ filtering analysis and the synthesis of general non-linear stochastic systems with finite settling time. We assume that the system dynamic is modelled by Itô-type stochastic differential equations of which the state and the measurement are corrupted by state-dependent noises and exogenous disturbances. A sufficient condition for non-linear stochastic systems to have the finite-time H∞ performance with gain less than or equal to a prescribed positive number is established in terms of a certain Hamilton-Jacobi inequality. Based on this result, the existence of a finite-time H∞ filter is given for the general non-linear stochastic system by a second-order non-linear partial differential inequality, and the filter can be obtained by solving this inequality. The effectiveness of the obtained result is illustrated by a numerical example.

Existence of entire solutions of some non-linear differential-difference equations.

PubMed

Chen, Minfeng; Gao, Zongsheng; Du, Yunfei

2017-01-01

In this paper, we investigate the admissible entire solutions of finite order of the differential-difference equations [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] are two non-zero polynomials, [Formula: see text] is a polynomial and [Formula: see text]. In addition, we investigate the non-existence of entire solutions of finite order of the differential-difference equation [Formula: see text], where [Formula: see text], [Formula: see text] are two non-constant polynomials, [Formula: see text], m , n are positive integers and satisfy [Formula: see text] except for [Formula: see text], [Formula: see text].

Finite-horizon differential games for missile-target interception system using adaptive dynamic programming with input constraints

NASA Astrophysics Data System (ADS)

Sun, Jingliang; Liu, Chunsheng

2018-01-01

In this paper, the problem of intercepting a manoeuvring target within a fixed final time is posed in a non-linear constrained zero-sum differential game framework. The Nash equilibrium solution is found by solving the finite-horizon constrained differential game problem via adaptive dynamic programming technique. Besides, a suitable non-quadratic functional is utilised to encode the control constraints into a differential game problem. The single critic network with constant weights and time-varying activation functions is constructed to approximate the solution of associated time-varying Hamilton-Jacobi-Isaacs equation online. To properly satisfy the terminal constraint, an additional error term is incorporated in a novel weight-updating law such that the terminal constraint error is also minimised over time. By utilising Lyapunov's direct method, the closed-loop differential game system and the estimation weight error of the critic network are proved to be uniformly ultimately bounded. Finally, the effectiveness of the proposed method is demonstrated by using a simple non-linear system and a non-linear missile-target interception system, assuming first-order dynamics for the interceptor and target.

New numerical approximation of fractional derivative with non-local and non-singular kernel: Application to chaotic models

NASA Astrophysics Data System (ADS)

Toufik, Mekkaoui; Atangana, Abdon

2017-10-01

Recently a new concept of fractional differentiation with non-local and non-singular kernel was introduced in order to extend the limitations of the conventional Riemann-Liouville and Caputo fractional derivatives. A new numerical scheme has been developed, in this paper, for the newly established fractional differentiation. We present in general the error analysis. The new numerical scheme was applied to solve linear and non-linear fractional differential equations. We do not need a predictor-corrector to have an efficient algorithm, in this method. The comparison of approximate and exact solutions leaves no doubt believing that, the new numerical scheme is very efficient and converges toward exact solution very rapidly.

(N+1)-dimensional fractional reduced differential transform method for fractional order partial differential equations

NASA Astrophysics Data System (ADS)

Arshad, Muhammad; Lu, Dianchen; Wang, Jun

2017-07-01

In this paper, we pursue the general form of the fractional reduced differential transform method (DTM) to (N+1)-dimensional case, so that fractional order partial differential equations (PDEs) can be resolved effectively. The most distinct aspect of this method is that no prescribed assumptions are required, and the huge computational exertion is reduced and round-off errors are also evaded. We utilize the proposed scheme on some initial value problems and approximate numerical solutions of linear and nonlinear time fractional PDEs are obtained, which shows that the method is highly accurate and simple to apply. The proposed technique is thus an influential technique for solving the fractional PDEs and fractional order problems occurring in the field of engineering, physics etc. Numerical results are obtained for verification and demonstration purpose by using Mathematica software.

All-optical computation system for solving differential equations based on optical intensity differentiator.

PubMed

Tan, Sisi; Wu, Zhao; Lei, Lei; Hu, Shoujin; Dong, Jianji; Zhang, Xinliang

2013-03-25

We propose and experimentally demonstrate an all-optical differentiator-based computation system used for solving constant-coefficient first-order linear ordinary differential equations. It consists of an all-optical intensity differentiator and a wavelength converter, both based on a semiconductor optical amplifier (SOA) and an optical filter (OF). The equation is solved for various values of the constant-coefficient and two considered input waveforms, namely, super-Gaussian and Gaussian signals. An excellent agreement between the numerical simulation and the experimental results is obtained.

Modelling the Spread of an Oil-Slick with Irregular Information

ERIC Educational Resources Information Center

Winkel, Brian

2010-01-01

We describe a modelling activity for students in a course in which modelling with differential equations is appropriate. We have used this model in our coursework for years and have found that it enlightens students as to the model building process and parameter estimation for a linear, first-order, ordinary differential equation. The activity…

Program for solution of ordinary differential equations

NASA Technical Reports Server (NTRS)

Sloate, H.

1973-01-01

A program for the solution of linear and nonlinear first order ordinary differential equations is described and user instructions are included. The program contains a new integration algorithm for the solution of initial value problems which is particularly efficient for the solution of differential equations with a wide range of eigenvalues. The program in its present form handles up to ten state variables, but expansion to handle up to fifty state variables is being investigated.

Structure of Lie point and variational symmetry algebras for a class of odes

NASA Astrophysics Data System (ADS)

Ndogmo, J. C.

2018-04-01

It is known for scalar ordinary differential equations, and for systems of ordinary differential equations of order not higher than the third, that their Lie point symmetry algebras is of maximal dimension if and only if they can be reduced by a point transformation to the trivial equation y(n)=0. For arbitrary systems of ordinary differential equations of order n ≥ 3 reducible by point transformations to the trivial equation, we determine the complete structure of their Lie point symmetry algebras as well as that for their variational, and their divergence symmetry algebras. As a corollary, we obtain the maximal dimension of the Lie point symmetry algebra for any system of linear or nonlinear ordinary differential equations.

A multi-domain spectral method for time-fractional differential equations

NASA Astrophysics Data System (ADS)

Chen, Feng; Xu, Qinwu; Hesthaven, Jan S.

2015-07-01

This paper proposes an approach for high-order time integration within a multi-domain setting for time-fractional differential equations. Since the kernel is singular or nearly singular, two main difficulties arise after the domain decomposition: how to properly account for the history/memory part and how to perform the integration accurately. To address these issues, we propose a novel hybrid approach for the numerical integration based on the combination of three-term-recurrence relations of Jacobi polynomials and high-order Gauss quadrature. The different approximations used in the hybrid approach are justified theoretically and through numerical examples. Based on this, we propose a new multi-domain spectral method for high-order accurate time integrations and study its stability properties by identifying the method as a generalized linear method. Numerical experiments confirm hp-convergence for both time-fractional differential equations and time-fractional partial differential equations.

A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler-type kernel

NASA Astrophysics Data System (ADS)

Kumar, Devendra; Singh, Jagdev; Baleanu, Dumitru

2018-02-01

The mathematical model of breaking of non-linear dispersive water waves with memory effect is very important in mathematical physics. In the present article, we examine a novel fractional extension of the non-linear Fornberg-Whitham equation occurring in wave breaking. We consider the most recent theory of differentiation involving the non-singular kernel based on the extended Mittag-Leffler-type function to modify the Fornberg-Whitham equation. We examine the existence of the solution of the non-linear Fornberg-Whitham equation of fractional order. Further, we show the uniqueness of the solution. We obtain the numerical solution of the new arbitrary order model of the non-linear Fornberg-Whitham equation with the aid of the Laplace decomposition technique. The numerical outcomes are displayed in the form of graphs and tables. The results indicate that the Laplace decomposition algorithm is a very user-friendly and reliable scheme for handling such type of non-linear problems of fractional order.

Quantification and parametrization of non-linearity effects by higher-order sensitivity terms in scattered light differential optical absorption spectroscopy

NASA Astrophysics Data System (ADS)

Puķīte, Jānis; Wagner, Thomas

2016-05-01

We address the application of differential optical absorption spectroscopy (DOAS) of scattered light observations in the presence of strong absorbers (in particular ozone), for which the absorption optical depth is a non-linear function of the trace gas concentration. This is the case because Beer-Lambert law generally does not hold for scattered light measurements due to many light paths contributing to the measurement. While in many cases linear approximation can be made, for scenarios with strong absorptions non-linear effects cannot always be neglected. This is especially the case for observation geometries, for which the light contributing to the measurement is crossing the atmosphere under spatially well-separated paths differing strongly in length and location, like in limb geometry. In these cases, often full retrieval algorithms are applied to address the non-linearities, requiring iterative forward modelling of absorption spectra involving time-consuming wavelength-by-wavelength radiative transfer modelling. In this study, we propose to describe the non-linear effects by additional sensitivity parameters that can be used e.g. to build up a lookup table. Together with widely used box air mass factors (effective light paths) describing the linear response to the increase in the trace gas amount, the higher-order sensitivity parameters eliminate the need for repeating the radiative transfer modelling when modifying the absorption scenario even in the presence of a strong absorption background. While the higher-order absorption structures can be described as separate fit parameters in the spectral analysis (so-called DOAS fit), in practice their quantitative evaluation requires good measurement quality (typically better than that available from current measurements). Therefore, we introduce an iterative retrieval algorithm correcting for the higher-order absorption structures not yet considered in the DOAS fit as well as the absorption dependence on temperature and scattering processes.

Reformulating the Schrödinger equation as a Shabat-Zakharov system

NASA Astrophysics Data System (ADS)

Boonserm, Petarpa; Visser, Matt

2010-02-01

We reformulate the second-order Schrödinger equation as a set of two coupled first-order differential equations, a so-called "Shabat-Zakharov system" (sometimes called a "Zakharov-Shabat" system). There is considerable flexibility in this approach, and we emphasize the utility of introducing an "auxiliary condition" or "gauge condition" that is used to cut down the degrees of freedom. Using this formalism, we derive the explicit (but formal) general solution to the Schrödinger equation. The general solution depends on three arbitrarily chosen functions, and a path-ordered exponential matrix. If one considers path ordering to be an "elementary" process, then this represents complete quadrature, albeit formal, of the second-order linear ordinary differential equation.

Lumped Model Generation and Evaluation: Sensitivity and Lie Algebraic Techniques with Applications to Combustion

DTIC Science & Technology

1989-03-03

address global parameter space mapping issues for first order differential equations. The rigorous criteria for the existence of exact lumping by linear projective transformations was also established.

Stability analysis of gyroscopic systems with delay via decomposition

NASA Astrophysics Data System (ADS)

Aleksandrov, A. Yu.; Zhabko, A. P.; Chen, Y.

2018-05-01

A mechanical system describing by the second order linear differential equations with a positive parameter at the velocity forces and with time delay in the positional forces is studied. Using the decomposition method and Lyapunov-Krasovskii functionals, conditions are obtained under which from the asymptotic stability of two auxiliary first order subsystems it follows that, for sufficiently large values of the parameter, the original system is also asymptotically stable. Moreover, it is shown that the proposed approach can be applied to the stability investigation of linear gyroscopic systems with switched positional forces.

Polynomial mixture method of solving ordinary differential equations

NASA Astrophysics Data System (ADS)

Shahrir, Mohammad Shazri; Nallasamy, Kumaresan; Ratnavelu, Kuru; Kamali, M. Z. M.

2017-11-01

In this paper, a numerical solution of fuzzy quadratic Riccati differential equation is estimated using a proposed new approach that provides mixture of polynomials where iteratively the right mixture will be generated. This mixture provide a generalized formalism of traditional Neural Networks (NN). Previous works have shown reliable results using Runge-Kutta 4th order (RK4). This can be achieved by solving the 1st Order Non-linear Differential Equation (ODE) that is found commonly in Riccati differential equation. Research has shown improved results relatively to the RK4 method. It can be said that Polynomial Mixture Method (PMM) shows promising results with the advantage of continuous estimation and improved accuracy that can be produced over Mabood et al, RK-4, Multi-Agent NN and Neuro Method (NM).

Aspects of decision support in water management--example Berlin and Potsdam (Germany) I--spatially differentiated evaluation.

PubMed

Simon, Ute; Brüggemann, Rainer; Pudenz, Stefan

2004-04-01

Decisions about sustainable development demand spatially differentiated evaluations. As an example, we demonstrate the evaluation of water management strategies in the cities of Berlin and Potsdam (Germany) with respect to their ecological effects in 14 sections of the surface water system. Two decision support systems were compared, namely PROMETHEE, which is designed to obtain a clear decision (linear ranking), and Hasse Diagram Technique (HDT), normally providing more than one favourable solution (partial order). By PROMETHEE, the spatial differentiation had unwanted effects on the result, negating the stakeholders determined weighting of indicators. Therefore, the stakeholder can barely benefit from the convenience of obtaining a clear decision (linear ranking). In contrast, the result obtained by HDT was not influenced by spatial differentiation. Furthermore, HDT provided helpful tools to analyse the evaluation result, such as the concept of antagonistic indicators to discover conflicts in the evaluation process.

Differential operators on the supercircle S1|2 and symbol map

NASA Astrophysics Data System (ADS)

Hamza, Raouafi; Selmi, Zeineb; Boujelben, Jamel

2017-09-01

We consider the supercircle S1|2 equipped with the standard contact structure. The conformal Lie superalgebra 𝒦(2) acts on S1|2 as the Lie superalgebra of contact vector fields; it contains the Möbius superalgebra 𝔬𝔰𝔭(2|2). We study the space of linear differential operators on weighted densities as a module over 𝔬𝔰𝔭(2|2). We introduce the canonical isomorphism between this space and the corresponding space of symbols. This result allows us to give, in contrast to the classical setting, a classification of the 𝒦(2)-modules 𝔇λ,μk of linear differential operators of order k acting on the superspaces of weighted densities. This work is the simplest superization of a result by Gargoubi and Ovsienko [Modules of differential operators on the real line, Funct. Anal. Appl. 35(1) (2001) 13-18.

A case of "order insensitivity"? Natural and artificial language processing in a man with primary progressive aphasia.

PubMed

Zimmerer, Vitor C; Varley, Rosemary A

2015-08-01

Processing of linear word order (linear configuration) is important for virtually all languages and essential to languages such as English which have little functional morphology. Damage to systems underpinning configurational processing may specifically affect word-order reliant sentence structures. We explore order processing in WR, a man with primary progressive aphasia (PPA). In a previous report, we showed how WR showed impaired processing of actives, which rely strongly on word order, but not passives where functional morphology signals thematic roles. Using the artificial grammar learning (AGL) paradigm, we examined WR's ability to process order in non-verbal, visual sequences and compared his profile to that of healthy controls, and aphasic participants with and without severe syntactic disorder. Results suggested that WR, like some other patients with severe syntactic impairment, was unable to detect linear configurational structure. The data are consistent with the notion that disruption of possibly domain-general linearization systems differentially affects processing of active and passive sentence structures. Further research is needed to test this account, and we suggest hypotheses for future studies. Copyright © 2015 Elsevier Ltd. All rights reserved.

GHM method for obtaining rationalsolutions of nonlinear differential equations.

PubMed

Vazquez-Leal, Hector; Sarmiento-Reyes, Arturo

2015-01-01

In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30.

Fluorescent biopsy of biological tissues in differentiation of benign and malignant tumors of prostate

NASA Astrophysics Data System (ADS)

Trifoniuk, L. I.; Ushenko, Yu. A.; Sidor, M. I.; Minzer, O. P.; Gritsyuk, M. V.; Novakovskaya, O. Y.

2014-08-01

The work consists of investigation results of diagnostic efficiency of a new azimuthally stable Mueller-matrix method of analysis of laser autofluorescence coordinate distributions of biological tissues histological sections. A new model of generalized optical anisotropy of biological tissues protein networks is proposed in order to define the processes of laser autofluorescence. The influence of complex mechanisms of both phase anisotropy (linear birefringence and optical activity) and linear (circular) dichroism is taken into account. The interconnections between the azimuthally stable Mueller-matrix elements characterizing laser autofluorescence and different mechanisms of optical anisotropy are determined. The statistic analysis of coordinate distributions of such Mueller-matrix rotation invariants is proposed. Thereupon the quantitative criteria (statistic moments of the 1st to the 4th order) of differentiation of histological sections of uterus wall tumor - group 1 (dysplasia) and group 2 (adenocarcinoma) are estimated.

Multi-off-grid methods in multi-step integration of ordinary differential equations

NASA Technical Reports Server (NTRS)

Beaudet, P. R.

1974-01-01

Description of methods of solving first- and second-order systems of differential equations in which all derivatives are evaluated at off-grid locations in order to circumvent the Dahlquist stability limitation on the order of on-grid methods. The proposed multi-off-grid methods require off-grid state predictors for the evaluation of the n derivatives at each step. Progressing forward in time, the off-grid states are predicted using a linear combination of back on-grid state values and off-grid derivative evaluations. A comparison is made between the proposed multi-off-grid methods and the corresponding Adams and Cowell on-grid integration techniques in integrating systems of ordinary differential equations, showing a significant reduction in the error at larger step sizes in the case of the multi-off-grid integrator.

First integrals and parametric solutions of third-order ODEs admitting {\\mathfrak{sl}(2, {R})}

NASA Astrophysics Data System (ADS)

Ruiz, A.; Muriel, C.

2017-05-01

A complete set of first integrals for any third-order ordinary differential equation admitting a Lie symmetry algebra isomorphic to sl(2, {R}) is explicitly computed. These first integrals are derived from two linearly independent solutions of a linear second-order ODE, without additional integration. The general solution in parametric form can be obtained by using the computed first integrals. The study includes a parallel analysis of the four inequivalent realizations of sl(2, {R}) , and it is applied to several particular examples. These include the generalized Chazy equation, as well as an example of an equation which admits the most complicated of the four inequivalent realizations.

Solution of second order quasi-linear boundary value problems by a wavelet method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhang, Lei; Zhou, Youhe; Wang, Jizeng, E-mail: jzwang@lzu.edu.cn

2015-03-10

A wavelet Galerkin method based on expansions of Coiflet-like scaling function bases is applied to solve second order quasi-linear boundary value problems which represent a class of typical nonlinear differential equations. Two types of typical engineering problems are selected as test examples: one is about nonlinear heat conduction and the other is on bending of elastic beams. Numerical results are obtained by the proposed wavelet method. Through comparing to relevant analytical solutions as well as solutions obtained by other methods, we find that the method shows better efficiency and accuracy than several others, and the rate of convergence can evenmore » reach orders of 5.8.« less

New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods

NASA Astrophysics Data System (ADS)

Doha, E. H.; Abd-Elhameed, W. M.; Bassuony, M. A.

2013-03-01

This paper is concerned with spectral Galerkin algorithms for solving high even-order two point boundary value problems in one dimension subject to homogeneous and nonhomogeneous boundary conditions. The proposed algorithms are extended to solve two-dimensional high even-order differential equations. The key to the efficiency of these algorithms is to construct compact combinations of Chebyshev polynomials of the third and fourth kinds as basis functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. Numerical examples are included to demonstrate the validity and applicability of the proposed algorithms, and some comparisons with some other methods are made.

On the singular perturbations for fractional differential equation.

PubMed

Atangana, Abdon

2014-01-01

The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.

Differential-Drive Mobile Robot Control Design based-on Linear Feedback Control Law

NASA Astrophysics Data System (ADS)

Nurmaini, Siti; Dewi, Kemala; Tutuko, Bambang

2017-04-01

This paper deals with the problem of how to control differential driven mobile robot with simple control law. When mobile robot moves from one position to another to achieve a position destination, it always produce some errors. Therefore, a mobile robot requires a certain control law to drive the robot’s movement to the position destination with a smallest possible error. In this paper, in order to reduce position error, a linear feedback control is proposed with pole placement approach to regulate the polynoms desired. The presented work leads to an improved understanding of differential-drive mobile robot (DDMR)-based kinematics equation, which will assist to design of suitable controllers for DDMR movement. The result show by using the linier feedback control method with pole placement approach the position error is reduced and fast convergence is achieved.

Compact tunable silicon photonic differential-equation solver for general linear time-invariant systems.

PubMed

Wu, Jiayang; Cao, Pan; Hu, Xiaofeng; Jiang, Xinhong; Pan, Ting; Yang, Yuxing; Qiu, Ciyuan; Tremblay, Christine; Su, Yikai

2014-10-20

We propose and experimentally demonstrate an all-optical temporal differential-equation solver that can be used to solve ordinary differential equations (ODEs) characterizing general linear time-invariant (LTI) systems. The photonic device implemented by an add-drop microring resonator (MRR) with two tunable interferometric couplers is monolithically integrated on a silicon-on-insulator (SOI) wafer with a compact footprint of ~60 μm × 120 μm. By thermally tuning the phase shifts along the bus arms of the two interferometric couplers, the proposed device is capable of solving first-order ODEs with two variable coefficients. The operation principle is theoretically analyzed, and system testing of solving ODE with tunable coefficients is carried out for 10-Gb/s optical Gaussian-like pulses. The experimental results verify the effectiveness of the fabricated device as a tunable photonic ODE solver.

A physically based connection between fractional calculus and fractal geometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Butera, Salvatore, E-mail: sg.butera@gmail.com; Di Paola, Mario, E-mail: mario.dipaola@unipa.it

2014-11-15

We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalousmore » dimension of such geometry, as well as to the model used to describe the physics involved. By linearizing the non linear dependence of the response of the system at hand to a proper forcing action then, exploiting the Boltzmann superposition principle, a fractional differential equation is found, describing the dynamics of the system itself. The order of such equation is again related to the anomalous dimension of the underlying geometry.« less

Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation

NASA Astrophysics Data System (ADS)

Bokhari, Ashfaque H.; Mahomed, F. M.; Zaman, F. D.

2010-05-01

The complete symmetry group classification of the fourth-order Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry.

General solution of the Bagley-Torvik equation with fractional-order derivative

NASA Astrophysics Data System (ADS)

Wang, Z. H.; Wang, X.

2010-05-01

This paper investigates the general solution of the Bagley-Torvik equation with 1/2-order derivative or 3/2-order derivative. This fractional-order differential equation is changed into a sequential fractional-order differential equation (SFDE) with constant coefficients. Then the general solution of the SFDE is expressed as the linear combination of fundamental solutions that are in terms of α-exponential functions, a kind of functions that play the same role of the classical exponential function. Because the number of fundamental solutions of the SFDE is greater than 2, the general solution of the SFDE depends on more than two free (independent) constants. This paper shows that the general solution of the Bagley-Torvik equation involves actually two free constants only, and it can be determined fully by the initial displacement and initial velocity.

Software Reviews.

ERIC Educational Resources Information Center

Teles, Elizabeth, Ed.; And Others

1990-01-01

Reviewed are two computer software packages for Macintosh microcomputers including "Phase Portraits," an exploratory graphics tool for studying first-order planar systems; and "MacMath," a set of programs for exploring differential equations, linear algebra, and other mathematical topics. Features, ease of use, cost, availability, and hardware…

An Algebraic Construction of the First Integrals of the Stationary KdV Hierarchy

NASA Astrophysics Data System (ADS)

Matsushima, Masatomo; Ohmiya, Mayumi

2009-09-01

The stationary KdV hierarchy is constructed using a kind of recursion operator called Λ-operator. The notion of the maximal solution of the n-th stationary KdV equation is introduced. Using this maximal solution, a specific differential polynomial with the auxiliary spectral parameter called the spectral M-function is constructed as the quadratic form of the fundamental system of the eigenvalue problem for the 2-nd order linear ordinary differential equation which is related to the linearizing operator of the hierarchy. By calculating a perfect square condition of the quadratic form by an elementary algebraic method, the complete set of first integrals of this hierarchy is constructed.

Pseudospectral collocation methods for fourth order differential equations

NASA Technical Reports Server (NTRS)

Malek, Alaeddin; Phillips, Timothy N.

1994-01-01

Collocation schemes are presented for solving linear fourth order differential equations in one and two dimensions. The variational formulation of the model fourth order problem is discretized by approximating the integrals by a Gaussian quadrature rule generalized to include the values of the derivative of the integrand at the boundary points. Collocation schemes are derived which are equivalent to this discrete variational problem. An efficient preconditioner based on a low-order finite difference approximation to the same differential operator is presented. The corresponding multidomain problem is also considered and interface conditions are derived. Pseudospectral approximations which are C1 continuous at the interfaces are used in each subdomain to approximate the solution. The approximations are also shown to be C3 continuous at the interfaces asymptotically. A complete analysis of the collocation scheme for the multidomain problem is provided. The extension of the method to the biharmonic equation in two dimensions is discussed and results are presented for a problem defined in a nonrectangular domain.

An efficient computer based wavelets approximation method to solve Fuzzy boundary value differential equations

NASA Astrophysics Data System (ADS)

Alam Khan, Najeeb; Razzaq, Oyoon Abdul

2016-03-01

In the present work a wavelets approximation method is employed to solve fuzzy boundary value differential equations (FBVDEs). Essentially, a truncated Legendre wavelets series together with the Legendre wavelets operational matrix of derivative are utilized to convert FB- VDE into a simple computational problem by reducing it into a system of fuzzy algebraic linear equations. The capability of scheme is investigated on second order FB- VDE considered under generalized H-differentiability. Solutions are represented graphically showing competency and accuracy of this method.

Preliminary Planar Formation: Flight Dynamics Near Sun-Earth L2 Point

NASA Technical Reports Server (NTRS)

Segerman, Alan M.; Zedd, Michael F.

2003-01-01

NASA's Goddard Space Flight Center is planning a series of missions in the vicinity of the Sun-Earth L2 libration point. Some of these projects will involve a distributed space system of telescope spacecraft acting together as a single telescope for high-resolution. The individual telescopes will be configured in a plane, surrounding a hub, where the telescope plane can be aimed toward various astronomical targets of interest. In preparation for these missions, it is necessary to develop an improved understanding of the dynamical behavior of objects in a planar configuration near L2. The classical circular restricted three body problem is taken as the basis for the analysis. At first order, the motion of such a telescope relative to the hub is described by a system of linear second order differential equations. These equations are identical to the circular restricted problem's linear equations describing the hub motion about L2. Therefore, the fundamental frequencies, both parallel to and normal to the ecliptic plane, are the same for the relative telescope motion as for the hub motion. To maintain the telescope plane for the duration necessary for the planned observations, a halo-type orbit of the telescopes about the hub is investigated. By using a halo orbit, the individual telescopes remain in approximately the same plane over the observation duration. For such an orbit, the fundamental periods parallel to and normal to the ecliptic plane are forced to be the same by careful selection of the initial conditions in order to adjust the higher order forces. The relative amplitudes of the resulting oscillations are associated with the orientation of the telescope plane relative to the ecliptic. As in the circular restricted problem, initial conditions for the linearized equations must be selected so as not to excite the convergent or divergent linear modes. In a higher order analysis, the telescope relative motion equations include the effects of the position of the hub relative to L2. In this paper, the differential equations are developed through second order in the distance of the hub from the libration point. A modified Lindstedt-Poincad perturbation method is employed to construct the solution of these differential equations through that same order of magnitude. In the course of the solution process, relationships are determined between the initial conditions of the telescopes, selected in order to avoid resonance excitation. As the differential equations include the hub position, it is necessary to simultaneously develop the solution for the hub. As has been done in past analyses of the circular restricted problem, the hub position is written in a power series formulation in terms of its distance from L2. Then, in order to be included in the telescope equations, the hub solution is cast in terms of the nonlinear frequency of the relative telescope motion. In the course of the analysis, it is determined that the hub should also maintain a halo orbit - about L2. Additionally, relationships are formed between the initial conditions of the telescopes and the hub. These relationships may be used to associate sets of initial conditions with particular orientations of the telescope plane. The accuracy of the analytical solution is verified through various simulations and comparison to numerical integration of the differential equations. The results of the simulations are presented, along with a graphical representation of the relationships between the initial conditions of the telescopes and hub.

On a family of nonoscillatory equations y double prime = phi(x)y

NASA Technical Reports Server (NTRS)

Gingold, H.

1988-01-01

The oscillation or nonoscillation of a class of second-order linear differential equations is investigated analytically, with a focus on cases in which the functions phi(x) and y are complex-valued. Two linear transformations are introduced, and an asymptotic-decomposition procedure involving Shur triangularization is applied. The relationship of the present analysis to the nonoscillation criterion of Kneser (1896) and other more recent results is explored in two examples.

Investigation of a Nonlinear Control System

NASA Technical Reports Server (NTRS)

Flugge-Lotz, I; Taylor, C F; Lindberg, H E

1958-01-01

A discontinuous variation of coefficients of the differential equation describing the linear control system before nonlinear elements are added is studied in detail. The nonlinear feedback is applied to a second-order system. Simulation techniques are used to study performance of the nonlinear control system and to compare it with the linear system for a wide variety of inputs. A detailed quantitative study of the influence of relay delays and of a transport delay is presented.

Qualitative properties of large buckled states of spherical shells

NASA Technical Reports Server (NTRS)

Shih, K. G.; Antman, S. S.

1985-01-01

A system of 6th-order quasi-linear Ordinary Differential Equations is analyzed to show the global existence of axisymmetrically buckled states. A surprising nodal property is obtained which shows that everywhere along a branch of solutions that bifurcates from a simple eigenvalue of the linearized equation, the number of simultaneously vanishing points of both shear resultant and circumferential bending moment resultant remains invariant, provided that a certain auxiliary condition is satisfied.

Bimolecular Recombination Kinetics of an Exciton-Trion Gas

DTIC Science & Technology

2015-07-01

3-D systems. Whereas a linear time-dependent system of first-order differential equations has only trivial steady- state solutions (all carrier...derivatives to zero, which reduces the system (Eq. 9) to the following set of 3 algebraic equations: ( ) ( ) ( ) ( ) 1 2 210 2 110...crossover around 20 ns. The exciton curve is nearly linear over a wide range from 10 ns to 50 ns. Fig. 2 Time dependence of carrier species for Λ = 4

Local uncontrollability for affine control systems with jumps

NASA Astrophysics Data System (ADS)

Treanţă, Savin

2017-09-01

This paper investigates affine control systems with jumps for which the ideal If(g1, …, gm) generated by the drift vector field f in the Lie algebra L(f, g1, …, gm) can be imbedded as a kernel of a linear first-order partial differential equation. It will lead us to uncontrollable affine control systems with jumps for which the corresponding reachable sets are included in explicitly described differentiable manifolds.

Hipergeometric solutions to some nonhomogeneous equations of fractional order

NASA Astrophysics Data System (ADS)

Olivares, Jorge; Martin, Pablo; Maass, Fernando

2017-12-01

In this paper a study is performed to the solution of the linear non homogeneous fractional order alpha differential equation equal to I 0(x), where I 0(x) is the modified Bessel function of order zero, the initial condition is f(0)=0 and 0 < alpha < 1. Caputo definition for the fractional derivatives is considered. Fractional derivatives have become important in physical and chemical phenomena as visco-elasticity and visco-plasticity, anomalous diffusion and electric circuits. In particular in this work the values of alpha=1/2, 1/4 and 3/4. are explicitly considered . In these cases Laplace transform is applied, and later the inverse Laplace transform leads to the solutions of the differential equation, which become hypergeometric functions.

On the Singular Perturbations for Fractional Differential Equation

PubMed Central

Atangana, Abdon

2014-01-01

The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method. PMID:24683357

All-optical 1st- and 2nd-order differential equation solvers with large tuning ranges using Fabry-Pérot semiconductor optical amplifiers.

PubMed

Chen, Kaisheng; Hou, Jie; Huang, Zhuyang; Cao, Tong; Zhang, Jihua; Yu, Yuan; Zhang, Xinliang

2015-02-09

We experimentally demonstrate an all-optical temporal computation scheme for solving 1st- and 2nd-order linear ordinary differential equations (ODEs) with tunable constant coefficients by using Fabry-Pérot semiconductor optical amplifiers (FP-SOAs). By changing the injection currents of FP-SOAs, the constant coefficients of the differential equations are practically tuned. A quite large constant coefficient tunable range from 0.0026/ps to 0.085/ps is achieved for the 1st-order differential equation. Moreover, the constant coefficient p of the 2nd-order ODE solver can be continuously tuned from 0.0216/ps to 0.158/ps, correspondingly with the constant coefficient q varying from 0.0000494/ps(2) to 0.006205/ps(2). Additionally, a theoretical model that combining the carrier density rate equation of the semiconductor optical amplifier (SOA) with the transfer function of the Fabry-Pérot (FP) cavity is exploited to analyze the solving processes. For both 1st- and 2nd-order solvers, excellent agreements between the numerical simulations and the experimental results are obtained. The FP-SOAs based all-optical differential-equation solvers can be easily integrated with other optical components based on InP/InGaAsP materials, such as laser, modulator, photodetector and waveguide, which can motivate the realization of the complicated optical computing on a single integrated chip.

Exact solutions to the time-fractional differential equations via local fractional derivatives

NASA Astrophysics Data System (ADS)

Guner, Ozkan; Bekir, Ahmet

2018-01-01

This article utilizes the local fractional derivative and the exp-function method to construct the exact solutions of nonlinear time-fractional differential equations (FDEs). For illustrating the validity of the method, it is applied to the time-fractional Camassa-Holm equation and the time-fractional-generalized fifth-order KdV equation. Moreover, the exact solutions are obtained for the equations which are formed by different parameter values related to the time-fractional-generalized fifth-order KdV equation. This method is an reliable and efficient mathematical tool for solving FDEs and it can be applied to other non-linear FDEs.

Synchronization of an Inertial Neural Network With Time-Varying Delays and Its Application to Secure Communication.

PubMed

Lakshmanan, Shanmugam; Prakash, Mani; Lim, Chee Peng; Rakkiyappan, Rajan; Balasubramaniam, Pagavathigounder; Nahavandi, Saeid

2018-01-01

In this paper, synchronization of an inertial neural network with time-varying delays is investigated. Based on the variable transformation method, we transform the second-order differential equations into the first-order differential equations. Then, using suitable Lyapunov-Krasovskii functionals and Jensen's inequality, the synchronization criteria are established in terms of linear matrix inequalities. Moreover, a feedback controller is designed to attain synchronization between the master and slave models, and to ensure that the error model is globally asymptotically stable. Numerical examples and simulations are presented to indicate the effectiveness of the proposed method. Besides that, an image encryption algorithm is proposed based on the piecewise linear chaotic map and the chaotic inertial neural network. The chaotic signals obtained from the inertial neural network are utilized for the encryption process. Statistical analyses are provided to evaluate the effectiveness of the proposed encryption algorithm. The results ascertain that the proposed encryption algorithm is efficient and reliable for secure communication applications.

Linearized Model of an Actively Controlled Cable for a Carlina Diluted Telescope

NASA Astrophysics Data System (ADS)

Andersen, T.; Le Coroller, H.; Owner-Petersen, M.; Dejonghe, J.

2014-04-01

The Carlina thinned pupil telescope has a focal unit (``gondola'') suspended by cables over the primary mirror. To predict the structural behavior of the gondola system, a simulation building block of a single cable is needed. A preloaded cable is a strongly non-linear system and can be modeled either with partial differential equations or non-linear finite elements. Using the latter, we set up an iteration procedure for determination of the static cable form and we formulate the necessary second-order differential equations for such a model. We convert them to a set of first-order differential equations (an ``ABCD''-model). Symmetrical in-plane eigenmodes and ``axial'' eigenmodes are the only eigenmodes that play a role in practice for a taut cable. Using the model and a generic suspension, a parameter study is made to find the influence of various design parameters. We conclude that the cable should be as stiff and thick as practically possible with a fairly high preload. Steel or Aramid are suitable materials. Further, placing the cable winches on the gondola and not on the ground does not provide significant advantages. Finally, it seems that use of reaction-wheels and/or reaction-masses will make the way for more accurate control of the gondola position under wind load. An adaptive stage with tip/tilt/piston correction for subapertures together with a focus and guiding system for freezing the fringes must also be studied.

High-Order Automatic Differentiation of Unmodified Linear Algebra Routines via Nilpotent Matrices

NASA Astrophysics Data System (ADS)

Dunham, Benjamin Z.

This work presents a new automatic differentiation method, Nilpotent Matrix Differentiation (NMD), capable of propagating any order of mixed or univariate derivative through common linear algebra functions--most notably third-party sparse solvers and decomposition routines, in addition to basic matrix arithmetic operations and power series--without changing data-type or modifying code line by line; this allows differentiation across sequences of arbitrarily many such functions with minimal implementation effort. NMD works by enlarging the matrices and vectors passed to the routines, replacing each original scalar with a matrix block augmented by derivative data; these blocks are constructed with special sparsity structures, termed "stencils," each designed to be isomorphic to a particular multidimensional hypercomplex algebra. The algebras are in turn designed such that Taylor expansions of hypercomplex function evaluations are finite in length and thus exactly track derivatives without approximation error. Although this use of the method in the "forward mode" is unique in its own right, it is also possible to apply it to existing implementations of the (first-order) discrete adjoint method to find high-order derivatives with lowered cost complexity; for example, for a problem with N inputs and an adjoint solver whose cost is independent of N--i.e., O(1)--the N x N Hessian can be found in O(N) time, which is comparable to existing second-order adjoint methods that require far more problem-specific implementation effort. Higher derivatives are likewise less expensive--e.g., a N x N x N rank-three tensor can be found in O(N2). Alternatively, a Hessian-vector product can be found in O(1) time, which may open up many matrix-based simulations to a range of existing optimization or surrogate modeling approaches. As a final corollary in parallel to the NMD-adjoint hybrid method, the existing complex-step differentiation (CD) technique is also shown to be capable of finding the Hessian-vector product. All variants are implemented on a stochastic diffusion problem and compared in-depth with various cost and accuracy metrics.

Numerical solution of system of boundary value problems using B-spline with free parameter

NASA Astrophysics Data System (ADS)

Gupta, Yogesh

2017-01-01

This paper deals with method of B-spline solution for a system of boundary value problems. The differential equations are useful in various fields of science and engineering. Some interesting real life problems involve more than one unknown function. These result in system of simultaneous differential equations. Such systems have been applied to many problems in mathematics, physics, engineering etc. In present paper, B-spline and B-spline with free parameter methods for the solution of a linear system of second-order boundary value problems are presented. The methods utilize the values of cubic B-spline and its derivatives at nodal points together with the equations of the given system and boundary conditions, ensuing into the linear matrix equation.

Noise suppression for the differential detection in nuclear magnetic resonance gyroscope

NASA Astrophysics Data System (ADS)

Yang, Dan; Zhou, Binquan; Chen, LinLin; Jia, YuChen; Lu, QiLin

2017-10-01

The nuclear magnetic resonance gyroscope is based on spin-exchange optical pumping of noble gases to detect and measure the angular velocity of the carrier, but it would be challenging to measure the precession signal of noble gas nuclei directly. To solve the problem, the primary detection method utilizes alkali atoms, the precession of nuclear magnetization modulates the alkali atoms at the Larmor frequency of nuclei, relatively speaking, and it is easier to detect the precession signal of alkali atoms. The precession frequency of alkali atoms is detected by the rotation angle of linearly polarized probe light; and differential detection method is commonly used in NMRG in order to detect the linearly polarized light rotation angle. Thus, the detection accuracy of differential detection system will affect the sensitivity of the NMRG. For the purpose of further improvement of the sensitivity level of the NMRG, this paper focuses on the aspects of signal detection, and aims to do an error analysis as well as an experimental research of the linearly light rotation angle detection. Through the theoretical analysis and the experimental illustration, we found that the extinction ratio σ2 and DC bias are the factors that will produce detective noise in the differential detection method.

Chosen interval methods for solving linear interval systems with special type of matrix

NASA Astrophysics Data System (ADS)

Szyszka, Barbara

2013-10-01

The paper is devoted to chosen direct interval methods for solving linear interval systems with special type of matrix. This kind of matrix: band matrix with a parameter, from finite difference problem is obtained. Such linear systems occur while solving one dimensional wave equation (Partial Differential Equations of hyperbolic type) by using the central difference interval method of the second order. Interval methods are constructed so as the errors of method are enclosed in obtained results, therefore presented linear interval systems contain elements that determining the errors of difference method. The chosen direct algorithms have been applied for solving linear systems because they have no errors of method. All calculations were performed in floating-point interval arithmetic.

Fourier Series and Elliptic Functions

ERIC Educational Resources Information Center

Fay, Temple H.

2003-01-01

Non-linear second-order differential equations whose solutions are the elliptic functions "sn"("t, k"), "cn"("t, k") and "dn"("t, k") are investigated. Using "Mathematica", high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these non-elementary functions that are…

Oscillation of two-dimensional linear second-order differential systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kwong, M.K.; Kaper, H.G.

This article is concerned with the oscillatory behavior at infinity of the solution y: (a, infinity) ..-->.. R/sup 2/ of a system of two second-order differential equations, y''(t) + Q(t) y(t) = 0, t epsilon(a, infinity); Q is a continuous matrix-valued function on (a, infinity) whose values are real symmetric matrices of order 2. It is shown that the solution is oscillatory at infinity if the largest eigenvalue of the matrix integral/sub a//sup t/ Q(s) ds tends to infinity as t ..-->.. infinity. This proves a conjecture of D. Hinton and R.T. Lewis for the two-dimensional case. Furthermore, it ismore » shown that considerably weaker forms of the condition still suffice for oscillatory behavior at infinity. 7 references.« less

Non-symmetric forms of non-linear vibrations of flexible cylindrical panels and plates under longitudinal load and additive white noise

NASA Astrophysics Data System (ADS)

Krysko, V. A.; Awrejcewicz, J.; Krylova, E. Yu; Papkova, I. V.; Krysko, A. V.

2018-06-01

Parametric non-linear vibrations of flexible cylindrical panels subjected to additive white noise are studied. The governing Marguerre equations are investigated using the finite difference method (FDM) of the second-order accuracy and the Runge-Kutta method. The considered mechanical structural member is treated as a system of many/infinite number of degrees of freedom (DoF). The dependence of chaotic vibrations on the number of DoFs is investigated. Reliability of results is guaranteed by comparing the results obtained using two qualitatively different methods to reduce the problem of PDEs (partial differential equations) to ODEs (ordinary differential equations), i.e. the Faedo-Galerkin method in higher approximations and the 4th and 6th order FDM. The Cauchy problem obtained by the FDM is eventually solved using the 4th-order Runge-Kutta methods. The numerical experiment yielded, for a certain set of parameters, the non-symmetric vibration modes/forms with and without white noise. In particular, it has been illustrated and discussed that action of white noise on chaotic vibrations implies quasi-periodicity, whereas the previously non-symmetric vibration modes are closer to symmetric ones.

Linear stability analysis of the Vlasov-Poisson equations in high density plasmas in the presence of crossed fields and density gradients

NASA Technical Reports Server (NTRS)

Kaup, D. J.; Hansen, P. J.; Choudhury, S. Roy; Thomas, Gary E.

1986-01-01

The equations for the single-particle orbits in a nonneutral high density plasma in the presence of inhomogeneous crossed fields are obtained. Using these orbits, the linearized Vlasov equation is solved as an expansion in the orbital radii in the presence of inhomogeneities and density gradients. A model distribution function is introduced whose cold-fluid limit is exactly the same as that used in many previous studies of the cold-fluid equations. This model function is used to reduce the linearized Vlasov-Poisson equations to a second-order ordinary differential equation for the linearized electrostatic potential whose eigenvalue is the perturbation frequency.

A canonical form of the equation of motion of linear dynamical systems

NASA Astrophysics Data System (ADS)

Kawano, Daniel T.; Salsa, Rubens Goncalves; Ma, Fai; Morzfeld, Matthias

2018-03-01

The equation of motion of a discrete linear system has the form of a second-order ordinary differential equation with three real and square coefficient matrices. It is shown that, for almost all linear systems, such an equation can always be converted by an invertible transformation into a canonical form specified by two diagonal coefficient matrices associated with the generalized acceleration and displacement. This canonical form of the equation of motion is unique up to an equivalence class for non-defective systems. As an important by-product, a damped linear system that possesses three symmetric and positive definite coefficients can always be recast as an undamped and decoupled system.

Application of the enhanced homotopy perturbation method to solve the fractional-order Bagley-Torvik differential equation

NASA Astrophysics Data System (ADS)

Zolfaghari, M.; Ghaderi, R.; Sheikhol Eslami, A.; Ranjbar, A.; Hosseinnia, S. H.; Momani, S.; Sadati, J.

2009-10-01

The enhanced homotopy perturbation method (EHPM) is applied for finding improved approximate solutions of the well-known Bagley-Torvik equation for three different cases. The main characteristic of the EHPM is using a stabilized linear part, which guarantees the stability and convergence of the overall solution. The results are finally compared with the Adams-Bashforth-Moulton numerical method, the Adomian decomposition method (ADM) and the fractional differential transform method (FDTM) to verify the performance of the EHPM.

Pseudo almost periodic solutions to some systems of nonlinear hyperbolic second-order partial differential equations

NASA Astrophysics Data System (ADS)

Al-Islam, Najja Shakir

In this Dissertation, the existence of pseudo almost periodic solutions to some systems of nonlinear hyperbolic second-order partial differential equations is established. For that, (Al-Islam [4]) is first studied and then obtained under some suitable assumptions. That is, the existence of pseudo almost periodic solutions to a hyperbolic second-order partial differential equation with delay. The second-order partial differential equation (1) represents a mathematical model for the dynamics of gas absorption, given by uxt+a x,tux=Cx,t,u x,t , u0,t=4 t, 1 where a : [0, L] x RR , C : [0, L] x R x RR , and ϕ : RR are (jointly) continuous functions ( t being the greatest integer function) and L > 0. The results in this Dissertation generalize those of Poorkarimi and Wiener [22]. Secondly, a generalization of the above-mentioned system consisting of the non-linear hyperbolic second-order partial differential equation uxt+a x,tux+bx,t ut+cx,tu=f x,t,u, x∈ 0,L,t∈ R, 2 equipped with the boundary conditions ux,0 =40x, u0,t=u 0t, uxx,0=y 0x, x∈0,L, t∈R, 3 where a, b, c : [0, L ] x RR and f : [0, L] x R x RR are (jointly) continuous functions is studied. Under some suitable assumptions, the existence and uniqueness of pseudo almost periodic solutions to particular cases, as well as the general case of the second-order hyperbolic partial differential equation (2) are studied. The results of all studies contained within this text extend those obtained by Aziz and Meyers [6] in the periodic setting.

Global exponential synchronization of inertial memristive neural networks with time-varying delay via nonlinear controller.

PubMed

Gong, Shuqing; Yang, Shaofu; Guo, Zhenyuan; Huang, Tingwen

2018-06-01

The paper is concerned with the synchronization problem of inertial memristive neural networks with time-varying delay. First, by choosing a proper variable substitution, inertial memristive neural networks described by second-order differential equations can be transformed into first-order differential equations. Then, a novel controller with a linear diffusive term and discontinuous sign term is designed. By using the controller, the sufficient conditions for assuring the global exponential synchronization of the derive and response neural networks are derived based on Lyapunov stability theory and some inequality techniques. Finally, several numerical simulations are provided to substantiate the effectiveness of the theoretical results. Copyright © 2018 Elsevier Ltd. All rights reserved.

Numerical analysis for trajectory controllability of a coupled multi-order fractional delay differential system via the shifted Jacobi method

NASA Astrophysics Data System (ADS)

Priya, B. Ganesh; Muthukumar, P.

2018-02-01

This paper deals with the trajectory controllability for a class of multi-order fractional linear systems subject to a constant delay in state vector. The solution for the coupled fractional delay differential equation is established by the Mittag-Leffler function. The necessary and sufficient condition for the trajectory controllability is formulated and proved by the generalized Gronwall's inequality. The approximate trajectory for the proposed system is obtained through the shifted Jacobi operational matrix method. The numerical simulation of the approximate solution shows the theoretical results. Finally, some remarks and comments on the existing results of constrained controllability for the fractional dynamical system are also presented.

Absorbing boundary conditions for second-order hyperbolic equations

NASA Technical Reports Server (NTRS)

Jiang, Hong; Wong, Yau Shu

1989-01-01

A uniform approach to construct absorbing artificial boundary conditions for second-order linear hyperbolic equations is proposed. The nonlocal boundary condition is given by a pseudodifferential operator that annihilates travelling waves. It is obtained through the dispersion relation of the differential equation by requiring that the initial-boundary value problem admits the wave solutions travelling in one direction only. Local approximation of this global boundary condition yields an nth-order differential operator. It is shown that the best approximations must be in the canonical forms which can be factorized into first-order operators. These boundary conditions are perfectly absorbing for wave packets propagating at certain group velocities. A hierarchy of absorbing boundary conditions is derived for transonic small perturbation equations of unsteady flows. These examples illustrate that the absorbing boundary conditions are easy to derive, and the effectiveness is demonstrated by the numerical experiments.

Unconditionally marginal stability of harmonic electron hole equilibria in current-driven plasmas

NASA Astrophysics Data System (ADS)

Schamel, Hans

2018-06-01

Two forms of the linearized eigenvalue problem with respect to linear perturbations of a privileged cnoidal electron hole as a structural nonlinear equilibrium element are established. Whereas its integral form involves integrations along the characteristics or unperturbed particle orbits, the differential form has to cope with a differential operator of infinite order. Both are hence faced with difficulties to obtain a solution. A first successful attempt is, however, made by addressing a single harmonic wave as a nonlinear equilibrium structure. By this microscopic nonlinear approach, its marginal stability against linear perturbations in both linear stability regimes, the sub- and super-critical one, is shown independent of the mobility of ions and in favor with recent observations. Responsible for vanishing damping (growth) is the microscopic distortion of the resonant distribution function. The macroscopic form of the trapping nonlinearity—the 3/2 power term of the electrostatic potential in the density—which disappears in the monochromatic harmonic wave limit is consequently necessary for the occurrence of a nonlinear plasma instability in the sub-critical regime.

Stokes polarimetry imaging of dog prostate tissue

NASA Astrophysics Data System (ADS)

Kim, Jihoon; Johnston, William K., III; Walsh, Joseph T., Jr.

2010-02-01

Prostate cancer is the second leading cause of death in the United States in 2009. Radical prostatectomy (complete removal of the prostate) is the most common treatment for prostate cancer, however, differentiating prostate tissue from adjacent bladder, nerves, and muscle is difficult. Improved visualization could improve oncologic outcomes and decrease damage to adjacent nerves and muscle important for preservation of potency and continence. A novel Stokes polarimetry imaging (SPI) system was developed and evaluated using a dog prostate specimen in order to examine the feasibility of the system to differentiate prostate from bladder. The degree of linear polarization (DOLP) image maps from linearly polarized light illumination at different visible wavelengths (475, 510, and 650 nm) were constructed. The SPI system used the polarization property of the prostate tissue. The DOLP images allowed advanced differentiation by distinguishing glandular tissue of prostate from the muscular-stromal tissue in the bladder. The DOLP image at 650 nm effectively differentiated prostate and bladder by strong DOLP in bladder. SPI system has the potential to improve surgical outcomes in open or robotic-assisted laparoscopic removal of the prostate. Further in vivo testing is warranted.

Computer simulation of two-dimensional unsteady flows in estuaries and embayments by the method of characteristics : basic theory and the formulation of the numerical method

USGS Publications Warehouse

Lai, Chintu

1977-01-01

Two-dimensional unsteady flows of homogeneous density in estuaries and embayments can be described by hyperbolic, quasi-linear partial differential equations involving three dependent and three independent variables. A linear combination of these equations leads to a parametric equation of characteristic form, which consists of two parts: total differentiation along the bicharacteristics and partial differentiation in space. For its numerical solution, the specified-time-interval scheme has been used. The unknown, partial space-derivative terms can be eliminated first by suitable combinations of difference equations, converted from the corresponding differential forms and written along four selected bicharacteristics and a streamline. Other unknowns are thus made solvable from the known variables on the current time plane. The computation is carried to the second-order accuracy by using trapezoidal rule of integration. Means to handle complex boundary conditions are developed for practical application. Computer programs have been written and a mathematical model has been constructed for flow simulation. The favorable computer outputs suggest further exploration and development of model worthwhile. (Woodard-USGS)

Exact finite difference schemes for the non-linear unidirectional wave equation

NASA Technical Reports Server (NTRS)

Mickens, R. E.

1985-01-01

Attention is given to the construction of exact finite difference schemes for the nonlinear unidirectional wave equation that describes the nonlinear propagation of a wave motion in the positive x-direction. The schemes constructed for these equations are compared with those obtained by using the usual procedures of numerical analysis. It is noted that the order of the exact finite difference models is equal to the order of the differential equation.

An algorithm for solving the perturbed gas dynamic equations

NASA Technical Reports Server (NTRS)

Davis, Sanford

1993-01-01

The present application of a compact, higher-order central-difference approximation to the linearized Euler equations illustrates the multimodal character of these equations by means of computations for acoustic, vortical, and entropy waves. Such dissipationless central-difference methods are shown to propagate waves exhibiting excellent phase and amplitude resolution on the basis of relatively large time-steps; they can be applied to wave problems governed by systems of first-order partial differential equations.

Linear Back-Drive Differentials

NASA Technical Reports Server (NTRS)

Waydo, Peter

2003-01-01

Linear back-drive differentials have been proposed as alternatives to conventional gear differentials for applications in which there is only limited rotational motion (e.g., oscillation). The finite nature of the rotation makes it possible to optimize a linear back-drive differential in ways that would not be possible for gear differentials or other differentials that are required to be capable of unlimited rotation. As a result, relative to gear differentials, linear back-drive differentials could be more compact and less massive, could contain fewer complex parts, and could be less sensitive to variations in the viscosities of lubricants. Linear back-drive differentials would operate according to established principles of power ball screws and linear-motion drives, but would utilize these principles in an innovative way. One major characteristic of such mechanisms that would be exploited in linear back-drive differentials is the possibility of designing them to drive or back-drive with similar efficiency and energy input: in other words, such a mechanism can be designed so that a rotating screw can drive a nut linearly or the linear motion of the nut can cause the screw to rotate. A linear back-drive differential (see figure) would include two collinear shafts connected to two parts that are intended to engage in limited opposing rotations. The linear back-drive differential would also include a nut that would be free to translate along its axis but not to rotate. The inner surface of the nut would be right-hand threaded at one end and left-hand threaded at the opposite end to engage corresponding right- and left-handed threads on the shafts. A rotation and torque introduced into the system via one shaft would drive the nut in linear motion. The nut, in turn, would back-drive the other shaft, creating a reaction torque. Balls would reduce friction, making it possible for the shaft/nut coupling on each side to operate with 90 percent efficiency.

Exact linearized Coulomb collision operator in the moment expansion

DOE PAGES

Ji, Jeong -Young; Held, Eric D.

2006-10-05

In the moment expansion, the Rosenbluth potentials, the linearized Coulomb collision operators, and the moments of the collision operators are analytically calculated for any moment. The explicit calculation of Rosenbluth potentials converts the integro-differential form of the Coulomb collision operator into a differential operator, which enables one to express the collision operator in a simple closed form for any arbitrary mass and temperature ratios. In addition, it is shown that gyrophase averaging the collision operator acting on arbitrary distribution functions is the same as the collision operator acting on the corresponding gyrophase averaged distribution functions. The moments of the collisionmore » operator are linear combinations of the fluid moments with collision coefficients parametrized by mass and temperature ratios. Furthermore, useful forms involving the small mass-ratio approximation are easily found since the collision operators and their moments are expressed in terms of the mass ratio. As an application, the general moment equations are explicitly written and the higher order heat flux equation is derived.« less

Analysis of control system responses for aircraft stability and efficient numerical techniques for real-time simulations

NASA Astrophysics Data System (ADS)

Stroe, Gabriela; Andrei, Irina-Carmen; Frunzulica, Florin

2017-01-01

The objectives of this paper are the study and the implementation of both aerodynamic and propulsion models, as linear interpolations using look-up tables in a database. The aerodynamic and propulsion dependencies on state and control variable have been described by analytic polynomial models. Some simplifying hypotheses were made in the development of the nonlinear aircraft simulations. The choice of a certain technique to use depends on the desired accuracy of the solution and the computational effort to be expended. Each nonlinear simulation includes the full nonlinear dynamics of the bare airframe, with a scaled direct connection from pilot inputs to control surface deflections to provide adequate pilot control. The engine power dynamic response was modeled with an additional state equation as first order lag in the actual power level response to commanded power level was computed as a function of throttle position. The number of control inputs and engine power states varied depending on the number of control surfaces and aircraft engines. The set of coupled, nonlinear, first-order ordinary differential equations that comprise the simulation model can be represented by the vector differential equation. A linear time-invariant (LTI) system representing aircraft dynamics for small perturbations about a reference trim condition is given by the state and output equations present. The gradients are obtained numerically by perturbing each state and control input independently and recording the changes in the trimmed state and output equations. This is done using the numerical technique of central finite differences, including the perturbations of the state and control variables. For a reference trim condition of straight and level flight, linearization results in two decoupled sets of linear, constant-coefficient differential equations for longitudinal and lateral / directional motion. The linearization is valid for small perturbations about the reference trim condition. Experimental aerodynamic and thrust data are used to model the applied aerodynamic and propulsion forces and moments for arbitrary states and controls. There is no closed form solution to such problems, so the equations must be solved using numerical integration. Techniques for solving this initial value problem for ordinary differential equations are employed to obtain approximate solutions at discrete points along the aircraft state trajectory.

A Fifth-order Symplectic Trigonometrically Fitted Partitioned Runge-Kutta Method

NASA Astrophysics Data System (ADS)

Kalogiratou, Z.; Monovasilis, Th.; Simos, T. E.

2007-09-01

Trigonometrically fitted symplectic Partitioned Runge Kutta (EFSPRK) methods for the numerical integration of Hamoltonian systems with oscillatory solutions are derived. These methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions sin(wx),cos(wx), w∈R. We modify a fifth order symplectic PRK method with six stages so to derive an exponentially fitted SPRK method. The methods are tested on the numerical integration of the two body problem.

Trees, B-series and G-symplectic methods

NASA Astrophysics Data System (ADS)

Butcher, J. C.

2017-07-01

The order conditions for Runge-Kutta methods are intimately connected with the graphs known as rooted trees. The conditions can be expressed in terms of Taylor expansions written as weighted sums of elementary differentials, that is as B-series. Polish notation provides a unifying structure for representing many of the quantities appearing in this theory. Applications include the analysis of general linear methods with special reference to G-symplectic methods. A new order 6 method has recently been constructed.

Method and system of Jones-matrix mapping of blood plasma films with "fuzzy" analysis in differentiation of breast pathology changes

NASA Astrophysics Data System (ADS)

Zabolotna, Natalia I.; Radchenko, Kostiantyn O.; Karas, Oleksandr V.

2018-01-01

A fibroadenoma diagnosing of breast using statistical analysis (determination and analysis of statistical moments of the 1st-4th order) of the obtained polarization images of Jones matrix imaginary elements of the optically thin (attenuation coefficient τ <= 0,1 ) blood plasma films with further intellectual differentiation based on the method of "fuzzy" logic and discriminant analysis were proposed. The accuracy of the intellectual differentiation of blood plasma samples to the "norm" and "fibroadenoma" of breast was 82.7% by the method of linear discriminant analysis, and by the "fuzzy" logic method is 95.3%. The obtained results allow to confirm the potentially high level of reliability of the method of differentiation by "fuzzy" analysis.

PAN AIR: A Computer Program for Predicting Subsonic or Supersonic Linear Potential Flows About Arbitrary Configurations Using a Higher Order Panel Method. Volume 1; Theory Document (Version 1.1)

NASA Technical Reports Server (NTRS)

Magnus, Alfred E.; Epton, Michael A.

1981-01-01

An outline of the derivation of the differential equation governing linear subsonic and supersonic potential flow is given. The use of Green's Theorem to obtain an integral equation over the boundary surface is discussed. The engineering techniques incorporated in the PAN AIR (Panel Aerodynamics) program (a discretization method which solves the integral equation for arbitrary first order boundary conditions) are then discussed in detail. Items discussed include the construction of the compressibility transformations, splining techniques, imposition of the boundary conditions, influence coefficient computation (including the concept of the finite part of an integral), computation of pressure coefficients, and computation of forces and moments.

On differential operators generating iterative systems of linear ODEs of maximal symmetry algebra

NASA Astrophysics Data System (ADS)

Ndogmo, J. C.

2017-06-01

Although every iterative scalar linear ordinary differential equation is of maximal symmetry algebra, the situation is different and far more complex for systems of linear ordinary differential equations, and an iterative system of linear equations need not be of maximal symmetry algebra. We illustrate these facts by examples and derive families of vector differential operators whose iterations are all linear systems of equations of maximal symmetry algebra. Some consequences of these results are also discussed.

Computing the Lyapunov spectrum of a dynamical system from an observed time series

NASA Technical Reports Server (NTRS)

Brown, Reggie; Bryant, Paul; Abarbanel, Henry D. I.

1991-01-01

The paper examines the problem of accurately determining, from an observed time series, the Liapunov exponents for the dynamical system generating the data. It is shown that, even with very large data sets, it is clearly advantageous to utilize local neighborhood-to-neighborhood mappings with higher-order Taylor series rather than just local linear maps. This procedure is demonstrated on the Henon and Ikeda maps of the plane itself, the Lorenz system of three ordinary differential equations, and the Mackey-Glass delay differential equation.

Schwarz maps of algebraic linear ordinary differential equations

NASA Astrophysics Data System (ADS)

Sanabria Malagón, Camilo

2017-12-01

A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard-Vessiot Theory.

Local Discontinuous Galerkin Methods for Partial Differential Equations with Higher Order Derivatives

NASA Technical Reports Server (NTRS)

Yan, Jue; Shu, Chi-Wang; Bushnell, Dennis M. (Technical Monitor)

2002-01-01

In this paper we review the existing and develop new continuous Galerkin methods for solving time dependent partial differential equations with higher order derivatives in one and multiple space dimensions. We review local discontinuous Galerkin methods for convection diffusion equations involving second derivatives and for KdV type equations involving third derivatives. We then develop new local discontinuous Galerkin methods for the time dependent bi-harmonic type equations involving fourth derivatives, and partial differential equations involving fifth derivatives. For these new methods we present correct interface numerical fluxes and prove L(exp 2) stability for general nonlinear problems. Preliminary numerical examples are shown to illustrate these methods. Finally, we present new results on a post-processing technique, originally designed for methods with good negative-order error estimates, on the local discontinuous Galerkin methods applied to equations with higher derivatives. Numerical experiments show that this technique works as well for the new higher derivative cases, in effectively doubling the rate of convergence with negligible additional computational cost, for linear as well as some nonlinear problems, with a local uniform mesh.

Numerical method for the solution of large systems of differential equations of the boundary layer type

NASA Technical Reports Server (NTRS)

Green, M. J.; Nachtsheim, P. R.

1972-01-01

A numerical method for the solution of large systems of nonlinear differential equations of the boundary-layer type is described. The method is a modification of the technique for satisfying asymptotic boundary conditions. The present method employs inverse interpolation instead of the Newton method to adjust the initial conditions of the related initial-value problem. This eliminates the so-called perturbation equations. The elimination of the perturbation equations not only reduces the user's preliminary work in the application of the method, but also reduces the number of time-consuming initial-value problems to be numerically solved at each iteration. For further ease of application, the solution of the overdetermined system for the unknown initial conditions is obtained automatically by applying Golub's linear least-squares algorithm. The relative ease of application of the proposed numerical method increases directly as the order of the differential-equation system increases. Hence, the method is especially attractive for the solution of large-order systems. After the method is described, it is applied to a fifth-order problem from boundary-layer theory.

An almost symmetric Strang splitting scheme for nonlinear evolution equations.

PubMed

Einkemmer, Lukas; Ostermann, Alexander

2014-07-01

In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow cannot be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described, the classic Strang splitting scheme, while still being a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation.

An almost symmetric Strang splitting scheme for nonlinear evolution equations☆

PubMed Central

Einkemmer, Lukas; Ostermann, Alexander

2014-01-01

In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow cannot be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described, the classic Strang splitting scheme, while still being a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation. PMID:25844017

O(t-α)-synchronization and Mittag-Leffler synchronization for the fractional-order memristive neural networks with delays and discontinuous neuron activations.

PubMed

Chen, Jiejie; Chen, Boshan; Zeng, Zhigang

2018-04-01

This paper investigates O(t -α )-synchronization and adaptive Mittag-Leffler synchronization for the fractional-order memristive neural networks with delays and discontinuous neuron activations. Firstly, based on the framework of Filippov solution and differential inclusion theory, using a Razumikhin-type method, some sufficient conditions ensuring the global O(t -α )-synchronization of considered networks are established via a linear-type discontinuous control. Next, a new fractional differential inequality is established and two new discontinuous adaptive controller is designed to achieve Mittag-Leffler synchronization between the drive system and the response systems using this inequality. Finally, two numerical simulations are given to show the effectiveness of the theoretical results. Our approach and theoretical results have a leading significance in the design of synchronized fractional-order memristive neural networks circuits involving discontinuous activations and time-varying delays. Copyright © 2018 Elsevier Ltd. All rights reserved.

Asymptotic integration algorithms for first-order ODEs with application to viscoplasticity

NASA Technical Reports Server (NTRS)

Freed, Alan D.; Yao, Minwu; Walker, Kevin P.

1992-01-01

When constructing an algorithm for the numerical integration of a differential equation, one must first convert the known ordinary differential equation (ODE), which is defined at a point, into an ordinary difference equation (O(delta)E), which is defined over an interval. Asymptotic, generalized, midpoint, and trapezoidal, O(delta)E algorithms are derived for a nonlinear first order ODE written in the form of a linear ODE. The asymptotic forward (typically underdamped) and backward (typically overdamped) integrators bound these midpoint and trapezoidal integrators, which tend to cancel out unwanted numerical damping by averaging, in some sense, the forward and backward integrations. Viscoplasticity presents itself as a system of nonlinear, coupled first-ordered ODE's that are mathematically stiff, and therefore, difficult to numerically integrate. They are an excellent application for the asymptotic integrators. Considering a general viscoplastic structure, it is demonstrated that one can either integrate the viscoplastic stresses or their associated eigenstrains.

Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions

NASA Astrophysics Data System (ADS)

Zeng, Fanhai; Zhang, Zhongqiang; Karniadakis, George Em

2017-12-01

Starting with the asymptotic expansion of the error equation of the shifted Gr\\"{u}nwald--Letnikov formula, we derive a new modified weighted shifted Gr\\"{u}nwald--Letnikov (WSGL) formula by introducing appropriate correction terms. We then apply one special case of the modified WSGL formula to solve multi-term fractional ordinary and partial differential equations, and we prove the linear stability and second-order convergence for both smooth and non-smooth solutions. We show theoretically and numerically that numerical solutions up to certain accuracy can be obtained with only a few correction terms. Moreover, the correction terms can be tuned according to the fractional derivative orders without explicitly knowing the analytical solutions. Numerical simulations verify the theoretical results and demonstrate that the new formula leads to better performance compared to other known numerical approximations with similar resolution.

On analyticity of linear waves scattered by a layered medium

NASA Astrophysics Data System (ADS)

Nicholls, David P.

2017-10-01

The scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering. In this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of High-Order Perturbation of Surfaces methods for simulating such waves. More specifically, we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a multiply layered periodic structure in three dimensions. This result provides hypotheses under which a rigorous numerical analysis could be conducted for recent generalizations to the methods of Operator Expansions, Field Expansions, and Transformed Field Expansions.

A highly linear baseband Gm—C filter for WLAN application

NASA Astrophysics Data System (ADS)

Lijun, Yang; Zheng, Gong; Yin, Shi; Zhiming, Chen

2011-09-01

A low voltage, highly linear transconductan—C (Gm—C) low-pass filter for wireless local area network (WLAN) transceiver application is proposed. This transmitter (Tx) filter adopts a 9.8 MHz 3rd-order Chebyshev low pass prototype and achieves 35 dB stop-band attenuation at 30 MHz frequency. By utilizing pseudo-differential linear-region MOS transconductors, the filter IIP3 is measured to be as high as 9.5 dBm. Fabricated in a 0.35 μm standard CMOS technology, the proposed filter chip occupies a 0.41 × 0.17 mm2 die area and consumes 3.36 mA from a 3.3-V power supply.

Gaussian closure technique applied to the hysteretic Bouc model with non-zero mean white noise excitation

NASA Astrophysics Data System (ADS)

Waubke, Holger; Kasess, Christian H.

2016-11-01

Devices that emit structure-borne sound are commonly decoupled by elastic components to shield the environment from acoustical noise and vibrations. The elastic elements often have a hysteretic behavior that is typically neglected. In order to take hysteretic behavior into account, Bouc developed a differential equation for such materials, especially joints made of rubber or equipped with dampers. In this work, the Bouc model is solved by means of the Gaussian closure technique based on the Kolmogorov equation. Kolmogorov developed a method to derive probability density functions for arbitrary explicit first-order vector differential equations under white noise excitation using a partial differential equation of a multivariate conditional probability distribution. Up to now no analytical solution of the Kolmogorov equation in conjunction with the Bouc model exists. Therefore a wide range of approximate solutions, especially the statistical linearization, were developed. Using the Gaussian closure technique that is an approximation to the Kolmogorov equation assuming a multivariate Gaussian distribution an analytic solution is derived in this paper for the Bouc model. For the stationary case the two methods yield equivalent results, however, in contrast to statistical linearization the presented solution allows to calculate the transient behavior explicitly. Further, stationary case leads to an implicit set of equations that can be solved iteratively with a small number of iterations and without instabilities for specific parameter sets.

Infrared spectroscopy as a tool to characterise starch ordered structure--a joint FTIR-ATR, NMR, XRD and DSC study.

PubMed

Warren, Frederick J; Gidley, Michael J; Flanagan, Bernadine M

2016-03-30

Starch has a heterogeneous, semi-crystalline granular structure and the degree of ordered structure can affect its behaviour in foods and bioplastics. A range of methodologies are employed to study starch structure; differential scanning calorimetry, (13)C nuclear magnetic resonance, X-ray diffraction and Fourier transform infrared spectroscopy (FTIR). Despite the appeal of FTIR as a rapid, non-destructive methodology, there is currently no systematically defined quantitative relationship between FTIR spectral features and other starch structural measures. Here, we subject 61 starch samples to structural analysis, and systematically correlate FTIR spectra with other measures of starch structure. A hydration dependent peak position shift in the FTIR spectra of starch is observed, resulting from increased molecular order, but with complex, non-linear behaviour. We demonstrate that FTIR is a tool that can quantitatively probe short range interactions in starch structure. However, the assumptions of linear relationships between starch ordered structure and peak ratios are overly simplistic. Copyright © 2015 Elsevier Ltd. All rights reserved.

Long-period fiber gratings as ultrafast optical differentiators.

PubMed

Kulishov, Mykola; Azaña, José

2005-10-15

It is demonstrated that a single, uniform long-period fiber grating (LPFG) working in the linear regime inherently behaves as an ultrafast optical temporal differentiator. Specifically, we show that the output temporal waveform in the core mode of a LPFG providing full energy coupling into the cladding mode is proportional to the first derivative of the optical temporal signal (e.g., optical pulse) launched at the input of the LPFG. Moreover, a LPFG providing full energy recoupling back from the cladding mode into the core mode inherently implements second-order temporal differentiation. Our numerical results have confirmed the feasibility of this simple, all-fiber approach to processing optical signals with temporal features in the picosecond and subpicosecond ranges.

On the connection coefficients and recurrence relations arising from expansions in series of Laguerre polynomials

NASA Astrophysics Data System (ADS)

Doha, E. H.

2003-05-01

A formula expressing the Laguerre coefficients of a general-order derivative of an infinitely differentiable function in terms of its original coefficients is proved, and a formula expressing explicitly the derivatives of Laguerre polynomials of any degree and for any order as a linear combination of suitable Laguerre polynomials is deduced. A formula for the Laguerre coefficients of the moments of one single Laguerre polynomial of certain degree is given. Formulae for the Laguerre coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its Laguerre coefficients are also obtained. A simple approach in order to build and solve recursively for the connection coefficients between Jacobi-Laguerre and Hermite-Laguerre polynomials is described. An explicit formula for these coefficients between Jacobi and Laguerre polynomials is given, of which the ultra-spherical polynomials of the first and second kinds and Legendre polynomials are important special cases. An analytical formula for the connection coefficients between Hermite and Laguerre polynomials is also obtained.

White noise analysis of Phycomyces light growth response system. I. Normal intensity range.

PubMed Central

Lipson, E D

1975-01-01

The Wiener-Lee-Schetzen method for the identification of a nonlinear system through white gaussian noise stimulation was applied to the transient light growth response of the sporangiophore of Phycomyces. In order to cover a moderate dynamic range of light intensity I, the imput variable was defined to be log I. The experiments were performed in the normal range of light intensity, centered about I0 = 10(-6) W/cm2. The kernels of the Wierner functionals were computed up to second order. Within the range of a few decades the system is reasonably linear with log I. The main nonlinear feature of the second-order kernel corresponds to the property of rectification. Power spectral analysis reveals that the slow dynamics of the system are of at least fifth order. The system can be represented approximately by a linear transfer function, including a first-order high-pass (adaptation) filter with a 4 min time constant and an underdamped fourth-order low-pass filter. Accordingly a linear electronic circuit was constructed to simulate the small scale response characteristics. In terms of the adaptation model of Delbrück and Reichardt (1956, in Cellular Mechanisms in Differentiation and Growth, Princeton University Press), kernels were deduced for the dynamic dependence of the growth velocity (output) on the "subjective intensity", a presumed internal variable. Finally the linear electronic simulator above was generalized to accommodate the large scale nonlinearity of the adaptation model and to serve as a tool for deeper test of the model. PMID:1203444

Unsteady Solution of Non-Linear Differential Equations Using Walsh Function Series

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.

2015-01-01

Walsh functions form an orthonormal basis set consisting of square waves. The discontinuous nature of square waves make the system well suited for representing functions with discontinuities. The product of any two Walsh functions is another Walsh function - a feature that can radically change an algorithm for solving non-linear partial differential equations (PDEs). The solution algorithm of non-linear differential equations using Walsh function series is unique in that integrals and derivatives may be computed using simple matrix multiplication of series representations of functions. Solutions to PDEs are derived as functions of wave component amplitude. Three sample problems are presented to illustrate the Walsh function series approach to solving unsteady PDEs. These include an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the use of the Walsh function solution algorithms, exploiting Fast Walsh Transforms in multi-dimensions (O(Nlog(N))). Details of a Fast Walsh Reciprocal, defined here for the first time, enable inversion of aWalsh Symmetric Matrix in O(Nlog(N)) operations. Walsh functions have been derived using a fractal recursion algorithm and these fractal patterns are observed in the progression of pairs of wave number amplitudes in the solutions. These patterns are most easily observed in a remapping defined as a fractal fingerprint (FFP). A prolongation of existing solutions to the next highest order exploits these patterns. The algorithms presented here are considered a work in progress that provide new alternatives and new insights into the solution of non-linear PDEs.

Recursive linearization of multibody dynamics equations of motion

NASA Technical Reports Server (NTRS)

Lin, Tsung-Chieh; Yae, K. Harold

1989-01-01

The equations of motion of a multibody system are nonlinear in nature, and thus pose a difficult problem in linear control design. One approach is to have a first-order approximation through the numerical perturbations at a given configuration, and to design a control law based on the linearized model. Here, a linearized model is generated analytically by following the footsteps of the recursive derivation of the equations of motion. The equations of motion are first written in a Newton-Euler form, which is systematic and easy to construct; then, they are transformed into a relative coordinate representation, which is more efficient in computation. A new computational method for linearization is obtained by applying a series of first-order analytical approximations to the recursive kinematic relationships. The method has proved to be computationally more efficient because of its recursive nature. It has also turned out to be more accurate because of the fact that analytical perturbation circumvents numerical differentiation and other associated numerical operations that may accumulate computational error, thus requiring only analytical operations of matrices and vectors. The power of the proposed linearization algorithm is demonstrated, in comparison to a numerical perturbation method, with a two-link manipulator and a seven degrees of freedom robotic manipulator. Its application to control design is also demonstrated.

Existence of liouvillian solutions in the problem of motion of a rotationally symmetric body on a sphere

NASA Astrophysics Data System (ADS)

Kuleshov, Alexander S.; Katasonova, Vera A.

2018-05-01

The problem of rolling without slipping of a rotationally symmetric rigid body on a sphere is considered. The rolling body is assumed to be subjected to the forces, the resultant of which is directed from the center of mass G of the body to the center O of the sphere, and depends only on the distance between G and O. In this case the solution of this problem is reduced to solving the second order linear differential equation over the projection of the angular velocity of the body onto its axis of symmetry. Using the Kovacic algorithm we search for liouvillian solutions of the corresponding second order differential equation in the case, when the rolling body is a dynamically symmetric ball.

Linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

NASA Astrophysics Data System (ADS)

Camporesi, Roberto

2011-06-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and the variation of constants method. The approach presented here can be used in a first course on differential equations for science and engineering majors.

Nonautonomous linear system of the terrestrial carbon cycle

NASA Astrophysics Data System (ADS)

Luo, Y.

2012-12-01

Carbon cycle has been studied by uses of observation through various networks, field and laboratory experiments, and simulation models. Much less has been done on theoretical thinking and analysis to understand fundament properties of carbon cycle and then guide observatory, experimental, and modeling research. This presentation is to explore what would be the theoretical properties of terrestrial carbon cycle and how those properties can be used to make observatory, experimental, and modeling research more effective. Thousands of published data sets from litter decomposition and soil incubation studies almost all indicate that decay processes of litter and soil organic carbon can be well described by first order differential equations with one or more pools. Carbon pool dynamics in plants and soil after disturbances (e.g., wildfire, clear-cut of forests, and plows of soil for cropping) and during natural recovery or ecosystem restoration also exhibit characteristics of first-order linear systems. Thus, numerous lines of empirical evidence indicate that the terrestrial carbon cycle can be adequately described as a nonautonomous linear system. The linearity reflects the nature of the carbon cycle that carbon, once fixed by photosynthesis, is linearly transferred among pools within an ecosystem. The linear carbon transfer, however, is modified by nonlinear functions of external forcing variables. In addition, photosynthetic carbon influx is also nonlinearly influenced by external variables. This nonautonomous linear system can be mathematically expressed by a first-order linear ordinary matrix equation. We have recently used this theoretical property of terrestrial carbon cycle to develop a semi-analytic solution of spinup. The new methods have been applied to five global land models, including NCAR's CLM and CABLE models and can computationally accelerate spinup by two orders of magnitude. We also use this theoretical property to develop an analytic framework to decompose modeled carbon cycle into a few traceable components so as to facilitate model intercompsirosn, benchmark analysis, and data assimilation of global land models.

Fast solution of elliptic partial differential equations using linear combinations of plane waves.

PubMed

Pérez-Jordá, José M

2016-02-01

Given an arbitrary elliptic partial differential equation (PDE), a procedure for obtaining its solution is proposed based on the method of Ritz: the solution is written as a linear combination of plane waves and the coefficients are obtained by variational minimization. The PDE to be solved is cast as a system of linear equations Ax=b, where the matrix A is not sparse, which prevents the straightforward application of standard iterative methods in order to solve it. This sparseness problem can be circumvented by means of a recursive bisection approach based on the fast Fourier transform, which makes it possible to implement fast versions of some stationary iterative methods (such as Gauss-Seidel) consuming O(NlogN) memory and executing an iteration in O(Nlog(2)N) time, N being the number of plane waves used. In a similar way, fast versions of Krylov subspace methods and multigrid methods can also be implemented. These procedures are tested on Poisson's equation expressed in adaptive coordinates. It is found that the best results are obtained with the GMRES method using a multigrid preconditioner with Gauss-Seidel relaxation steps.

Modeling of aircraft unsteady aerodynamic characteristics. Part 1: Postulated models

NASA Technical Reports Server (NTRS)

Klein, Vladislav; Noderer, Keith D.

1994-01-01

A short theoretical study of aircraft aerodynamic model equations with unsteady effects is presented. The aerodynamic forces and moments are expressed in terms of indicial functions or internal state variables. The first representation leads to aircraft integro-differential equations of motion; the second preserves the state-space form of the model equations. The formulations of unsteady aerodynamics is applied in two examples. The first example deals with a one-degree-of-freedom harmonic motion about one of the aircraft body axes. In the second example, the equations for longitudinal short-period motion are developed. In these examples, only linear aerodynamic terms are considered. The indicial functions are postulated as simple exponentials and the internal state variables are governed by linear, time-invariant, first-order differential equations. It is shown that both approaches to the modeling of unsteady aerodynamics lead to identical models.

Axisymmetric Powell-Eyring fluid flow over a stretching sheet with a convective boundary condition and suction effects

NASA Astrophysics Data System (ADS)

Nasir, Nor Ain Azeany Mohd; Ishak, Anuar; Pop, Ioan

2018-04-01

In this paper, the heat and mass transfer of an axisymmetric Powell-Eyring fluid flow over a stretching sheet with a convective boundary condition and suction effects are investigated. An appropriate similarity transformation is used to reduce the highly non-linear partial differential equation into second and third order non-linear ordinary differential equations. Numerical solutions of the reduced governing equations are computed numerically by utilizing the MATLAB's built-in boundary value problem solver, bvp4c. The physical significance of various parameters such as Biot number, fluid parameters and Prandtl number on the velocity and temperature evolution profiles are illustrated graphically. The effects of these governing parameters on the skin friction coefficient and the local Nusselt number are also displayed graphically. It is noticed that the Powell-Eyring fluid parameter gives significant influence on the rates of heat and mass transfer of the fluid.

The surface-induced spatial-temporal structures in confined binary alloys

NASA Astrophysics Data System (ADS)

Krasnyuk, Igor B.; Taranets, Roman M.; Chugunova, Marina

2014-12-01

This paper examines surface-induced ordering in confined binary alloys. The hyperbolic initial boundary value problem (IBVP) is used to describe a scenario of spatiotemporal ordering in a disordered phase for concentration of one component of binary alloy and order parameter with non-linear dynamic boundary conditions. This hyperbolic model consists of two coupled second order differential equations for order parameter and concentration. It also takes into account effects of the “memory” on the ordering of atoms and their densities in the alloy. The boundary conditions characterize surface velocities of order parameter and concentration changing which is due to surface (super)cooling on walls confining the binary alloy. It is shown that for large times there are three classes of dynamic non-linear boundary conditions which lead to three different types of attractor’s elements for the IBVP. Namely, the elements of attractor are the limit periodic simple shock waves with fronts of “discontinuities” Γ. If Γ is finite, then the attractor contains spatiotemporal functions of relaxation type. If Γ is infinite and countable then we observe the functions of pre-turbulent type. If Γ is infinite and uncountable then we obtain the functions of turbulent type.

Rationale for the Definition of the Particular Solution to an Initial Value Problem: A Unique Solution Is Guaranteed

ERIC Educational Resources Information Center

Perna, James

2016-01-01

The purpose of this article is to examine the reasoning behind the wording of the definition of the particular solution to an initial value problem. This article will be of practical importance for students taking a first year calculus course that includes the study of first order linear separable differential equations.

Kinetic modeling and fitting software for interconnected reaction schemes: VisKin.

PubMed

Zhang, Xuan; Andrews, Jared N; Pedersen, Steen E

2007-02-15

Reaction kinetics for complex, highly interconnected kinetic schemes are modeled using analytical solutions to a system of ordinary differential equations. The algorithm employs standard linear algebra methods that are implemented using MatLab functions in a Visual Basic interface. A graphical user interface for simple entry of reaction schemes facilitates comparison of a variety of reaction schemes. To ensure microscopic balance, graph theory algorithms are used to determine violations of thermodynamic cycle constraints. Analytical solutions based on linear differential equations result in fast comparisons of first order kinetic rates and amplitudes as a function of changing ligand concentrations. For analysis of higher order kinetics, we also implemented a solution using numerical integration. To determine rate constants from experimental data, fitting algorithms that adjust rate constants to fit the model to imported data were implemented using the Levenberg-Marquardt algorithm or using Broyden-Fletcher-Goldfarb-Shanno methods. We have included the ability to carry out global fitting of data sets obtained at varying ligand concentrations. These tools are combined in a single package, which we have dubbed VisKin, to guide and analyze kinetic experiments. The software is available online for use on PCs.

A regularity result for fixed points, with applications to linear response

NASA Astrophysics Data System (ADS)

Sedro, Julien

2018-04-01

In this paper, we show a series of abstract results on fixed point regularity with respect to a parameter. They are based on a Taylor development taking into account a loss of regularity phenomenon, typically occurring for composition operators acting on spaces of functions with finite regularity. We generalize this approach to higher order differentiability, through the notion of an n-graded family. We then give applications to the fixed point of a nonlinear map, and to linear response in the context of (uniformly) expanding dynamics (theorem 3 and corollary 2), in the spirit of Gouëzel-Liverani.

Higher symmetries and exact solutions of linear and nonlinear Schr{umlt o}dinger equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Fushchych, W.I.; Nikitin, A.G.

1997-11-01

A new approach for the analysis of partial differential equations is developed which is characterized by a simultaneous use of higher and conditional symmetries. Higher symmetries of the Schr{umlt o}dinger equation with an arbitrary potential are investigated. Nonlinear determining equations for potentials are solved using reductions to Weierstrass, Painlev{acute e}, and Riccati forms. Algebraic properties of higher order symmetry operators are analyzed. Combinations of higher and conditional symmetries are used to generate families of exact solutions of linear and nonlinear Schr{umlt o}dinger equations. {copyright} {ital 1997 American Institute of Physics.}

PAN AIR: A computer program for predicting subsonic or supersonic linear potential flows about arbitrary configurations using a higher order panel method. Volume 1: Theory document (version 1.1)

NASA Technical Reports Server (NTRS)

Magnus, A. E.; Epton, M. A.

1981-01-01

Panel aerodynamics (PAN AIR) is a system of computer programs designed to analyze subsonic and supersonic inviscid flows about arbitrary configurations. A panel method is a program which solves a linear partial differential equation by approximating the configuration surface by a set of panels. An overview of the theory of potential flow in general and PAN AIR in particular is given along with detailed mathematical formulations. Fluid dynamics, the Navier-Stokes equation, and the theory of panel methods were also discussed.

Numerical analysis of MHD Carreau fluid flow over a stretching cylinder with homogenous-heterogeneous reactions

NASA Astrophysics Data System (ADS)

Khan, Imad; Ullah, Shafquat; Malik, M. Y.; Hussain, Arif

2018-06-01

The current analysis concentrates on the numerical solution of MHD Carreau fluid flow over a stretching cylinder under the influences of homogeneous-heterogeneous reactions. Modelled non-linear partial differential equations are converted into ordinary differential equations by using suitable transformations. The resulting system of equations is solved with the aid of shooting algorithm supported by fifth order Runge-Kutta integration scheme. The impact of non-dimensional governing parameters on the velocity, temperature, skin friction coefficient and local Nusselt number are comprehensively delineated with the help of graphs and tables.

Lie algebras and linear differential equations.

NASA Technical Reports Server (NTRS)

Brockett, R. W.; Rahimi, A.

1972-01-01

Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.

On the Singularity Structure of WKB Solution of the Boosted Whittaker Equation: its Relevance to Resurgent Functions with Essential Singularities

NASA Astrophysics Data System (ADS)

Kamimoto, Shingo; Kawai, Takahiro; Koike, Tatsuya

2016-12-01

Inspired by the symbol calculus of linear differential operators of infinite order applied to the Borel transformed WKB solutions of simple-pole type equation [Kamimoto et al. (RIMS Kôkyûroku Bessatsu B 52:127-146, 2014)], which is summarized in Section 1, we introduce in Section 2 the space of simple resurgent functions depending on a parameter with an infra-exponential type growth order, and then we define the assigning operator A which acts on the space and produces resurgent functions with essential singularities. In Section 3, we apply the operator A to the Borel transforms of the Voros coefficient and its exponentiation for the Whittaker equation with a large parameter so that we may find the Borel transforms of the Voros coefficient and its exponentiation for the boosted Whittaker equation with a large parameter. In Section 4, we use these results to find the explicit form of the alien derivatives of the Borel transformed WKB solutions of the boosted Whittaker equation with a large parameter. The results in this paper manifest the importance of resurgent functions with essential singularities in developing the exact WKB analysis, the WKB analysis based on the resurgent function theory. It is also worth emphasizing that the concrete form of essential singularities we encounter is expressed by the linear differential operators of infinite order.

Time and frequency domain analysis of sampled data controllers via mixed operation equations

NASA Technical Reports Server (NTRS)

Frisch, H. P.

1981-01-01

Specification of the mathematical equations required to define the dynamic response of a linear continuous plant, subject to sampled data control, is complicated by the fact that the digital components of the control system cannot be modeled via linear ordinary differential equations. This complication can be overcome by introducing two new mathematical operations; namely, the operation of zero order hold and digial delay. It is shown that by direct utilization of these operations, a set of linear mixed operation equations can be written and used to define the dynamic response characteristics of the controlled system. It also is shown how these linear mixed operation equations lead, in an automatable manner, directly to a set of finite difference equations which are in a format compatible with follow on time and frequency domain analysis methods.

Quasi-Newton methods for parameter estimation in functional differential equations

NASA Technical Reports Server (NTRS)

Brewer, Dennis W.

1988-01-01

A state-space approach to parameter estimation in linear functional differential equations is developed using the theory of linear evolution equations. A locally convergent quasi-Newton type algorithm is applied to distributed systems with particular emphasis on parameters that induce unbounded perturbations of the state. The algorithm is computationally implemented on several functional differential equations, including coefficient and delay estimation in linear delay-differential equations.

Newton's method: A link between continuous and discrete solutions of nonlinear problems

NASA Technical Reports Server (NTRS)

Thurston, G. A.

1980-01-01

Newton's method for nonlinear mechanics problems replaces the governing nonlinear equations by an iterative sequence of linear equations. When the linear equations are linear differential equations, the equations are usually solved by numerical methods. The iterative sequence in Newton's method can exhibit poor convergence properties when the nonlinear problem has multiple solutions for a fixed set of parameters, unless the iterative sequences are aimed at solving for each solution separately. The theory of the linear differential operators is often a better guide for solution strategies in applying Newton's method than the theory of linear algebra associated with the numerical analogs of the differential operators. In fact, the theory for the differential operators can suggest the choice of numerical linear operators. In this paper the method of variation of parameters from the theory of linear ordinary differential equations is examined in detail in the context of Newton's method to demonstrate how it might be used as a guide for numerical solutions.

A highly linear fully integrated powerline filter for biopotential acquisition systems.

PubMed

Alzaher, Hussain A; Tasadduq, Noman; Mahnashi, Yaqub

2013-10-01

Powerline interference is one of the most dominant problems in detection and processing of biopotential signals. This work presents a new fully integrated notch filter exhibiting high linearity and low power consumption. High filter linearity is preserved utilizing active-RC approach while IC implementation is achieved through replacing passive resistors by R-2R ladders achieving area saving of approximately 120 times. The filter design is optimized for low power operation using an efficient circuit topology and an ultra-low power operational amplifier. Fully differential implementation of the proposed filter shows notch depth of 43 dB (78 dB for 4th-order) with THD of better than -70 dB while consuming about 150 nW from 1.5 V supply.

Differentially pumped dual linear quadrupole ion trap mass spectrometer

DOE Office of Scientific and Technical Information (OSTI.GOV)

Owen, Benjamin C.; Kenttamaa, Hilkka I.

The present disclosure provides a new tandem mass spectrometer and methods of using the same for analyzing charged particles. The differentially pumped dual linear quadrupole ion trap mass spectrometer of the present disclose includes a combination of two linear quadrupole (LQIT) mass spectrometers with differentially pumped vacuum chambers.

A Novel Fractional Order Model for the Dynamic Hysteresis of Piezoelectrically Actuated Fast Tool Servo

PubMed Central

Zhu, Zhiwei; Zhou, Xiaoqin

2012-01-01

The main contribution of this paper is the development of a linearized model for describing the dynamic hysteresis behaviors of piezoelectrically actuated fast tool servo (FTS). A linearized hysteresis force model is proposed and mathematically described by a fractional order differential equation. Combining the dynamic modeling of the FTS mechanism, a linearized fractional order dynamic hysteresis (LFDH) model for the piezoelectrically actuated FTS is established. The unique features of the LFDH model could be summarized as follows: (a) It could well describe the rate-dependent hysteresis due to its intrinsic characteristics of frequency-dependent nonlinear phase shifts and amplitude modulations; (b) The linearization scheme of the LFDH model would make it easier to implement the inverse dynamic control on piezoelectrically actuated micro-systems. To verify the effectiveness of the proposed model, a series of experiments are conducted. The toolpaths of the FTS for creating two typical micro-functional surfaces involving various harmonic components with different frequencies and amplitudes are scaled and employed as command signals for the piezoelectric actuator. The modeling errors in the steady state are less than ±2.5% within the full span range which is much smaller than certain state-of-the-art modeling methods, demonstrating the efficiency and superiority of the proposed model for modeling dynamic hysteresis effects. Moreover, it indicates that the piezoelectrically actuated micro systems would be more suitably described as a fractional order dynamic system.

Simultaneous source and attenuation reconstruction in SPECT using ballistic and single scattering data

NASA Astrophysics Data System (ADS)

Courdurier, M.; Monard, F.; Osses, A.; Romero, F.

2015-09-01

In medical single-photon emission computed tomography (SPECT) imaging, we seek to simultaneously obtain the internal radioactive sources and the attenuation map using not only ballistic measurements but also first-order scattering measurements and assuming a very specific scattering regime. The problem is modeled using the radiative transfer equation by means of an explicit non-linear operator that gives the ballistic and scattering measurements as a function of the radioactive source and attenuation distributions. First, by differentiating this non-linear operator we obtain a linearized inverse problem. Then, under regularity hypothesis for the source distribution and attenuation map and considering small attenuations, we rigorously prove that the linear operator is invertible and we compute its inverse explicitly. This allows proof of local uniqueness for the non-linear inverse problem. Finally, using the previous inversion result for the linear operator, we propose a new type of iterative algorithm for simultaneous source and attenuation recovery for SPECT based on the Neumann series and a Newton-Raphson algorithm.

Systems of conservation laws with third-order Hamiltonian structures

NASA Astrophysics Data System (ADS)

Ferapontov, Evgeny V.; Pavlov, Maxim V.; Vitolo, Raffaele F.

2018-06-01

We investigate n-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in P^{n+2} satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space W of dimension n+2, classify n-tuples of skew-symmetric 2-forms A^{α } \\in Λ^2(W) such that φ _{β γ }A^{β }\\wedge A^{γ }=0, for some non-degenerate symmetric φ.

Intrinsic problems of the gravitational baryogenesis

NASA Astrophysics Data System (ADS)

Arbuzova, E. V.; Dolgov, A. D.

2017-06-01

Modification of gravity due to the curvature dependent term in the gravitational baryogenesis scenario is considered. It is shown that this term leads to the fourth order differential equation of motion for the curvature scalar instead of the algebraic one of General Relativity (GR). The fourth order gravitational equations are generically unstable with respect to small perturbations. Non-linear in curvature terms may stabilize the solution but the magnitude of the stabilized curvature scalar would be much larger than that dictated by GR, so the standard cosmology would be strongly distorted.

Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory

NASA Astrophysics Data System (ADS)

Ansari, R.; Faraji Oskouie, M.; Gholami, R.

2016-01-01

In recent decades, mathematical modeling and engineering applications of fractional-order calculus have been extensively utilized to provide efficient simulation tools in the field of solid mechanics. In this paper, a nonlinear fractional nonlocal Euler-Bernoulli beam model is established using the concept of fractional derivative and nonlocal elasticity theory to investigate the size-dependent geometrically nonlinear free vibration of fractional viscoelastic nanobeams. The non-classical fractional integro-differential Euler-Bernoulli beam model contains the nonlocal parameter, viscoelasticity coefficient and order of the fractional derivative to interpret the size effect, viscoelastic material and fractional behavior in the nanoscale fractional viscoelastic structures, respectively. In the solution procedure, the Galerkin method is employed to reduce the fractional integro-partial differential governing equation to a fractional ordinary differential equation in the time domain. Afterwards, the predictor-corrector method is used to solve the nonlinear fractional time-dependent equation. Finally, the influences of nonlocal parameter, order of fractional derivative and viscoelasticity coefficient on the nonlinear time response of fractional viscoelastic nanobeams are discussed in detail. Moreover, comparisons are made between the time responses of linear and nonlinear models.

Closed-form solution for static pull-in voltage of electrostatically actuated clamped-clamped micro/nano beams under the effect of fringing field and van der Waals force

NASA Astrophysics Data System (ADS)

Bhojawala, V. M.; Vakharia, D. P.

2017-12-01

This investigation provides an accurate prediction of static pull-in voltage for clamped-clamped micro/nano beams based on distributed model. The Euler-Bernoulli beam theory is used adapting geometric non-linearity of beam, internal (residual) stress, van der Waals force, distributed electrostatic force and fringing field effects for deriving governing differential equation. The Galerkin discretisation method is used to make reduced-order model of the governing differential equation. A regime plot is presented in the current work for determining the number of modes required in reduced-order model to obtain completely converged pull-in voltage for micro/nano beams. A closed-form relation is developed based on the relationship obtained from curve fitting of pull-in instability plots and subsequent non-linear regression for the proposed relation. The output of regression analysis provides Chi-square (χ 2) tolerance value equals to 1  ×  10-9, adjusted R-square value equals to 0.999 29 and P-value equals to zero, these statistical parameters indicate the convergence of non-linear fit, accuracy of fitted data and significance of the proposed model respectively. The closed-form equation is validated using available data of experimental and numerical results. The relative maximum error of 4.08% in comparison to several available experimental and numerical data proves the reliability of the proposed closed-form equation.

Generalized nonlinear Schrödinger equation and ultraslow optical solitons in a cold four-state atomic system.

PubMed

Hang, Chao; Huang, Guoxiang; Deng, L

2006-03-01

We investigate the influence of high-order dispersion and nonlinearity on the propagation of ultraslow optical solitons in a lifetime broadened four-state atomic system under a Raman excitation. Using a standard method of multiple-scales we derive a generalized nonlinear Schrödinger equation and show that for realistic physical parameters and at the pulse duration of 10(-6)s, the effects of third-order linear dispersion, nonlinear dispersion, and delay in nonlinear refractive index can be significant and may not be considered as perturbations. We provide exact soliton solutions for the generalized nonlinear Schrödinger equation and demonstrate that optical solitons obtained may still have ultraslow propagating velocity. Numerical simulations on the stability and interaction of these ultraslow optical solitons in the presence of linear and differential absorptions are also presented.

Stability of elastic bending and torsion of uniform cantilever rotor blades in hover with variable structural coupling

NASA Technical Reports Server (NTRS)

Hodges, D. H., Roberta.

1976-01-01

The stability of elastic flap bending, lead-lag bending, and torsion of uniform, untwisted, cantilever rotor blades without chordwise offsets between the elastic, mass, tension, and areodynamic center axes is investigated for the hovering flight condition. The equations of motion are obtained by simplifying the general, nonlinear, partial differential equations of motion of an elastic rotating cantilever blade. The equations are adapted for a linearized stability analysis in the hovering flight condition by prescribing aerodynamic forces, applying Galerkin's method, and linearizing the resulting ordinary differential equations about the equilibrium operating condition. The aerodynamic forces are obtained from strip theory based on a quasi-steady approximation of two-dimensional unsteady airfoil theory. Six coupled mode shapes, calculated from free vibration about the equilibrium operating condition, are used in the linearized stability analysis. The study emphasizes the effects of two types of structural coupling that strongly influence the stability of hingeless rotor blades. The first structural coupling is the linear coupling between flap and lead-lag bending of the rotor blade. The second structural coupling is a nonlinear coupling between flap bending, lead-lag bending, and torsion deflections. Results are obtained for a wide variety of hingeless rotor configurations and operating conditions in order to provide a reasonably complete picture of hingeless rotor blade stability characteristics.

Approximate reduction of linear population models governed by stochastic differential equations: application to multiregional models.

PubMed

Sanz, Luis; Alonso, Juan Antonio

2017-12-01

In this work we develop approximate aggregation techniques in the context of slow-fast linear population models governed by stochastic differential equations and apply the results to the treatment of populations with spatial heterogeneity. Approximate aggregation techniques allow one to transform a complex system involving many coupled variables and in which there are processes with different time scales, by a simpler reduced model with a fewer number of 'global' variables, in such a way that the dynamics of the former can be approximated by that of the latter. In our model we contemplate a linear fast deterministic process together with a linear slow process in which the parameters are affected by additive noise, and give conditions for the solutions corresponding to positive initial conditions to remain positive for all times. By letting the fast process reach equilibrium we build a reduced system with a lesser number of variables, and provide results relating the asymptotic behaviour of the first- and second-order moments of the population vector for the original and the reduced system. The general technique is illustrated by analysing a multiregional stochastic system in which dispersal is deterministic and the rate growth of the populations in each patch is affected by additive noise.

Fifth-order superintegrable quantum systems separating in Cartesian coordinates: Doubly exotic potentials

NASA Astrophysics Data System (ADS)

Abouamal, Ismail; Winternitz, Pavel

2018-02-01

We consider a two-dimensional quantum Hamiltonian separable in Cartesian coordinates and allowing a fifth-order integral of motion. We impose the superintegrablity condition and find all doubly exotic superintegrable potentials (i.e., potentials V(x, y) = V1(x) + V2(y), where neither V1(x) nor V2(y) satisfy a linear ordinary differential equation), allowing the existence of such an integral. All of these potentials are found to have the Painlevé property. Most of them are expressed in terms of known Painlevé transcendents or elliptic functions but some may represent new higher order Painlevé transcendents.

Non-linear optical measurements using a scanned, Bessel beam

NASA Astrophysics Data System (ADS)

Collier, Bradley B.; Awasthi, Samir; Lieu, Deborah K.; Chan, James W.

2015-03-01

Oftentimes cells are removed from the body for disease diagnosis or cellular research. This typically requires fluorescent labeling followed by sorting with a flow cytometer; however, possible disruption of cellular function or even cell death due to the presence of the label can occur. This may be acceptable for ex vivo applications, but as cells are more frequently moving from the lab to the body, label-free methods of cell sorting are needed to eliminate these issues. This is especially true of the growing field of stem cell research where specialized cells are needed for treatments. Because differentiation processes are not completely efficient, cells must be sorted to eliminate any unwanted cells (i.e. un-differentiated or differentiated into an unwanted cell type). In order to perform label-free measurements, non-linear optics (NLO) have been increasingly utilized for single cell analysis because of their ability to not disrupt cellular function. An optical system was developed for the measurement of NLO in a microfluidic channel similar to a flow cytometer. In order to improve the excitation efficiency of NLO, a scanned Bessel beam was utilized to create a light-sheet across the channel. The system was tested by monitoring twophoton fluorescence from polystyrene microbeads of different sizes. Fluorescence intensity obtained from light-sheet measurements were significantly greater than measurements made using a static Gaussian beam. In addition, the increase in intensity from larger sized beads was more evident for the light-sheet system.

The shear-Hall instability in newborn neutron stars

NASA Astrophysics Data System (ADS)

Kondić, T.; Rüdiger, G.; Hollerbach, R.

2011-11-01

Aims: In the first few minutes of a newborn neutron star's life the Hall effect and differential rotation may both be important. We demonstrate that these two ingredients are sufficient for generating a "shear-Hall instability" and for studying its excitation conditions, growth rates, and characteristic magnetic field patterns. Methods: We numerically solve the induction equation in a spherical shell, with a kinematically prescribed differential rotation profile Ω(s), where s is the cylindrical radius. The Hall term is linearized about an imposed uniform axial field. The linear stability of individual azimuthal modes, both axisymmetric and non-axisymmetric, is then investigated. Results: For the shear-Hall instability to occur, the axial field must be parallel to the rotation axis if Ω(s) decreases outward, whereas if Ω(s) increases outward it must be anti-parallel. The instability draws its energy from the differential rotation, and occurs on the short rotational timescale rather than on the much longer Hall timescale. It operates most efficiently if the Hall time is comparable to the diffusion time. Depending on the precise field strengths B0, either axisymmetric or non-axisymmetric modes may be the most unstable. Conclusions: Even if the differential rotation in newborn neutron stars is quenched within minutes, the shear-Hall instability may nevertheless amplify any seed magnetic fields by many orders of magnitude.

Cosmological perturbations in mimetic Horndeski gravity

NASA Astrophysics Data System (ADS)

Arroja, Frederico; Bartolo, Nicola; Karmakar, Purnendu; Matarrese, Sabino

2016-04-01

We study linear scalar perturbations around a flat FLRW background in mimetic Horndeski gravity. In the absence of matter, we show that the Newtonian potential satisfies a second-order differential equation with no spatial derivatives. This implies that the sound speed for scalar perturbations is exactly zero on this background. We also show that in mimetic G3 theories the sound speed is equally zero. We obtain the equation of motion for the comoving curvature perturbation (first order differential equation) and solve it to find that the comoving curvature perturbation is constant on all scales in mimetic Horndeski gravity. We find solutions for the Newtonian potential evolution equation in two simple models. Finally we show that the sound speed is zero on all backgrounds and therefore the system does not have any wave-like scalar degrees of freedom.

Application of the comparison principle to analysis of nonlinear systems. [using Lipschitz condition and differential equations

NASA Technical Reports Server (NTRS)

Gunderson, R. W.

1975-01-01

A comparison principle based on a Kamke theorem and Lipschitz conditions is presented along with its possible applications and modifications. It is shown that the comparison lemma can be used in the study of such areas as classical stability theory, higher order trajectory derivatives, Liapunov functions, boundary value problems, approximate dynamic systems, linear and nonlinear systems, and bifurcation analysis.

Advanced Control Systems for Aircraft Powerplants

DTIC Science & Technology

1980-02-01

production of high- integrity software. 1.0 INTRODUCTION Work on full-authority digital control for gas turbines was started at Rolls- Royce Limited... INTRODUCTION In order to fully understand the operation of the Secondary Power System Control Unit - abbreviated SPSCU - we must first take a close look at...Only Memory EPROM -- Erasable Read Only Memory PLA -- Power Lever Angle LVDT -- Linear Variable Differential Transformer INTRODUCTION Preliminary design

Exponential integration algorithms applied to viscoplasticity

NASA Technical Reports Server (NTRS)

Freed, Alan D.; Walker, Kevin P.

1991-01-01

Four, linear, exponential, integration algorithms (two implicit, one explicit, and one predictor/corrector) are applied to a viscoplastic model to assess their capabilities. Viscoplasticity comprises a system of coupled, nonlinear, stiff, first order, ordinary differential equations which are a challenge to integrate by any means. Two of the algorithms (the predictor/corrector and one of the implicits) give outstanding results, even for very large time steps.

Generalized Functions for the Fractional Calculus

NASA Technical Reports Server (NTRS)

Lorenzo, Carl F.; Hartley, Tom T.

1999-01-01

Previous papers have used two important functions for the solution of fractional order differential equations, the Mittag-Leffler functionE(sub q)[at(exp q)](1903a, 1903b, 1905), and the F-function F(sub q)[a,t] of Hartley & Lorenzo (1998). These functions provided direct solution and important understanding for the fundamental linear fractional order differential equation and for the related initial value problem (Hartley and Lorenzo, 1999). This paper examines related functions and their Laplace transforms. Presented for consideration are two generalized functions, the R-function and the G-function, useful in analysis and as a basis for computation in the fractional calculus. The R-function is unique in that it contains all of the derivatives and integrals of the F-function. The R-function also returns itself on qth order differ-integration. An example application of the R-function is provided. A further generalization of the R-function, called the G-function brings in the effects of repeated and partially repeated fractional poles.

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

NASA Astrophysics Data System (ADS)

Jiao, Yujian; Wang, Li-Lian; Huang, Can

2016-01-01

The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order μ ∈ (0 , 1) to compute that of any order k + μ with integer k ≥ 0, while in the modified RL case, it is only necessary to evaluate a fractional integral matrix of order μ ∈ (0 , 1). Secondly, we introduce suitable fractional JGL Birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties: (i) the matrix generated from the new basis yields the exact inverse of F-PSDM at "interior" JGL points; (ii) the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal; and (iii) the resulted linear system is well-conditioned in the Caputo case, while in the modified RL case, the eigenvalues of the coefficient matrix are highly concentrated. In both cases, the linear systems of the collocation schemes using the new basis can be solved by an iterative solver within a few iterations. Notably, the inverse can be computed in a very stable manner, so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations (FDEs). It is also noteworthy that the choice of certain special JGL points with parameters related to the order of the equations can ease the implementation. We highlight that the use of the Bateman's fractional integral formulas and fast transforms between Jacobi polynomials with different parameters, is essential for our algorithm development.

Linear Ordinary Differential Equations with Constant Coefficients. Revisiting the Impulsive Response Method Using Factorization

ERIC Educational Resources Information Center

Camporesi, Roberto

2011-01-01

We present an approach to the impulsive response method for solving linear constant-coefficient ordinary differential equations based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of…

The Riemann-Lanczos equations in general relativity and their integrability

NASA Astrophysics Data System (ADS)

Dolan, P.; Gerber, A.

2008-06-01

The aim of this paper is to examine the Riemann-Lanczos equations and how they can be made integrable. They consist of a system of linear first-order partial differential equations that arise in general relativity, whereby the Riemann curvature tensor is generated by an unknown third-order tensor potential field called the Lanczos tensor. Our approach is based on the theory of jet bundles, where all field variables and all their partial derivatives of all relevant orders are treated as independent variables alongside the local manifold coordinates (xa) on the given space-time manifold M. This approach is adopted in (a) Cartan's method of exterior differential systems, (b) Vessiot's dual method using vector field systems, and (c) the Janet-Riquier theory of systems of partial differential equations. All three methods allow for the most general situations under which integrability conditions can be found. They give equivalent results, namely, that involutivity is always achieved at all generic points of the jet manifold M after a finite number of prolongations. Two alternative methods that appear in the general relativity literature to find integrability conditions for the Riemann-Lanczos equations generate new partial differential equations for the Lanczos potential that introduce a source term, which is nonlinear in the components of the Riemann tensor. We show that such sources do not occur when either of method (a), (b), or (c) are used.

Homotopy approach to optimal, linear quadratic, fixed architecture compensation

NASA Technical Reports Server (NTRS)

Mercadal, Mathieu

1991-01-01

Optimal linear quadratic Gaussian compensators with constrained architecture are a sensible way to generate good multivariable feedback systems meeting strict implementation requirements. The optimality conditions obtained from the constrained linear quadratic Gaussian are a set of highly coupled matrix equations that cannot be solved algebraically except when the compensator is centralized and full order. An alternative to the use of general parameter optimization methods for solving the problem is to use homotopy. The benefit of the method is that it uses the solution to a simplified problem as a starting point and the final solution is then obtained by solving a simple differential equation. This paper investigates the convergence properties and the limitation of such an approach and sheds some light on the nature and the number of solutions of the constrained linear quadratic Gaussian problem. It also demonstrates the usefulness of homotopy on an example of an optimal decentralized compensator.

Conservation laws and conserved quantities for (1+1)D linearized Boussinesq equations

NASA Astrophysics Data System (ADS)

Carvalho, Cindy; Harley, Charis

2017-05-01

Conservation laws and physical conserved quantities for the (1+1)D linearized Boussinesq equations at a constant water depth are presented. These equations describe incompressible, inviscid, irrotational fluid flow in the form of a non steady solitary wave. A systematic multiplier approach is used to obtain the conservation laws of the system of third order partial differential equations (PDEs) in dimensional form. Physical conserved quantities are derived by integrating the conservation laws in the direction of wave propagation and imposing decaying boundary conditions in the horizontal direction. One of these is a newly discovered conserved quantity which relates to an energy flux density.

Methods and means of 3D diffuse Mueller-matrix tomography of depolarizing optically anisotropic biological layers

NASA Astrophysics Data System (ADS)

Dubolazov, O. V.; Ushenko, V. O.; Trifoniuk, L.; Ushenko, Yu. O.; Zhytaryuk, V. G.; Prydiy, O. G.; Grytsyuk, M.; Kushnerik, L.; Meglinskiy, I.

2017-09-01

A new technique of Mueller-matrix mapping of polycrystalline structure of histological sections of biological tissues is suggested. The algorithms of reconstruction of distribution of parameters of linear and circular birefringence of prostate histological sections are found. The interconnections between such distributions and parameters of linear and circular birefringence of prostate tissue histological sections are defined. The comparative investigations of coordinate distributions of phase anisotropy parameters formed by fibrillar networks of prostate tissues of different pathological states (adenoma and carcinoma) are performed. The values and ranges of change of the statistical (moments of the 1st - 4th order) parameters of coordinate distributions of the value of linear and circular birefringence are defined. The objective criteria of cause of Benign and malignant conditions differentiation are determined.

Laplace transform homotopy perturbation method for the approximation of variational problems.

PubMed

Filobello-Nino, U; Vazquez-Leal, H; Rashidi, M M; Sedighi, H M; Perez-Sesma, A; Sandoval-Hernandez, M; Sarmiento-Reyes, A; Contreras-Hernandez, A D; Pereyra-Diaz, D; Hoyos-Reyes, C; Jimenez-Fernandez, V M; Huerta-Chua, J; Castro-Gonzalez, F; Laguna-Camacho, J R

2016-01-01

This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in order to find analytical approximate solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the approximate solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems.

The structure of polarization maps of skin histological sections in the Fourier domain for the tasks of benign and malignant formations differentiation

NASA Astrophysics Data System (ADS)

Ushenko, V. A.; Dubolazov, A. V.; Savich, V. O.; Novakovskaya, O. Y.; Olar, O. V.; Marchuk, Y. F.

2015-02-01

The optical model of birefringent networks of biological tissues is presented. The technique of Fourier polarimetry for selection of manifestations of linear and circular birefringence of protein fibrils is suggested. The results of investigations of statistical (statistical moments of the 1st-4th orders), correlation (dispersion and excess of autocorrelation functions) and scalar-self-similar (logarithmic dependencies of power spectra) structure of Fourier spectra of polarization azimuths distribution of laser images of skin samples are presented. The criteria of differentiation of postoperative biopsy of benign (keratoma) and malignant (adenocarcinoma) skin tumors are determined.

A semigroup approach to the strong ergodic theorem of the multistate stable population process.

PubMed

Inaba, H

1988-01-01

"In this paper we first formulate the dynamics of multistate stable population processes as a partial differential equation. Next, we rewrite this equation as an abstract differential equation in a Banach space, and solve it by using the theory of strongly continuous semigroups of bounded linear operators. Subsequently, we investigate the asymptotic behavior of this semigroup to show the strong ergodic theorem which states that there exists a stable distribution independent of the initial distribution. Finally, we introduce the dual problem in order to obtain a logical definition for the reproductive value and we discuss its applications." (SUMMARY IN FRE) excerpt

A Numerical Scheme for Ordinary Differential Equations Having Time Varying and Nonlinear Coefficients Based on the State Transition Matrix

NASA Technical Reports Server (NTRS)

Bartels, Robert E.

2002-01-01

A variable order method of integrating initial value ordinary differential equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. While it is more complex than most other methods, it produces exact solutions at arbitrary time step size when the time variation of the system can be modeled exactly by a polynomial. Solutions to several nonlinear problems exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with an exact solution and with solutions obtained by established methods.

A higher-order conservation element solution element method for solving hyperbolic differential equations on unstructured meshes

NASA Astrophysics Data System (ADS)

Bilyeu, David

This dissertation presents an extension of the Conservation Element Solution Element (CESE) method from second- to higher-order accuracy. The new method retains the favorable characteristics of the original second-order CESE scheme, including (i) the use of the space-time integral equation for conservation laws, (ii) a compact mesh stencil, (iii) the scheme will remain stable up to a CFL number of unity, (iv) a fully explicit, time-marching integration scheme, (v) true multidimensionality without using directional splitting, and (vi) the ability to handle two- and three-dimensional geometries by using unstructured meshes. This algorithm has been thoroughly tested in one, two and three spatial dimensions and has been shown to obtain the desired order of accuracy for solving both linear and non-linear hyperbolic partial differential equations. The scheme has also shown its ability to accurately resolve discontinuities in the solutions. Higher order unstructured methods such as the Discontinuous Galerkin (DG) method and the Spectral Volume (SV) methods have been developed for one-, two- and three-dimensional application. Although these schemes have seen extensive development and use, certain drawbacks of these methods have been well documented. For example, the explicit versions of these two methods have very stringent stability criteria. This stability criteria requires that the time step be reduced as the order of the solver increases, for a given simulation on a given mesh. The research presented in this dissertation builds upon the work of Chang, who developed a fourth-order CESE scheme to solve a scalar one-dimensional hyperbolic partial differential equation. The completed research has resulted in two key deliverables. The first is a detailed derivation of a high-order CESE methods on unstructured meshes for solving the conservation laws in two- and three-dimensional spaces. The second is the code implementation of these numerical methods in a computer code. For code development, a one-dimensional solver for the Euler equations was developed. This work is an extension of Chang's work on the fourth-order CESE method for solving a one-dimensional scalar convection equation. A generic formulation for the nth-order CESE method, where n ≥ 4, was derived. Indeed, numerical implementation of the scheme confirmed that the order of convergence was consistent with the order of the scheme. For the two- and three-dimensional solvers, SOLVCON was used as the basic framework for code implementation. A new solver kernel for the fourth-order CESE method has been developed and integrated into the framework provided by SOLVCON. The main part of SOLVCON, which deals with unstructured meshes and parallel computing, remains intact. The SOLVCON code for data transmission between computer nodes for High Performance Computing (HPC). To validate and verify the newly developed high-order CESE algorithms, several one-, two- and three-dimensional simulations where conducted. For the arbitrary order, one-dimensional, CESE solver, three sets of governing equations were selected for simulation: (i) the linear convection equation, (ii) the linear acoustic equations, (iii) the nonlinear Euler equations. All three systems of equations were used to verify the order of convergence through mesh refinement. In addition the Euler equations were used to solve the Shu-Osher and Blastwave problems. These two simulations demonstrated that the new high-order CESE methods can accurately resolve discontinuities in the flow field.For the two-dimensional, fourth-order CESE solver, the Euler equation was employed in four different test cases. The first case was used to verify the order of convergence through mesh refinement. The next three cases demonstrated the ability of the new solver to accurately resolve discontinuities in the flows. This was demonstrated through: (i) the interaction between acoustic waves and an entropy pulse, (ii) supersonic flow over a circular blunt body, (iii) supersonic flow over a guttered wedge. To validate and verify the three-dimensional, fourth-order CESE solver, two different simulations where selected. The first used the linear convection equations to demonstrate fourth-order convergence. The second used the Euler equations to simulate supersonic flow over a spherical body to demonstrate the scheme's ability to accurately resolve shocks. All test cases used are well known benchmark problems and as such, there are multiple sources available to validate the numerical results. Furthermore, the simulations showed that the high-order CESE solver was stable at a CFL number near unity.

Phase comparator apparatus and method

DOEpatents

Coffield, F.E.

1985-02-01

This invention finds especially useful application for interferometer measurements made in plasma fusion devices (e.g., for measuring the line integral of electron density in the plasma). Such interferometers typically use very high intermediate frequencies (e.g., on the order of 10 to 70 MHz) and therefore the phase comparison circuitry should be a high speed circuit with a linear transfer characteristic so as to accurately differentiate between small fractions of interference fringes.

f(R)-gravity from Killing tensors

NASA Astrophysics Data System (ADS)

Paliathanasis, Andronikos

2016-04-01

We consider f(R)-gravity in a Friedmann-Lemaître-Robertson-Walker spacetime with zero spatial curvature. We apply the Killing tensors of the minisuperspace in order to specify the functional form of f(R) and for the field equations to be invariant under Lie-Bäcklund transformations, which are linear in momentum (contact symmetries). Consequently, the field equations to admit quadratic conservation laws given by Noether’s theorem. We find three new integrable f(R)-models, for which, with the application of the conservation laws, we reduce the field equations to a system of two first-order ordinary differential equations. For each model we study the evolution of the cosmological fluid. We find that for each integrable model the cosmological fluid has an equation of state parameter, in which there is linear behavior in terms of the scale factor which describes the Chevallier, Polarski and Linder parametric dark energy model.

Mueller-matrix of laser-induced autofluorescence of polycrystalline films of dried peritoneal fluid in diagnostics of endometriosis

NASA Astrophysics Data System (ADS)

Ushenko, Yuriy A.; Koval, Galina D.; Ushenko, Alexander G.; Dubolazov, Olexander V.; Ushenko, Vladimir A.; Novakovskaia, Olga Yu.

2016-07-01

This research presents investigation results of the diagnostic efficiency of an azimuthally stable Mueller-matrix method of analysis of laser autofluorescence of polycrystalline films of dried uterine cavity peritoneal fluid. A model of the generalized optical anisotropy of films of dried peritoneal fluid is proposed in order to define the processes of laser autofluorescence. The influence of complex mechanisms of both phase (linear and circular birefringence) and amplitude (linear and circular dichroism) anisotropies is taken into consideration. The interconnections between the azimuthally stable Mueller-matrix elements characterizing laser autofluorescence and different mechanisms of optical anisotropy are determined. The statistical analysis of coordinate distributions of such Mueller-matrix rotation invariants is proposed. Thereupon the quantitative criteria (statistic moments of the first to the fourth order) of differentiation of polycrystalline films of dried peritoneal fluid, group 1 (healthy donors) and group 2 (uterus endometriosis patients), are determined.

Methods and means of Fourier-Stokes polarimetry and the spatial-frequency filtering of phase anisotropy manifestations in endometriosis diagnostics

NASA Astrophysics Data System (ADS)

Ushenko, A. G.; Dubolazov, O. V.; Ushenko, Vladimir A.; Ushenko, Yu. A.; Sakhnovskiy, M. Yu.; Prydiy, O. G.; Lakusta, I. I.; Novakovskaya, O. Yu.; Melenko, S. R.

2016-12-01

This research presents investigation results of diagnostic efficiency of a new azimuthally stable Mueller-matrix method of laser autofluorescence coordinate distributions analysis of dried polycrystalline films of uterine cavity peritoneal fluid. A new model of generalized optical anisotropy of biological tissues protein networks is proposed in order to define the processes of laser autofluorescence. The influence of complex mechanisms of both phase anisotropy (linear birefringence and optical activity) and linear (circular) dichroism is taken into account. The interconnections between the azimuthally stable Mueller-matrix elements characterizing laser autofluorescence and different mechanisms of optical anisotropy are determined. The statistic analysis of coordinate distributions of such Mueller-matrix rotation invariants is proposed. Thereupon the quantitative criteria (statistic moments of the 1st to the 4th order) of differentiation of dried polycrystalline films of peritoneal fluid - group 1 (healthy donors) and group 2 (uterus endometriosis patients) are estimated.

Symmetries of the Space of Linear Symplectic Connections

NASA Astrophysics Data System (ADS)

Fox, Daniel J. F.

2017-01-01

There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen-Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle polynomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their! linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term.

A first-order Green's function approach to supersonic oscillatory flow: A mixed analytic and numeric treatment

NASA Technical Reports Server (NTRS)

Freedman, M. I.; Sipcic, S.; Tseng, K.

1985-01-01

A frequency domain Green's Function Method for unsteady supersonic potential flow around complex aircraft configurations is presented. The focus is on the supersonic range wherein the linear potential flow assumption is valid. In this range the effects of the nonlinear terms in the unsteady supersonic compressible velocity potential equation are negligible and therefore these terms will be omitted. The Green's function method is employed in order to convert the potential flow differential equation into an integral one. This integral equation is then discretized, through standard finite element technique, to yield a linear algebraic system of equations relating the unknown potential to its prescribed co-normalwash (boundary condition) on the surface of the aircraft. The arbitrary complex aircraft configuration (e.g., finite-thickness wing, wing-body-tail) is discretized into hyperboloidal (twisted quadrilateral) panels. The potential and co-normalwash are assumed to vary linearly within each panel. The long range goal is to develop a comprehensive theory for unsteady supersonic potential aerodynamic which is capable of yielding accurate results even in the low supersonic (i.e., high transonic) range.

Effective quadrature formula in solving linear integro-differential equations of order two

NASA Astrophysics Data System (ADS)

Eshkuvatov, Z. K.; Kammuji, M.; Long, N. M. A. Nik; Yunus, Arif A. M.

2017-08-01

In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed.

Differential Galois theory and non-integrability of planar polynomial vector fields

NASA Astrophysics Data System (ADS)

Acosta-Humánez, Primitivo B.; Lázaro, J. Tomás; Morales-Ruiz, Juan J.; Pantazi, Chara

2018-06-01

We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the "Risch algorithm". In this way we point out the connection of the non integrability with some higher transcendent functions, like the error function.

Robust fast controller design via nonlinear fractional differential equations.

PubMed

Zhou, Xi; Wei, Yiheng; Liang, Shu; Wang, Yong

2017-07-01

A new method for linear system controller design is proposed whereby the closed-loop system achieves both robustness and fast response. The robustness performance considered here means the damping ratio of closed-loop system can keep its desired value under system parameter perturbation, while the fast response, represented by rise time of system output, can be improved by tuning the controller parameter. We exploit techniques from both the nonlinear systems control and the fractional order systems control to derive a novel nonlinear fractional order controller. For theoretical analysis of the closed-loop system performance, two comparison theorems are developed for a class of fractional differential equations. Moreover, the rise time of the closed-loop system can be estimated, which facilitates our controller design to satisfy the fast response performance and maintain the robustness. Finally, numerical examples are given to illustrate the effectiveness of our methods. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Stabilization and control of distributed systems with time-dependent spatial domains

NASA Technical Reports Server (NTRS)

Wang, P. K. C.

1990-01-01

This paper considers the problem of the stabilization and control of distributed systems with time-dependent spatial domains. The evolution of the spatial domains with time is described by a finite-dimensional system of ordinary differential equations, while the distributed systems are described by first-order or second-order linear evolution equations defined on appropriate Hilbert spaces. First, results pertaining to the existence and uniqueness of solutions of the system equations are presented. Then, various optimal control and stabilization problems are considered. The paper concludes with some examples which illustrate the application of the main results.

Improved Filon-type asymptotic methods for highly oscillatory differential equations with multiple time scales

NASA Astrophysics Data System (ADS)

Wang, Bin; Wu, Xinyuan

2014-11-01

In this paper we consider multi-frequency highly oscillatory second-order differential equations x″ (t) + Mx (t) = f (t , x (t) ,x‧ (t)) where high-frequency oscillations are generated by the linear part Mx (t), and M is positive semi-definite (not necessarily nonsingular). It is known that Filon-type methods are effective approach to numerically solving highly oscillatory problems. Unfortunately, however, existing Filon-type asymptotic methods fail to apply to the highly oscillatory second-order differential equations when M is singular. We study and propose an efficient improvement on the existing Filon-type asymptotic methods, so that the improved Filon-type asymptotic methods can be able to numerically solving this class of multi-frequency highly oscillatory systems with a singular matrix M. The improved Filon-type asymptotic methods are designed by combining Filon-type methods with the asymptotic methods based on the variation-of-constants formula. We also present one efficient and practical improved Filon-type asymptotic method which can be performed at lower cost. Accompanying numerical results show the remarkable efficiency.

[Spectroscopic analysis of the interaction of ethanol and acid phosphatase from wheat germ].

PubMed

Xu, Dong-mei; Liu, Guang-shen; Wang, Li-ming; Liu, Wei-ping

2004-11-01

Conformational and activity changes of acid phosphatase from wheat germ in ethanol solutions of different concentrations were measured by fluorescence spectra and differential UV-absorption spectra. The effect of ethanol on kinetics of acid phosphatase was determined by using the double reciprocal plot. The results indicate the ethanol has a significant effect on the activity and conformation of acid phosphatase. The activity of acid phosphatase decreased linearly with increasing the concentration of ethanol. Differential UV-absorption spectra of the enzyme denatured in ethanol solutions showed two positive peaks at 213 and 234 nm, respectively. The peaks on the differential UV-absorption spectra suggested that the conformation of enzyme molecule changed from orderly structure to out-of-order crispation. The fluorescence emission peak intensity of the enzyme gradually strengthened with increasing ethanol concentration, which is in concordance with the conformational change of the microenvironments of tyrosine and tryptophan residues. The results indicate that the expression of the enzyme activity correlates with the stability and integrity of the enzyme conformation to a great degree. Ethanol is uncompetitive inhibitor of acid phosphatase.

Global dynamics for switching systems and their extensions by linear differential equations

NASA Astrophysics Data System (ADS)

Huttinga, Zane; Cummins, Bree; Gedeon, Tomáš; Mischaikow, Konstantin

2018-03-01

Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.

Convergence and stability of the exponential Euler method for semi-linear stochastic delay differential equations.

PubMed

Zhang, Ling

2017-01-01

The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs). It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order [Formula: see text] to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.

Global dynamics for switching systems and their extensions by linear differential equations.

PubMed

Huttinga, Zane; Cummins, Bree; Gedeon, Tomáš; Mischaikow, Konstantin

2018-03-15

Switching systems use piecewise constant nonlinearities to model gene regulatory networks. This choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of Morse graphs associated to nodes of a parameter graph. The parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical Morse graphs. However, there are many cellular processes that do not exhibit threshold-like behavior and thus are not well described by a switching system. We consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations. We show that the parameter graphs associated to the switching system and any of its extensions are identical. For each parameter graph node, there is an order-preserving map from the Morse graph of the switching system to the Morse graph of any of its extensions. We provide counterexamples that show why possible stronger relationships between the Morse graphs are not valid.

A structure-preserving split finite element discretization of the split 1D linear shallow-water equations

NASA Astrophysics Data System (ADS)

Bauer, Werner; Behrens, Jörn

2017-04-01

We present a locally conservative, low-order finite element (FE) discretization of the covariant 1D linear shallow-water equations written in split form (cf. tet{[1]}). The introduction of additional differential forms (DF) that build pairs with the original ones permits a splitting of these equations into topological momentum and continuity equations and metric-dependent closure equations that apply the Hodge-star. Our novel discretization framework conserves this geometrical structure, in particular it provides for all DFs proper FE spaces such that the differential operators (here gradient and divergence) hold in strong form. The discrete topological equations simply follow by trivial projections onto piecewise constant FE spaces without need to partially integrate. The discrete Hodge-stars operators, representing the discretized metric equations, are realized by nontrivial Galerkin projections (GP). Here they follow by projections onto either a piecewise constant (GP0) or a piecewise linear (GP1) space. Our framework thus provides essentially three different schemes with significantly different behavior. The split scheme using twice GP1 is unstable and shares the same discrete dispersion relation and similar second-order convergence rates as the conventional P1-P1 FE scheme that approximates both velocity and height variables by piecewise linear spaces. The split scheme that applies both GP1 and GP0 is stable and shares the dispersion relation of the conventional P1-P0 FE scheme that approximates the velocity by a piecewise linear and the height by a piecewise constant space with corresponding second- and first-order convergence rates. Exhibiting for both velocity and height fields second-order convergence rates, we might consider the split GP1-GP0 scheme though as stable versions of the conventional P1-P1 FE scheme. For the split scheme applying twice GP0, we are not aware of a corresponding conventional formulation to compare with. Though exhibiting larger absolute error values, it shows similar convergence rates as the other split schemes, but does not provide a satisfactory approximation of the dispersion relation as short waves are propagated much to fast. Despite this, the finding of this new scheme illustrates the potential of our discretization framework as a toolbox to find and to study new FE schemes based on new combinations of FE spaces. [1] Bauer, W. [2016], A new hierarchically-structured n-dimensional covariant form of rotating equations of geophysical fluid dynamics, GEM - International Journal on Geomathematics, 7(1), 31-101.

Three dimensional rotating flow of Powell-Eyring nanofluid with non-Fourier's heat flux and non-Fick's mass flux theory

NASA Astrophysics Data System (ADS)

Ibrahim, Wubshet

2018-03-01

This article numerically examines three dimensional boundary layer flow of a rotating Powell-Eyring nanofluid. In modeling heat transfer processes, non-Fourier heat flux theory and for mass transfer non-Fick's mass flux theory are employed. This theory is recently re-initiated and it becomes the active research area to resolves some drawback associated with the famous Fourier heat flux and mass flux theory. The mathematical model of the flow problem is a system of non-linear partial differential equations which are obtained using the boundary layer analysis. The non-linear partial differential equations have been transformed into non-linear high order ordinary differential equations using similarity transformation. Employing bvp4c algorithm from matlab software routine, the numerical solution of the transformed ordinary differential equations is obtained. The governing equations are constrained by parameters such as rotation parameter λ , the non-Newtonian parameter N, dimensionless thermal relaxation and concentration relaxation parameters δt and δc . The impacts of these parameters have been discussed thoroughly and illustrated using graphs and tables. The findings show that thermal relaxation time δt reduces the thermal and concentration boundary layer thickness. Further, the results reveal that the rotational parameter λ has the effect of decreasing the velocity boundary layer thickness in both x and y directions. Further examination pinpoints that the skin friction coefficient along x-axis is an increasing and skin friction coefficient along y-axis is a decreasing function of rotation parameter λ . Furthermore, the non-Newtonian fluid parameter N has the characteristic of reducing the amount of local Nusselt numbers -f″ (0) and -g″ (0) both in x and y -directions.

Conformal and covariant Z4 formulation of the Einstein equations: Strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes

NASA Astrophysics Data System (ADS)

Dumbser, Michael; Guercilena, Federico; Köppel, Sven; Rezzolla, Luciano; Zanotti, Olindo

2018-04-01

We present a strongly hyperbolic first-order formulation of the Einstein equations based on the conformal and covariant Z4 system (CCZ4) with constraint-violation damping, which we refer to as FO-CCZ4. As CCZ4, this formulation combines the advantages of a conformal and traceless formulation, with the suppression of constraint violations given by the damping terms, but being first order in time and space, it is particularly suited for a discontinuous Galerkin (DG) implementation. The strongly hyperbolic first-order formulation has been obtained by making careful use of first and second-order ordering constraints. A proof of strong hyperbolicity is given for a selected choice of standard gauges via an analytical computation of the entire eigenstructure of the FO-CCZ4 system. The resulting governing partial differential equations system is written in nonconservative form and requires the evolution of 58 unknowns. A key feature of our formulation is that the first-order CCZ4 system decouples into a set of pure ordinary differential equations and a reduced hyperbolic system of partial differential equations that contains only linearly degenerate fields. We implement FO-CCZ4 in a high-order path-conservative arbitrary-high-order-method-using-derivatives (ADER)-DG scheme with adaptive mesh refinement and local time-stepping, supplemented with a third-order ADER-WENO subcell finite-volume limiter in order to deal with singularities arising with black holes. We validate the correctness of the formulation through a series of standard tests in vacuum, performed in one, two and three spatial dimensions, and also present preliminary results on the evolution of binary black-hole systems. To the best of our knowledge, these are the first successful three-dimensional simulations of moving punctures carried out with high-order DG schemes using a first-order formulation of the Einstein equations.

Fourth order difference methods for hyperbolic IBVP's

NASA Technical Reports Server (NTRS)

Gustafsson, Bertil; Olsson, Pelle

1994-01-01

Fourth order difference approximations of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second order viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth order methods with a standard second order one and with a third order TVD-method. The results show that the fourth order methods are the only ones that give good results for all the considered test problems.

A data processing method based on tracking light spot for the laser differential confocal component parameters measurement system

NASA Astrophysics Data System (ADS)

Shao, Rongjun; Qiu, Lirong; Yang, Jiamiao; Zhao, Weiqian; Zhang, Xin

2013-12-01

We have proposed the component parameters measuring method based on the differential confocal focusing theory. In order to improve the positioning precision of the laser differential confocal component parameters measurement system (LDDCPMS), the paper provides a data processing method based on tracking light spot. To reduce the error caused by the light point moving in collecting the axial intensity signal, the image centroiding algorithm is used to find and track the center of Airy disk of the images collected by the laser differential confocal system. For weakening the influence of higher harmonic noises during the measurement, Gaussian filter is used to process the axial intensity signal. Ultimately the zero point corresponding to the focus of the objective in a differential confocal system is achieved by linear fitting for the differential confocal axial intensity data. Preliminary experiments indicate that the method based on tracking light spot can accurately collect the axial intensity response signal of the virtual pinhole, and improve the anti-interference ability of system. Thus it improves the system positioning accuracy.

Symbolic Solution of Linear Differential Equations

NASA Technical Reports Server (NTRS)

Feinberg, R. B.; Grooms, R. G.

1981-01-01

An algorithm for solving linear constant-coefficient ordinary differential equations is presented. The computational complexity of the algorithm is discussed and its implementation in the FORMAC system is described. A comparison is made between the algorithm and some classical algorithms for solving differential equations.

Modeling non‐linear kinetics of hyperpolarized [1‐13C] pyruvate in the crystalloid‐perfused rat heart

PubMed Central

Mariotti, E.; Orton, M. R.; Eerbeek, O.; Ashruf, J. F.; Zuurbier, C. J.; Southworth, R.

2016-01-01

Hyperpolarized 13C MR measurements have the potential to display non‐linear kinetics. We have developed an approach to describe possible non‐first‐order kinetics of hyperpolarized [1‐13C] pyruvate employing a system of differential equations that agrees with the principle of conservation of mass of the hyperpolarized signal. Simultaneous fitting to a second‐order model for conversion of [1‐13C] pyruvate to bicarbonate, lactate and alanine was well described in the isolated rat heart perfused with Krebs buffer containing glucose as sole energy substrate, or glucose supplemented with pyruvate. Second‐order modeling yielded significantly improved fits of pyruvate–bicarbonate kinetics compared with the more traditionally used first‐order model and suggested time‐dependent decreases in pyruvate–bicarbonate flux. Second‐order modeling gave time‐dependent changes in forward and reverse reaction kinetics of pyruvate–lactate exchange and pyruvate–alanine exchange in both groups of hearts during the infusion of pyruvate; however, the fits were not significantly improved with respect to a traditional first‐order model. The mechanism giving rise to second‐order pyruvate dehydrogenase (PDH) kinetics was explored experimentally using surface fluorescence measurements of nicotinamide adenine dinucleotide reduced form (NADH) performed under the same conditions, demonstrating a significant increase of NADH during pyruvate infusion. This suggests a simultaneous depletion of available mitochondrial NAD+ (the cofactor for PDH), consistent with the non‐linear nature of the kinetics. NADH levels returned to baseline following cessation of the pyruvate infusion, suggesting this to be a transient effect. © 2016 The Authors. NMR in Biomedicine published by John Wiley & Sons Ltd. PMID:26777799

Electrokinetics Models for Micro and Nano Fluidic Impedance Sensors

DTIC Science & Technology

2010-11-01

primitive Differential-Algebraic Equations (DAEs), used to process and interpret the experimentally measured electrical impedance data (Sun and Morgan...field, and species respectively. A second-order scheme was used to calculate the ionic species distribution. The linearized algebraic equations were...is governed by the Poisson equation 2 0 0 r i i i F z cε ε φ∇ + =∑ where ε0 and εr are, respectively, the electrical permittivity in the vacuum

Introduction to Communication Systems

DTIC Science & Technology

2013-08-18

nonlinear differential equations involved, and to compare the results with the linearized analysis. Nonlinear model for the first order PLL: Let us try to...approaches to scaling up data rates: increasing spatial reuse (i.e., using the same time -bandwidth resources at locations that are far enough apart), and... Even when this music is recorded onto a digital storage medium such as a CD ( using the digital communication framework outlined in Section 1.1.2), when

A Parallel Implicit Reconstructed Discontinuous Galerkin Method for Compressible Flows on Hybrid Grids

NASA Astrophysics Data System (ADS)

Xia, Yidong

The objective this work is to develop a parallel, implicit reconstructed discontinuous Galerkin (RDG) method using Taylor basis for the solution of the compressible Navier-Stokes equations on 3D hybrid grids. This third-order accurate RDG method is based on a hierarchical weighed essentially non- oscillatory reconstruction scheme, termed as HWENO(P1P 2) to indicate that a quadratic polynomial solution is obtained from the underlying linear polynomial DG solution via a hierarchical WENO reconstruction. The HWENO(P1P2) is designed not only to enhance the accuracy of the underlying DG(P1) method but also to ensure non-linear stability of the RDG method. In this reconstruction scheme, a quadratic polynomial (P2) solution is first reconstructed using a least-squares approach from the underlying linear (P1) discontinuous Galerkin solution. The final quadratic solution is then obtained using a Hermite WENO reconstruction, which is necessary to ensure the linear stability of the RDG method on 3D unstructured grids. The first derivatives of the quadratic polynomial solution are then reconstructed using a WENO reconstruction in order to eliminate spurious oscillations in the vicinity of strong discontinuities, thus ensuring the non-linear stability of the RDG method. The parallelization in the RDG method is based on a message passing interface (MPI) programming paradigm, where the METIS library is used for the partitioning of a mesh into subdomain meshes of approximately the same size. Both multi-stage explicit Runge-Kutta and simple implicit backward Euler methods are implemented for time advancement in the RDG method. In the implicit method, three approaches: analytical differentiation, divided differencing (DD), and automatic differentiation (AD) are developed and implemented to obtain the resulting flux Jacobian matrices. The automatic differentiation is a set of techniques based on the mechanical application of the chain rule to obtain derivatives of a function given as a computer program. By using an AD tool, the manpower can be significantly reduced for deriving the flux Jacobians, which can be quite complicated, tedious, and error-prone if done by hand or symbolic arithmetic software, depending on the complexity of the numerical flux scheme. In addition, the workload for code maintenance can also be largely reduced in case the underlying flux scheme is updated. The approximate system of linear equations arising from the Newton linearization is solved by the general minimum residual (GMRES) algorithm with lower-upper symmetric gauss-seidel (LUSGS) preconditioning. This GMRES+LU-SGS linear solver is the most robust and efficient for implicit time integration of the discretized Navier-Stokes equations when the AD-based flux Jacobians are provided other than the other two approaches. The developed HWENO(P1P2) method is used to compute a variety of well-documented compressible inviscid and viscous flow test cases on 3D hybrid grids, including some standard benchmark test cases such as the Sod shock tube, flow past a circular cylinder, and laminar flow past a at plate. The computed solutions are compared with either analytical solutions or experimental data, if available to assess the accuracy of the HWENO(P 1P2) method. Numerical results demonstrate that the HWENO(P 1P2) method is able to not only enhance the accuracy of the underlying HWENO(P1) method, but also ensure the linear and non-linear stability at the presence of strong discontinuities. An extensive study of grid convergence analysis on various types of elements: tetrahedron, prism, hexahedron, and hybrid prism/hexahedron, for a number of test cases indicates that the developed HWENO(P1P2) method is able to achieve the designed third-order accuracy of spatial convergence for smooth inviscid flows: one order higher than the underlying second-order DG(P1) method without significant increase in computing costs and storage requirements. The performance of the the developed GMRES+LU-SGS implicit method is compared with the multi-stage Runge-Kutta time stepping scheme for a number of test cases in terms of the timestep and CPU time. Numerical results indicate that the overall performance of the implicit method with AD-based Jacobians is order of magnitude better than the its explicit counterpart. Finally, a set of parallel scaling tests for both explicit and implicit methods is conducted on North Carolina State University's ARC cluster, demonstrating almost an ideal scalability of the RDG method. (Abstract shortened by UMI.)

An Automatic Orthonormalization Method for Solving Stiff Boundary-Value Problems

NASA Astrophysics Data System (ADS)

Davey, A.

1983-08-01

A new initial-value method is described, based on a remark by Drury, for solving stiff linear differential two-point cigenvalue and boundary-value problems. The method is extremely reliable, it is especially suitable for high-order differential systems, and it is capable of accommodating realms of stiffness which other methods cannot reach. The key idea behind the method is to decompose the stiff differential operator into two non-stiff operators, one of which is nonlinear. The nonlinear one is specially chosen so that it advances an orthonormal frame, indeed the method is essentially a kind of automatic orthonormalization; the second is auxiliary but it is needed to determine the required function. The usefulness of the method is demonstrated by calculating some eigenfunctions for an Orr-Sommerfeld problem when the Reynolds number is as large as 10°.

Optimal Design of Spring Characteristics of Damper for Subharmonic Vibration in Automatic Transmission Powertrain

NASA Astrophysics Data System (ADS)

Nakae, T.; Ryu, T.; Matsuzaki, K.; Rosbi, S.; Sueoka, A.; Takikawa, Y.; Ooi, Y.

2016-09-01

In the torque converter, the damper of the lock-up clutch is used to effectively absorb the torsional vibration. The damper is designed using a piecewise-linear spring with three stiffness stages. However, a nonlinear vibration, referred to as a subharmonic vibration of order 1/2, occurred around the switching point in the piecewise-linear restoring torque characteristics because of the nonlinearity. In the present study, we analyze vibration reduction for subharmonic vibration. The model used herein includes the torque converter, the gear train, and the differential gear. The damper is modeled by a nonlinear rotational spring of the piecewise-linear spring. We focus on the optimum design of the spring characteristics of the damper in order to suppress the subharmonic vibration. A piecewise-linear spring with five stiffness stages is proposed, and the effect of the distance between switching points on the subharmonic vibration is investigated. The results of our analysis indicate that the subharmonic vibration can be suppressed by designing a damper with five stiffness stages to have a small spring constant ratio between the neighboring springs. The distances between switching points must be designed to be large enough that the amplitude of the main frequency component of the systems does not reach the neighboring switching point.

The Quantum Arnold Transformation for the damped harmonic oscillator: from the Caldirola-Kanai model toward the Bateman model

NASA Astrophysics Data System (ADS)

López-Ruiz, F. F.; Guerrero, J.; Aldaya, V.; Cossío, F.

2012-08-01

Using a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations (LSODE), including systems with friction linear in velocity such as the damped harmonic oscillator, can be related to the quantum free-particle dynamical system. This implies that symmetries and simple computations in the free particle can be exported to the LSODE-system. The quantum Arnold transformation is given explicitly for the damped harmonic oscillator, and an algebraic connection between the Caldirola-Kanai model for the damped harmonic oscillator and the Bateman system will be sketched out.

Stable static structures in models with higher-order derivatives

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bazeia, D., E-mail: bazeia@fisica.ufpb.br; Departamento de Física, Universidade Federal de Campina Grande, 58109-970 Campina Grande, PB; Lobão, A.S.

2015-09-15

We investigate the presence of static solutions in generalized models described by a real scalar field in four-dimensional space–time. We study models in which the scalar field engenders higher-order derivatives and spontaneous symmetry breaking, inducing the presence of domain walls. Despite the presence of higher-order derivatives, the models keep to equations of motion second-order differential equations, so we focus on the presence of first-order equations that help us to obtain analytical solutions and investigate linear stability on general grounds. We then illustrate the general results with some specific examples, showing that the domain wall may become compact and that themore » zero mode may split. Moreover, if the model is further generalized to include k-field behavior, it may contribute to split the static structure itself.« less

Stability analysis of a time-periodic 2-dof MEMS structure

NASA Astrophysics Data System (ADS)

Kniffka, Till Jochen; Welte, Johannes; Ecker, Horst

2012-11-01

Microelectromechanical systems (MEMS) are becoming important for all kinds of industrial applications. Among them are filters in communication devices, due to the growing demand for efficient and accurate filtering of signals. In recent developments single degree of freedom (1-dof) oscillators, that are operated at a parametric resonances, are employed for such tasks. Typically vibration damping is low in such MEM systems. While parametric excitation (PE) is used so far to take advantage of a parametric resonance, this contribution suggests to also exploit parametric anti-resonances in order to improve the damping behavior of such systems. Modeling aspects of a 2-dof MEM system and first results of the analysis of the non-linear and the linearized system are the focus of this paper. In principle the investigated system is an oscillating mechanical system with two degrees of freedom x = [x1x2]T that can be described by Mx+Cx+K1x+K3(x2)x+Fes(x,V(t)) = 0. The system is inherently non-linear because of the cubic mechanical stiffness K3 of the structure, but also because of electrostatic forces (1+cos(ωt))Fes(x) that act on the system. Electrostatic forces are generated by comb drives and are proportional to the applied time-periodic voltage V(t). These drives also provide the means to introduce time-periodic coefficients, i.e. parametric excitation (1+cos(ωt)) with frequency ω. For a realistic MEM system the coefficients of the non-linear set of differential equations need to be scaled for efficient numerical treatment. The final mathematical model is a set of four non-linear time-periodic homogeneous differential equations of first order. Numerical results are obtained from two different methods. The linearized time-periodic (LTP) system is studied by calculating the Monodromy matrix of the system. The eigenvalues of this matrix decide on the stability of the LTP-system. To study the unabridged non-linear system, the bifurcation software ManLab is employed. Continuation analysis including stability evaluations are executed and show the frequency ranges for which the 2-dof system becomes unstable due to parametric resonances. Moreover, the existence of frequency intervals are shown where enhanced damping for the system is observed for this MEMS. The results from the stability studies are confirmed by simulation results.

Semi-automatic sparse preconditioners for high-order finite element methods on non-uniform meshes

NASA Astrophysics Data System (ADS)

Austin, Travis M.; Brezina, Marian; Jamroz, Ben; Jhurani, Chetan; Manteuffel, Thomas A.; Ruge, John

2012-05-01

High-order finite elements often have a higher accuracy per degree of freedom than the classical low-order finite elements. However, in the context of implicit time-stepping methods, high-order finite elements present challenges to the construction of efficient simulations due to the high cost of inverting the denser finite element matrix. There are many cases where simulations are limited by the memory required to store the matrix and/or the algorithmic components of the linear solver. We are particularly interested in preconditioned Krylov methods for linear systems generated by discretization of elliptic partial differential equations with high-order finite elements. Using a preconditioner like Algebraic Multigrid can be costly in terms of memory due to the need to store matrix information at the various levels. We present a novel method for defining a preconditioner for systems generated by high-order finite elements that is based on a much sparser system than the original high-order finite element system. We investigate the performance for non-uniform meshes on a cube and a cubed sphere mesh, showing that the sparser preconditioner is more efficient and uses significantly less memory. Finally, we explore new methods to construct the sparse preconditioner and examine their effectiveness for non-uniform meshes. We compare results to a direct use of Algebraic Multigrid as a preconditioner and to a two-level additive Schwarz method.

Linear tearing mode stability equations for a low collisionality toroidal plasma

NASA Astrophysics Data System (ADS)

Connor, J. W.; Hastie, R. J.; Helander, P.

2009-01-01

Tearing mode stability is normally analysed using MHD or two-fluid Braginskii plasma models. However for present, or future, large hot tokamaks like JET or ITER the collisionality is such as to place them in the banana regime. Here we develop a linear stability theory for the resonant layer physics appropriate to such a regime. The outcome is a set of 'fluid' equations whose coefficients encapsulate all neoclassical physics: the neoclassical Ohm's law, enhanced ion inertia, cross-field transport of particles, heat and momentum all play a role. While earlier treatments have also addressed this type of neoclassical physics we differ in incorporating the more physically relevant 'semi-collisional fluid' regime previously considered in cylindrical geometry; semi-collisional effects tend to screen the resonant surface from the perturbed magnetic field, preventing reconnection. Furthermore we also include thermal physics, which may modify the results. While this electron description is of wide relevance and validity, the fluid treatment of the ions requires the ion banana orbit width to be less than the semi-collisional electron layer. This limits the application of the present theory to low magnetic shear—however, this is highly relevant to the sawtooth instability—or to colder ions. The outcome of the calculation is a set of one-dimensional radial differential equations of rather high order. However, various simplifications that reduce the computational task of solving these are discussed. In the collisional regime, when the set reduces to a single second-order differential equation, the theory extends previous work by Hahm et al (1988 Phys. Fluids 31 3709) to include diamagnetic-type effects arising from plasma gradients, both in Ohm's law and the ion inertia term of the vorticity equation. The more relevant semi-collisional regime pertaining to JET or ITER, is described by a pair of second-order differential equations, extending the cylindrical equations of Drake et al (1983 Phys. Fluids 26 2509) to toroidal geometry.

Boolean linear differential operators on elementary cellular automata

NASA Astrophysics Data System (ADS)

Martín Del Rey, Ángel

2014-12-01

In this paper, the notion of boolean linear differential operator (BLDO) on elementary cellular automata (ECA) is introduced and some of their more important properties are studied. Special attention is paid to those differential operators whose coefficients are the ECA with rule numbers 90 and 150.

Fast, purely growing collisionless reconnection as an eigenfunction problem related to but not involving linear whistler waves

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bellan, Paul M.

If either finite electron inertia or finite resistivity is included in 2D magnetic reconnection, the two-fluid equations become a pair of second-order differential equations coupling the out-of-plane magnetic field and vector potential to each other to form a fourth-order system. The coupling at an X-point is such that out-of-plane even-parity electric and odd-parity magnetic fields feed off each other to produce instability if the scale length on which the equilibrium magnetic field changes is less than the ion skin depth. The instability growth rate is given by an eigenvalue of the fourth-order system determined by boundary and symmetry conditions. Themore » instability is a purely growing mode, not a wave, and has growth rate of the order of the whistler frequency. The spatial profile of both the out-of-plane electric and magnetic eigenfunctions consists of an inner concave region having extent of the order of the electron skin depth, an intermediate convex region having extent of the order of the equilibrium magnetic field scale length, and a concave outer exponentially decaying region. If finite electron inertia and resistivity are not included, the inner concave region does not exist and the coupled pair of equations reduces to a second-order differential equation having non-physical solutions at an X-point.« less

Characteristics pertaining to a stiffness cross-coupled Jeffcott model

NASA Technical Reports Server (NTRS)

Spanyer, K. L.

1985-01-01

Rotordynamic studies of complex systems utilizing multiple degree-of-freedom analysis have been performed to understand response, loads, and stability. In order to understand the fundamental nature of rotordynamic response, the Jeffcott rotor model has received wide attention. The purpose of this paper is to provide a generic rotordynamic analysis of a stiffness cross-coupled Jeffcott rotor model to illustrate characteristics of a second order stiffness-coupled linear system. The particular characteristics investigated were forced response, force vector diagrams, response orbits, and stability. Numerical results were achieved through a fourth order Runge-Kutta method for solving differential equations and the Routh Hurwitz stability criterion. The numerical results were verified to an exact mathematical solution for the steady state response.

Item Purification in Differential Item Functioning Using Generalized Linear Mixed Models

ERIC Educational Resources Information Center

Liu, Qian

2011-01-01

For this dissertation, four item purification procedures were implemented onto the generalized linear mixed model for differential item functioning (DIF) analysis, and the performance of these item purification procedures was investigated through a series of simulations. Among the four procedures, forward and generalized linear mixed model (GLMM)…

Projection-Based Reduced Order Modeling for Spacecraft Thermal Analysis

NASA Technical Reports Server (NTRS)

Qian, Jing; Wang, Yi; Song, Hongjun; Pant, Kapil; Peabody, Hume; Ku, Jentung; Butler, Charles D.

2015-01-01

This paper presents a mathematically rigorous, subspace projection-based reduced order modeling (ROM) methodology and an integrated framework to automatically generate reduced order models for spacecraft thermal analysis. Two key steps in the reduced order modeling procedure are described: (1) the acquisition of a full-scale spacecraft model in the ordinary differential equation (ODE) and differential algebraic equation (DAE) form to resolve its dynamic thermal behavior; and (2) the ROM to markedly reduce the dimension of the full-scale model. Specifically, proper orthogonal decomposition (POD) in conjunction with discrete empirical interpolation method (DEIM) and trajectory piece-wise linear (TPWL) methods are developed to address the strong nonlinear thermal effects due to coupled conductive and radiative heat transfer in the spacecraft environment. Case studies using NASA-relevant satellite models are undertaken to verify the capability and to assess the computational performance of the ROM technique in terms of speed-up and error relative to the full-scale model. ROM exhibits excellent agreement in spatiotemporal thermal profiles (<0.5% relative error in pertinent time scales) along with salient computational acceleration (up to two orders of magnitude speed-up) over the full-scale analysis. These findings establish the feasibility of ROM to perform rational and computationally affordable thermal analysis, develop reliable thermal control strategies for spacecraft, and greatly reduce the development cycle times and costs.

Reduction by invariants and projection of linear representations of Lie algebras applied to the construction of nonlinear realizations

NASA Astrophysics Data System (ADS)

Campoamor-Stursberg, R.

2018-03-01

A procedure for the construction of nonlinear realizations of Lie algebras in the context of Vessiot-Guldberg-Lie algebras of first-order systems of ordinary differential equations (ODEs) is proposed. The method is based on the reduction of invariants and projection of lowest-dimensional (irreducible) representations of Lie algebras. Applications to the description of parameterized first-order systems of ODEs related by contraction of Lie algebras are given. In particular, the kinematical Lie algebras in (2 + 1)- and (3 + 1)-dimensions are realized simultaneously as Vessiot-Guldberg-Lie algebras of parameterized nonlinear systems in R3 and R4, respectively.

A Comparison of Two-Stage Approaches for Fitting Nonlinear Ordinary Differential Equation (ODE) Models with Mixed Effects

PubMed Central

Chow, Sy-Miin; Bendezú, Jason J.; Cole, Pamela M.; Ram, Nilam

2016-01-01

Several approaches currently exist for estimating the derivatives of observed data for model exploration purposes, including functional data analysis (FDA), generalized local linear approximation (GLLA), and generalized orthogonal local derivative approximation (GOLD). These derivative estimation procedures can be used in a two-stage process to fit mixed effects ordinary differential equation (ODE) models. While the performance and utility of these routines for estimating linear ODEs have been established, they have not yet been evaluated in the context of nonlinear ODEs with mixed effects. We compared properties of the GLLA and GOLD to an FDA-based two-stage approach denoted herein as functional ordinary differential equation with mixed effects (FODEmixed) in a Monte Carlo study using a nonlinear coupled oscillators model with mixed effects. Simulation results showed that overall, the FODEmixed outperformed both the GLLA and GOLD across all the embedding dimensions considered, but a novel use of a fourth-order GLLA approach combined with very high embedding dimensions yielded estimation results that almost paralleled those from the FODEmixed. We discuss the strengths and limitations of each approach and demonstrate how output from each stage of FODEmixed may be used to inform empirical modeling of young children’s self-regulation. PMID:27391255

A Comparison of Two-Stage Approaches for Fitting Nonlinear Ordinary Differential Equation Models with Mixed Effects.

PubMed

Chow, Sy-Miin; Bendezú, Jason J; Cole, Pamela M; Ram, Nilam

2016-01-01

Several approaches exist for estimating the derivatives of observed data for model exploration purposes, including functional data analysis (FDA; Ramsay & Silverman, 2005 ), generalized local linear approximation (GLLA; Boker, Deboeck, Edler, & Peel, 2010 ), and generalized orthogonal local derivative approximation (GOLD; Deboeck, 2010 ). These derivative estimation procedures can be used in a two-stage process to fit mixed effects ordinary differential equation (ODE) models. While the performance and utility of these routines for estimating linear ODEs have been established, they have not yet been evaluated in the context of nonlinear ODEs with mixed effects. We compared properties of the GLLA and GOLD to an FDA-based two-stage approach denoted herein as functional ordinary differential equation with mixed effects (FODEmixed) in a Monte Carlo (MC) study using a nonlinear coupled oscillators model with mixed effects. Simulation results showed that overall, the FODEmixed outperformed both the GLLA and GOLD across all the embedding dimensions considered, but a novel use of a fourth-order GLLA approach combined with very high embedding dimensions yielded estimation results that almost paralleled those from the FODEmixed. We discuss the strengths and limitations of each approach and demonstrate how output from each stage of FODEmixed may be used to inform empirical modeling of young children's self-regulation.

Double power series method for approximating cosmological perturbations

NASA Astrophysics Data System (ADS)

Wren, Andrew J.; Malik, Karim A.

2017-04-01

We introduce a double power series method for finding approximate analytical solutions for systems of differential equations commonly found in cosmological perturbation theory. The method was set out, in a noncosmological context, by Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases where perturbations are on subhorizon scales. The FSN method is essentially an extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding approximate analytical solutions for ordinary differential equations. The FSN method we use is applicable well beyond perturbation theory to solve systems of ordinary differential equations, linear in the derivatives, that also depend on a small parameter, which here we take to be related to the inverse wave-number. We use the FSN method to find new approximate oscillating solutions in linear order cosmological perturbation theory for a flat radiation-matter universe. Together with this model's well-known growing and decaying Mészáros solutions, these oscillating modes provide a complete set of subhorizon approximations for the metric potential, radiation and matter perturbations. Comparison with numerical solutions of the perturbation equations shows that our approximations can be made accurate to within a typical error of 1%, or better. We also set out a heuristic method for error estimation. A Mathematica notebook which implements the double power series method is made available online.

Mean, covariance, and effective dimension of stochastic distributed delay dynamics

NASA Astrophysics Data System (ADS)

René, Alexandre; Longtin, André

2017-11-01

Dynamical models are often required to incorporate both delays and noise. However, the inherently infinite-dimensional nature of delay equations makes formal solutions to stochastic delay differential equations (SDDEs) challenging. Here, we present an approach, similar in spirit to the analysis of functional differential equations, but based on finite-dimensional matrix operators. This results in a method for obtaining both transient and stationary solutions that is directly amenable to computation, and applicable to first order differential systems with either discrete or distributed delays. With fewer assumptions on the system's parameters than other current solution methods and no need to be near a bifurcation, we decompose the solution to a linear SDDE with arbitrary distributed delays into natural modes, in effect the eigenfunctions of the differential operator, and show that relatively few modes can suffice to approximate the probability density of solutions. Thus, we are led to conclude that noise makes these SDDEs effectively low dimensional, which opens the possibility of practical definitions of probability densities over their solution space.

Singular Hopf bifurcation in a differential equation with large state-dependent delay

PubMed Central

Kozyreff, G.; Erneux, T.

2014-01-01

We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol’s equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays. PMID:24511255

Singular Hopf bifurcation in a differential equation with large state-dependent delay.

PubMed

Kozyreff, G; Erneux, T

2014-02-08

We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol's equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays.

A computer program to generate equations of motion matrices, L217 (EOM). Volume 1: Engineering and usage

NASA Technical Reports Server (NTRS)

Kroll, R. I.; Clemmons, R. E.

1979-01-01

The equations of motion program L217 formulates the matrix coefficients for a set of second order linear differential equations that describe the motion of an airplane relative to its level equilibrium flight condition. Aerodynamic data from FLEXSTAB or Doublet Lattice (L216) programs can be used to derive the equations for quasi-steady or full unsteady aerodynamics. The data manipulation and the matrix coefficient formulation are described.

Probabilistic methods for rotordynamics analysis

NASA Technical Reports Server (NTRS)

Wu, Y.-T.; Torng, T. Y.; Millwater, H. R.; Fossum, A. F.; Rheinfurth, M. H.

1991-01-01

This paper summarizes the development of the methods and a computer program to compute the probability of instability of dynamic systems that can be represented by a system of second-order ordinary linear differential equations. Two instability criteria based upon the eigenvalues or Routh-Hurwitz test functions are investigated. Computational methods based on a fast probability integration concept and an efficient adaptive importance sampling method are proposed to perform efficient probabilistic analysis. A numerical example is provided to demonstrate the methods.

Recent Developments and Open Problems in the Mathematical Theory of Viscoelasticity.

DTIC Science & Technology

1984-11-01

integral terms . At each step of the iteration, we have to solve a linear parabolic equation with time-dependent coefficients. In Sobolevskii’s... parabolic Volterra integro- differential equation, SIAN J. Math. Anal. 13 (1982), ’ ~81-105. :-- 12. Heard, M. L., A class of hyperbolic Volterra ...then puts an n + 1 on the highest derivatives (the "principal terms " in the equation) and an n on lower order derivatives. Two things must then be

A counting-weighted calibration method for a field-programmable-gate-array-based time-to-digital converter

NASA Astrophysics Data System (ADS)

Chen, Yuan-Ho

2017-05-01

In this work, we propose a counting-weighted calibration method for field-programmable-gate-array (FPGA)-based time-to-digital converter (TDC) to provide non-linearity calibration for use in positron emission tomography (PET) scanners. To deal with the non-linearity in FPGA, we developed a counting-weighted delay line (CWD) to count the delay time of the delay cells in the TDC in order to reduce the differential non-linearity (DNL) values based on code density counts. The performance of the proposed CWD-TDC with regard to linearity far exceeds that of TDC with a traditional tapped delay line (TDL) architecture, without the need for nonlinearity calibration. When implemented in a Xilinx Vertix-5 FPGA device, the proposed CWD-TDC achieved time resolution of 60 ps with integral non-linearity (INL) and DNL of [-0.54, 0.24] and [-0.66, 0.65] least-significant-bit (LSB), respectively. This is a clear indication of the suitability of the proposed FPGA-based CWD-TDC for use in PET scanners.

The condition of regular degeneration for singularly perturbed systems of linear differential-difference equations.

NASA Technical Reports Server (NTRS)

Cooke, K. L.; Meyer, K. R.

1966-01-01

Extension of problem of singular perturbation for linear scalar constant coefficient differential- difference equation with single retardation to several retardations, noting degenerate equation solution

Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation.

PubMed

Zana, F; Klein, J C

2001-01-01

This paper presents an algorithm based on mathematical morphology and curvature evaluation for the detection of vessel-like patterns in a noisy environment. Such patterns are very common in medical images. Vessel detection is interesting for the computation of parameters related to blood flow. Its tree-like geometry makes it a usable feature for registration between images that can be of a different nature. In order to define vessel-like patterns, segmentation is performed with respect to a precise model. We define a vessel as a bright pattern, piece-wise connected, and locally linear, mathematical morphology is very well adapted to this description, however other patterns fit such a morphological description. In order to differentiate vessels from analogous background patterns, a cross-curvature evaluation is performed. They are separated out as they have a specific Gaussian-like profile whose curvature varies smoothly along the vessel. The detection algorithm that derives directly from this modeling is based on four steps: (1) noise reduction; (2) linear pattern with Gaussian-like profile improvement; (3) cross-curvature evaluation; (4) linear filtering. We present its theoretical background and illustrate it on real images of various natures, then evaluate its robustness and its accuracy with respect to noise.

Azimuthally invariant Mueller-matrix mapping of biological optically anisotropic network

NASA Astrophysics Data System (ADS)

Ushenko, Yu. O.; Vanchuliak, O.; Bodnar, G. B.; Ushenko, V. O.; Grytsyuk, M.; Pavlyukovich, N.; Pavlyukovich, O. V.; Antonyuk, O.

2017-09-01

A new technique of Mueller-matrix mapping of polycrystalline structure of histological sections of biological tissues is suggested. The algorithms of reconstruction of distribution of parameters of linear and circular dichroism of histological sections liver tissue of mice with different degrees of severity of diabetes are found. The interconnections between such distributions and parameters of linear and circular dichroism of liver of mice tissue histological sections are defined. The comparative investigations of coordinate distributions of parameters of amplitude anisotropy formed by Liver tissue with varying severity of diabetes (10 days and 24 days) are performed. The values and ranges of change of the statistical (moments of the 1st - 4th order) parameters of coordinate distributions of the value of linear and circular dichroism are defined. The objective criteria of cause of the degree of severity of the diabetes differentiation are determined.

Local energy decay for linear wave equations with variable coefficients

NASA Astrophysics Data System (ADS)

Ikehata, Ryo

2005-06-01

A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data, and its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561-568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata-Matsuyama [L2-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33-42] and Todorova-Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].

Chebyshev collocation spectral method for one-dimensional radiative heat transfer in linearly anisotropic-scattering cylindrical medium

NASA Astrophysics Data System (ADS)

Zhou, Rui-Rui; Li, Ben-Wen

2017-03-01

In this study, the Chebyshev collocation spectral method (CCSM) is developed to solve the radiative integro-differential transfer equation (RIDTE) for one-dimensional absorbing, emitting and linearly anisotropic-scattering cylindrical medium. The general form of quadrature formulas for Chebyshev collocation points is deduced. These formulas are proved to have the same accuracy as the Gauss-Legendre quadrature formula (GLQF) for the F-function (geometric function) in the RIDTE. The explicit expressions of the Lagrange basis polynomials and the differentiation matrices for Chebyshev collocation points are also given. These expressions are necessary for solving an integro-differential equation by the CCSM. Since the integrand in the RIDTE is continuous but non-smooth, it is treated by the segments integration method (SIM). The derivative terms in the RIDTE are carried out to improve the accuracy near the origin. In this way, a fourth order accuracy is achieved by the CCSM for the RIDTE, whereas it's only a second order one by the finite difference method (FDM). Several benchmark problems (BPs) with various combinations of optical thickness, medium temperature distribution, degree of anisotropy, and scattering albedo are solved. The results show that present CCSM is efficient to obtain high accurate results, especially for the optically thin medium. The solutions rounded to seven significant digits are given in tabular form, and show excellent agreement with the published data. Finally, the solutions of RIDTE are used as benchmarks for the solution of radiative integral transfer equations (RITEs) presented by Sutton and Chen (JQSRT 84 (2004) 65-103). A non-uniform grid refined near the wall is advised to improve the accuracy of RITEs solutions.

CFORM- LINEAR CONTROL SYSTEM DESIGN AND ANALYSIS: CLOSED FORM SOLUTION AND TRANSIENT RESPONSE OF THE LINEAR DIFFERENTIAL EQUATION

NASA Technical Reports Server (NTRS)

Jamison, J. W.

1994-01-01

CFORM was developed by the Kennedy Space Center Robotics Lab to assist in linear control system design and analysis using closed form and transient response mechanisms. The program computes the closed form solution and transient response of a linear (constant coefficient) differential equation. CFORM allows a choice of three input functions: the Unit Step (a unit change in displacement); the Ramp function (step velocity); and the Parabolic function (step acceleration). It is only accurate in cases where the differential equation has distinct roots, and does not handle the case for roots at the origin (s=0). Initial conditions must be zero. Differential equations may be input to CFORM in two forms - polynomial and product of factors. In some linear control analyses, it may be more appropriate to use a related program, Linear Control System Design and Analysis (KSC-11376), which uses root locus and frequency response methods. CFORM was written in VAX FORTRAN for a VAX 11/780 under VAX VMS 4.7. It has a central memory requirement of 30K. CFORM was developed in 1987.

Mathematics for Physics

NASA Astrophysics Data System (ADS)

Stone, Michael; Goldbart, Paul

2009-07-01

Preface; 1. Calculus of variations; 2. Function spaces; 3. Linear ordinary differential equations; 4. Linear differential operators; 5. Green functions; 6. Partial differential equations; 7. The mathematics of real waves; 8. Special functions; 9. Integral equations; 10. Vectors and tensors; 11. Differential calculus on manifolds; 12. Integration on manifolds; 13. An introduction to differential topology; 14. Group and group representations; 15. Lie groups; 16. The geometry of fibre bundles; 17. Complex analysis I; 18. Applications of complex variables; 19. Special functions and complex variables; Appendixes; Reference; Index.

Measurement of the differential cross section dσ/d(cosθ(t)) for Top-Quark Pair Production in pp Collisions at sqrt[s] = 1.96 TeV.

PubMed

Aaltonen, T; Amerio, S; Amidei, D; Anastassov, A; Annovi, A; Antos, J; Apollinari, G; Appel, J A; Arisawa, T; Artikov, A; Asaadi, J; Ashmanskas, W; Auerbach, B; Aurisano, A; Azfar, F; Badgett, W; Bae, T; Barbaro-Galtieri, A; Barnes, V E; Barnett, B A; Barria, P; Bartos, P; Bauce, M; Bedeschi, F; Behari, S; Bellettini, G; Bellinger, J; Benjamin, D; Beretvas, A; Bhatti, A; Bland, K R; Blumenfeld, B; Bocci, A; Bodek, A; Bortoletto, D; Boudreau, J; Boveia, A; Brigliadori, L; Bromberg, C; Brucken, E; Budagov, J; Budd, H S; Burkett, K; Busetto, G; Bussey, P; Butti, P; Buzatu, A; Calamba, A; Camarda, S; Campanelli, M; Canelli, F; Carls, B; Carlsmith, D; Carosi, R; Carrillo, S; Casal, B; Casarsa, M; Castro, A; Catastini, P; Cauz, D; Cavaliere, V; Cavalli-Sforza, M; Cerri, A; Cerrito, L; Chen, Y C; Chertok, M; Chiarelli, G; Chlachidze, G; Cho, K; Chokheli, D; Clark, A; Clarke, C; Convery, M E; Conway, J; Corbo, M; Cordelli, M; Cox, C A; Cox, D J; Cremonesi, M; Cruz, D; Cuevas, J; Culbertson, R; d'Ascenzo, N; Datta, M; de Barbaro, P; Demortier, L; Deninno, M; D'Errico, M; Devoto, F; Di Canto, A; Di Ruzza, B; Dittmann, J R; Donati, S; D'Onofrio, M; Dorigo, M; Driutti, A; Ebina, K; Edgar, R; Elagin, A; Erbacher, R; Errede, S; Esham, B; Farrington, S; Fernández Ramos, J P; Field, R; Flanagan, G; Forrest, R; Franklin, M; Freeman, J C; Frisch, H; Funakoshi, Y; Galloni, C; Garfinkel, A F; Garosi, P; Gerberich, H; Gerchtein, E; Giagu, S; Giakoumopoulou, V; Gibson, K; Ginsburg, C M; Giokaris, N; Giromini, P; Giurgiu, G; Glagolev, V; Glenzinski, D; Gold, M; Goldin, D; Golossanov, A; Gomez, G; Gomez-Ceballos, G; Goncharov, M; González López, O; Gorelov, I; Goshaw, A T; Goulianos, K; Gramellini, E; Grinstein, S; Grosso-Pilcher, C; Group, R C; Guimaraes da Costa, J; Hahn, S R; Han, J Y; Happacher, F; Hara, K; Hare, M; Harr, R F; Harrington-Taber, T; Hatakeyama, K; Hays, C; Heinrich, J; Herndon, M; Hocker, A; Hong, Z; Hopkins, W; Hou, S; Hughes, R E; Husemann, U; Hussein, M; Huston, J; Introzzi, G; Iori, M; Ivanov, A; James, E; Jang, D; Jayatilaka, B; Jeon, E J; Jindariani, S; Jones, M; Joo, K K; Jun, S Y; Junk, T R; Kambeitz, M; Kamon, T; Karchin, P E; Kasmi, A; Kato, Y; Ketchum, W; Keung, J; Kilminster, B; Kim, D H; Kim, H S; Kim, J E; Kim, M J; Kim, S H; Kim, S B; Kim, Y J; Kim, Y K; Kimura, N; Kirby, M; Knoepfel, K; Kondo, K; Kong, D J; Konigsberg, J; Kotwal, A V; Kreps, M; Kroll, J; Kruse, M; Kuhr, T; Kurata, M; Laasanen, A T; Lammel, S; Lancaster, M; Lannon, K; Latino, G; Lee, H S; Lee, J S; Leo, S; Leone, S; Lewis, J D; Limosani, A; Lipeles, E; Lister, A; Liu, H; Liu, Q; Liu, T; Lockwitz, S; Loginov, A; Lucchesi, D; Lucà, A; Lueck, J; Lujan, P; Lukens, P; Lungu, G; Lys, J; Lysak, R; Madrak, R; Maestro, P; Malik, S; Manca, G; Manousakis-Katsikakis, A; Marchese, L; Margaroli, F; Marino, P; Martínez, M; Matera, K; Mattson, M E; Mazzacane, A; Mazzanti, P; McNulty, R; Mehta, A; Mehtala, P; Mesropian, C; Miao, T; Mietlicki, D; Mitra, A; Miyake, H; Moed, S; Moggi, N; Moon, C S; Moore, R; Morello, M J; Mukherjee, A; Muller, Th; Murat, P; Mussini, M; Nachtman, J; Nagai, Y; Naganoma, J; Nakano, I; Napier, A; Nett, J; Neu, C; Nigmanov, T; Nodulman, L; Noh, S Y; Norniella, O; Oakes, L; Oh, S H; Oh, Y D; Oksuzian, I; Okusawa, T; Orava, R; Ortolan, L; Pagliarone, C; Palencia, E; Palni, P; Papadimitriou, V; Parker, W; Pauletta, G; Paulini, M; Paus, C; Phillips, T J; Piacentino, G; Pianori, E; Pilot, J; Pitts, K; Plager, C; Pondrom, L; Poprocki, S; Potamianos, K; Pranko, A; Prokoshin, F; Ptohos, F; Punzi, G; Ranjan, N; Redondo Fernández, I; Renton, P; Rescigno, M; Rimondi, F; Ristori, L; Robson, A; Rodriguez, T; Rolli, S; Ronzani, M; Roser, R; Rosner, J L; Ruffini, F; Ruiz, A; Russ, J; Rusu, V; Sakumoto, W K; Sakurai, Y; Santi, L; Sato, K; Saveliev, V; Savoy-Navarro, A; Schlabach, P; Schmidt, E E; Schwarz, T; Scodellaro, L; Scuri, F; Seidel, S; Seiya, Y; Semenov, A; Sforza, F; Shalhout, S Z; Shears, T; Shepard, P F; Shimojima, M; Shochet, M; Shreyber-Tecker, I; Simonenko, A; Sliwa, K; Smith, J R; Snider, F D; Song, H; Sorin, V; St Denis, R; Stancari, M; Stentz, D; Strologas, J; Sudo, Y; Sukhanov, A; Suslov, I; Takemasa, K; Takeuchi, Y; Tang, J; Tecchio, M; Teng, P K; Thom, J; Thomson, E; Thukral, V; Toback, D; Tokar, S; Tollefson, K; Tomura, T; Tonelli, D; Torre, S; Torretta, D; Totaro, P; Trovato, M; Ukegawa, F; Uozumi, S; Vázquez, F; Velev, G; Vellidis, C; Vernieri, C; Vidal, M; Vilar, R; Vizán, J; Vogel, M; Volpi, G; Wagner, P; Wallny, R; Wang, S M; Waters, D; Wester, W C; Whiteson, D; Wicklund, A B; Wilbur, S; Williams, H H; Wilson, J S; Wilson, P; Winer, B L; Wittich, P; Wolbers, S; Wolfe, H; Wright, T; Wu, X; Wu, Z; Yamamoto, K; Yamato, D; Yang, T; Yang, U K; Yang, Y C; Yao, W-M; Yeh, G P; Yi, K; Yoh, J; Yorita, K; Yoshida, T; Yu, G B; Yu, I; Zanetti, A M; Zeng, Y; Zhou, C; Zucchelli, S

2013-11-01

We report a measurement of the differential cross section dσ/d(cosθ(t)) for top-quark pair production as a function of the top-quark production angle in proton-antiproton collisions at sqrt[s] = 1.96 TeV. This measurement is performed using data collected with the CDF II detector at the Tevatron, corresponding to an integrated luminosity of 9.4 fb(-1). We employ the Legendre polynomials to characterize the shape of the differential cross section at the parton level. The observed Legendre coefficients are in good agreement with the prediction of the next-to-leading-order standard-model calculation, with the exception of an excess linear-term coefficient a(1) = 0.40 ± 0.12, compared to the standard-model prediction of a(1)=0.15(-0.03)(+0.07).

All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator.

PubMed

Yang, Ting; Dong, Jianji; Lu, Liangjun; Zhou, Linjie; Zheng, Aoling; Zhang, Xinliang; Chen, Jianping

2014-07-04

Photonic integrated circuits for photonic computing open up the possibility for the realization of ultrahigh-speed and ultra wide-band signal processing with compact size and low power consumption. Differential equations model and govern fundamental physical phenomena and engineering systems in virtually any field of science and engineering, such as temperature diffusion processes, physical problems of motion subject to acceleration inputs and frictional forces, and the response of different resistor-capacitor circuits, etc. In this study, we experimentally demonstrate a feasible integrated scheme to solve first-order linear ordinary differential equation with constant-coefficient tunable based on a single silicon microring resonator. Besides, we analyze the impact of the chirp and pulse-width of input signals on the computing deviation. This device can be compatible with the electronic technology (typically complementary metal-oxide semiconductor technology), which may motivate the development of integrated photonic circuits for optical computing.

All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator

PubMed Central

Yang, Ting; Dong, Jianji; Lu, Liangjun; Zhou, Linjie; Zheng, Aoling; Zhang, Xinliang; Chen, Jianping

2014-01-01

Photonic integrated circuits for photonic computing open up the possibility for the realization of ultrahigh-speed and ultra wide-band signal processing with compact size and low power consumption. Differential equations model and govern fundamental physical phenomena and engineering systems in virtually any field of science and engineering, such as temperature diffusion processes, physical problems of motion subject to acceleration inputs and frictional forces, and the response of different resistor-capacitor circuits, etc. In this study, we experimentally demonstrate a feasible integrated scheme to solve first-order linear ordinary differential equation with constant-coefficient tunable based on a single silicon microring resonator. Besides, we analyze the impact of the chirp and pulse-width of input signals on the computing deviation. This device can be compatible with the electronic technology (typically complementary metal-oxide semiconductor technology), which may motivate the development of integrated photonic circuits for optical computing. PMID:24993440

Probabilistic density function method for nonlinear dynamical systems driven by colored noise

DOE Office of Scientific and Technical Information (OSTI.GOV)

Barajas-Solano, David A.; Tartakovsky, Alexandre M.

2016-05-01

We present a probability density function (PDF) method for a system of nonlinear stochastic ordinary differential equations driven by colored noise. The method provides an integro-differential equation for the temporal evolution of the joint PDF of the system's state, which we close by means of a modified Large-Eddy-Diffusivity-type closure. Additionally, we introduce the generalized local linearization (LL) approximation for deriving a computable PDF equation in the form of the second-order partial differential equation (PDE). We demonstrate the proposed closure and localization accurately describe the dynamics of the PDF in phase space for systems driven by noise with arbitrary auto-correlation time.more » We apply the proposed PDF method to the analysis of a set of Kramers equations driven by exponentially auto-correlated Gaussian colored noise to study the dynamics and stability of a power grid.« less

Solutions to an advanced functional partial differential equation of the pantograph type

PubMed Central

Zaidi, Ali A.; Van Brunt, B.; Wake, G. C.

2015-01-01

A model for cells structured by size undergoing growth and division leads to an initial boundary value problem that involves a first-order linear partial differential equation with a functional term. Here, size can be interpreted as DNA content or mass. It has been observed experimentally and shown analytically that solutions for arbitrary initial cell distributions are asymptotic as time goes to infinity to a certain solution called the steady size distribution. The full solution to the problem for arbitrary initial distributions, however, is elusive owing to the presence of the functional term and the paucity of solution techniques for such problems. In this paper, we derive a solution to the problem for arbitrary initial cell distributions. The method employed exploits the hyperbolic character of the underlying differential operator, and the advanced nature of the functional argument to reduce the problem to a sequence of simple Cauchy problems. The existence of solutions for arbitrary initial distributions is established along with uniqueness. The asymptotic relationship with the steady size distribution is established, and because the solution is known explicitly, higher-order terms in the asymptotics can be readily obtained. PMID:26345391

Solutions to an advanced functional partial differential equation of the pantograph type.

PubMed

Zaidi, Ali A; Van Brunt, B; Wake, G C

2015-07-08

A model for cells structured by size undergoing growth and division leads to an initial boundary value problem that involves a first-order linear partial differential equation with a functional term. Here, size can be interpreted as DNA content or mass. It has been observed experimentally and shown analytically that solutions for arbitrary initial cell distributions are asymptotic as time goes to infinity to a certain solution called the steady size distribution. The full solution to the problem for arbitrary initial distributions, however, is elusive owing to the presence of the functional term and the paucity of solution techniques for such problems. In this paper, we derive a solution to the problem for arbitrary initial cell distributions. The method employed exploits the hyperbolic character of the underlying differential operator, and the advanced nature of the functional argument to reduce the problem to a sequence of simple Cauchy problems. The existence of solutions for arbitrary initial distributions is established along with uniqueness. The asymptotic relationship with the steady size distribution is established, and because the solution is known explicitly, higher-order terms in the asymptotics can be readily obtained.

Multiplicative noise removal through fractional order tv-based model and fast numerical schemes for its approximation

NASA Astrophysics Data System (ADS)

Ullah, Asmat; Chen, Wen; Khan, Mushtaq Ahmad

2017-07-01

This paper introduces a fractional order total variation (FOTV) based model with three different weights in the fractional order derivative definition for multiplicative noise removal purpose. The fractional-order Euler Lagrange equation which is a highly non-linear partial differential equation (PDE) is obtained by the minimization of the energy functional for image restoration. Two numerical schemes namely an iterative scheme based on the dual theory and majorization- minimization algorithm (MMA) are used. To improve the restoration results, we opt for an adaptive parameter selection procedure for the proposed model by applying the trial and error method. We report numerical simulations which show the validity and state of the art performance of the fractional-order model in visual improvement as well as an increase in the peak signal to noise ratio comparing to corresponding methods. Numerical experiments also demonstrate that MMAbased methodology is slightly better than that of an iterative scheme.

Interpreting experimental data on egg production--applications of dynamic differential equations.

PubMed

France, J; Lopez, S; Kebreab, E; Dijkstra, J

2013-09-01

This contribution focuses on applying mathematical models based on systems of ordinary first-order differential equations to synthesize and interpret data from egg production experiments. Models based on linear systems of differential equations are contrasted with those based on nonlinear systems. Regression equations arising from analytical solutions to linear compartmental schemes are considered as candidate functions for describing egg production curves, together with aspects of parameter estimation. Extant candidate functions are reviewed, a role for growth functions such as the Gompertz equation suggested, and a function based on a simple new model outlined. Structurally, the new model comprises a single pool with an inflow and an outflow. Compartmental simulation models based on nonlinear systems of differential equations, and thus requiring numerical solution, are next discussed, and aspects of parameter estimation considered. This type of model is illustrated in relation to development and evaluation of a dynamic model of calcium and phosphorus flows in layers. The model consists of 8 state variables representing calcium and phosphorus pools in the crop, stomachs, plasma, and bone. The flow equations are described by Michaelis-Menten or mass action forms. Experiments that measure Ca and P uptake in layers fed different calcium concentrations during shell-forming days are used to evaluate the model. In addition to providing a useful management tool, such a simulation model also provides a means to evaluate feeding strategies aimed at reducing excretion of potential pollutants in poultry manure to the environment.

A Performance Comparison of the Parallel Preconditioners for Iterative Methods for Large Sparse Linear Systems Arising from Partial Differential Equations on Structured Grids

NASA Astrophysics Data System (ADS)

Ma, Sangback

In this paper we compare various parallel preconditioners such as Point-SSOR (Symmetric Successive OverRelaxation), ILU(0) (Incomplete LU) in the Wavefront ordering, ILU(0) in the Multi-color ordering, Multi-Color Block SOR (Successive OverRelaxation), SPAI (SParse Approximate Inverse) and pARMS (Parallel Algebraic Recursive Multilevel Solver) for solving large sparse linear systems arising from two-dimensional PDE (Partial Differential Equation)s on structured grids. Point-SSOR is well-known, and ILU(0) is one of the most popular preconditioner, but it is inherently serial. ILU(0) in the Wavefront ordering maximizes the parallelism in the natural order, but the lengths of the wave-fronts are often nonuniform. ILU(0) in the Multi-color ordering is a simple way of achieving a parallelism of the order N, where N is the order of the matrix, but its convergence rate often deteriorates as compared to that of natural ordering. We have chosen the Multi-Color Block SOR preconditioner combined with direct sparse matrix solver, since for the Laplacian matrix the SOR method is known to have a nondeteriorating rate of convergence when used with the Multi-Color ordering. By using block version we expect to minimize the interprocessor communications. SPAI computes the sparse approximate inverse directly by least squares method. Finally, ARMS is a preconditioner recursively exploiting the concept of independent sets and pARMS is the parallel version of ARMS. Experiments were conducted for the Finite Difference and Finite Element discretizations of five two-dimensional PDEs with large meshsizes up to a million on an IBM p595 machine with distributed memory. Our matrices are real positive, i. e., their real parts of the eigenvalues are positive. We have used GMRES(m) as our outer iterative method, so that the convergence of GMRES(m) for our test matrices are mathematically guaranteed. Interprocessor communications were done using MPI (Message Passing Interface) primitives. The results show that in general ILU(0) in the Multi-Color ordering ahd ILU(0) in the Wavefront ordering outperform the other methods but for symmetric and nearly symmetric 5-point matrices Multi-Color Block SOR gives the best performance, except for a few cases with a small number of processors.

Finite-time synchronization of fractional-order memristor-based neural networks with time delays.

PubMed

Velmurugan, G; Rakkiyappan, R; Cao, Jinde

2016-01-01

In this paper, we consider the problem of finite-time synchronization of a class of fractional-order memristor-based neural networks (FMNNs) with time delays and investigated it potentially. By using Laplace transform, the generalized Gronwall's inequality, Mittag-Leffler functions and linear feedback control technique, some new sufficient conditions are derived to ensure the finite-time synchronization of addressing FMNNs with fractional order α:1<α<2 and 0<α<1. The results from the theory of fractional-order differential equations with discontinuous right-hand sides are used to investigate the problem under consideration. The derived results are extended to some previous related works on memristor-based neural networks. Finally, three numerical examples are presented to show the effectiveness of our proposed theoretical results. Copyright © 2015 Elsevier Ltd. All rights reserved.

Computational Aspects of Sensitivity Calculations in Linear Transient Structural Analysis. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Greene, William H.

1989-01-01

A study has been performed focusing on the calculation of sensitivities of displacements, velocities, accelerations, and stresses in linear, structural, transient response problems. One significant goal was to develop and evaluate sensitivity calculation techniques suitable for large-order finite element analyses. Accordingly, approximation vectors such as vibration mode shapes are used to reduce the dimensionality of the finite element model. Much of the research focused on the accuracy of both response quantities and sensitivities as a function of number of vectors used. Two types of sensitivity calculation techniques were developed and evaluated. The first type of technique is an overall finite difference method where the analysis is repeated for perturbed designs. The second type of technique is termed semianalytical because it involves direct, analytical differentiation of the equations of motion with finite difference approximation of the coefficient matrices. To be computationally practical in large-order problems, the overall finite difference methods must use the approximation vectors from the original design in the analyses of the perturbed models.

A high order accurate finite element algorithm for high Reynolds number flow prediction

NASA Technical Reports Server (NTRS)

Baker, A. J.

1978-01-01

A Galerkin-weighted residuals formulation is employed to establish an implicit finite element solution algorithm for generally nonlinear initial-boundary value problems. Solution accuracy, and convergence rate with discretization refinement, are quantized in several error norms, by a systematic study of numerical solutions to several nonlinear parabolic and a hyperbolic partial differential equation characteristic of the equations governing fluid flows. Solutions are generated using selective linear, quadratic and cubic basis functions. Richardson extrapolation is employed to generate a higher-order accurate solution to facilitate isolation of truncation error in all norms. Extension of the mathematical theory underlying accuracy and convergence concepts for linear elliptic equations is predicted for equations characteristic of laminar and turbulent fluid flows at nonmodest Reynolds number. The nondiagonal initial-value matrix structure introduced by the finite element theory is determined intrinsic to improved solution accuracy and convergence. A factored Jacobian iteration algorithm is derived and evaluated to yield a consequential reduction in both computer storage and execution CPU requirements while retaining solution accuracy.

Expectation propagation for large scale Bayesian inference of non-linear molecular networks from perturbation data.

PubMed

Narimani, Zahra; Beigy, Hamid; Ahmad, Ashar; Masoudi-Nejad, Ali; Fröhlich, Holger

2017-01-01

Inferring the structure of molecular networks from time series protein or gene expression data provides valuable information about the complex biological processes of the cell. Causal network structure inference has been approached using different methods in the past. Most causal network inference techniques, such as Dynamic Bayesian Networks and ordinary differential equations, are limited by their computational complexity and thus make large scale inference infeasible. This is specifically true if a Bayesian framework is applied in order to deal with the unavoidable uncertainty about the correct model. We devise a novel Bayesian network reverse engineering approach using ordinary differential equations with the ability to include non-linearity. Besides modeling arbitrary, possibly combinatorial and time dependent perturbations with unknown targets, one of our main contributions is the use of Expectation Propagation, an algorithm for approximate Bayesian inference over large scale network structures in short computation time. We further explore the possibility of integrating prior knowledge into network inference. We evaluate the proposed model on DREAM4 and DREAM8 data and find it competitive against several state-of-the-art existing network inference methods.

Collisionless kinetic theory of oblique tearing instabilities

DOE PAGES

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

2018-02-15

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. In this paper, we find that this stabilization is associated with the density-gradient-driven diamagnetic drift. Themore » analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. Finally, a simple analytic estimate for the stability criterion is provided.« less

Collisionless kinetic theory of oblique tearing instabilities

DOE Office of Scientific and Technical Information (OSTI.GOV)

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. In this paper, we find that this stabilization is associated with the density-gradient-driven diamagnetic drift. Themore » analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. Finally, a simple analytic estimate for the stability criterion is provided.« less

Collisionless kinetic theory of oblique tearing instabilities

NASA Astrophysics Data System (ADS)

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

2018-02-01

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. We find that this stabilization is associated with the density-gradient-driven diamagnetic drift. The analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. A simple analytic estimate for the stability criterion is provided.

Discontinuous Galerkin Methods for NonLinear Differential Systems

NASA Technical Reports Server (NTRS)

Barth, Timothy; Mansour, Nagi (Technical Monitor)

2001-01-01

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

A differential equation for the Generalized Born radii.

PubMed

Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro

2013-06-28

The Generalized Born (GB) model offers a convenient way of representing electrostatics in complex macromolecules like proteins or nucleic acids. The computation of atomic GB radii is currently performed by different non-local approaches involving volume or surface integrals. Here we obtain a non-linear second-order partial differential equation for the Generalized Born radius, which may be solved using local iterative algorithms. The equation is derived under the assumption that the usual GB approximation to the reaction field obeys Laplace's equation. The equation admits as particular solutions the correct GB radii for the sphere and the plane. The tests performed on a set of 55 different proteins show an overall agreement with other reference GB models and "perfect" Poisson-Boltzmann based values.

Exceptional point in a simple textbook example

NASA Astrophysics Data System (ADS)

Fernández, Francisco M.

2018-07-01

We propose to introduce the concept of exceptional points in intermediate courses on mathematics and classical mechanics by means of simple textbook examples. The first one is an ordinary second-order differential equation with constant coefficients. The second one is the well-known damped harmonic oscillator. From a strict mathematical viewpoint both are the same problem that enables one to connect the occurrence of linearly dependent exponential solutions with a defective matrix which cannot be diagonalized but can be transformed into a Jordan canonical form.

Random harmonic analysis program, L221 (TEV156). Volume 1: Engineering and usage

NASA Technical Reports Server (NTRS)

Miller, R. D.; Graham, M. L.

1979-01-01

A digital computer program capable of calculating steady state solutions for linear second order differential equations due to sinusoidal forcing functions is described. The field of application of the program, the analysis of airplane response and loads due to continuous random air turbulence, is discussed. Optional capabilities including frequency dependent input matrices, feedback damping, gradual gust penetration, multiple excitation forcing functions, and a static elastic solution are described. Program usage and a description of the analysis used are presented.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Luo, Yousong, E-mail: yousong.luo@rmit.edu.au

This paper deals with a class of optimal control problems governed by an initial-boundary value problem of a parabolic equation. The case of semi-linear boundary control is studied where the control is applied to the system via the Wentzell boundary condition. The differentiability of the state variable with respect to the control is established and hence a necessary condition is derived for the optimal solution in the case of both unconstrained and constrained problems. The condition is also sufficient for the unconstrained convex problems. A second order condition is also derived.

Validation of drift and diffusion coefficients from experimental data

NASA Astrophysics Data System (ADS)

Riera, R.; Anteneodo, C.

2010-04-01

Many fluctuation phenomena, in physics and other fields, can be modeled by Fokker-Planck or stochastic differential equations whose coefficients, associated with drift and diffusion components, may be estimated directly from the observed time series. Its correct characterization is crucial to determine the system quantifiers. However, due to the finite sampling rates of real data, the empirical estimates may significantly differ from their true functional forms. In the literature, low-order corrections, or even no corrections, have been applied to the finite-time estimates. A frequent outcome consists of linear drift and quadratic diffusion coefficients. For this case, exact corrections have been recently found, from Itô-Taylor expansions. Nevertheless, model validation constitutes a necessary step before determining and applying the appropriate corrections. Here, we exploit the consequences of the exact theoretical results obtained for the linear-quadratic model. In particular, we discuss whether the observed finite-time estimates are actually a manifestation of that model. The relevance of this analysis is put into evidence by its application to two contrasting real data examples in which finite-time linear drift and quadratic diffusion coefficients are observed. In one case the linear-quadratic model is readily rejected while in the other, although the model constitutes a very good approximation, low-order corrections are inappropriate. These examples give warning signs about the proper interpretation of finite-time analysis even in more general diffusion processes.

Linearized gravity in terms of differential forms

NASA Astrophysics Data System (ADS)

Baykal, Ahmet; Dereli, Tekin

2017-01-01

A technique to linearize gravitational field equations is developed in which the perturbation metric coefficients are treated as second rank, symmetric, 1-form fields belonging to the Minkowski background spacetime by using the exterior algebra of differential forms.

A new third order finite volume weighted essentially non-oscillatory scheme on tetrahedral meshes

NASA Astrophysics Data System (ADS)

Zhu, Jun; Qiu, Jianxian

2017-11-01

In this paper a third order finite volume weighted essentially non-oscillatory scheme is designed for solving hyperbolic conservation laws on tetrahedral meshes. Comparing with other finite volume WENO schemes designed on tetrahedral meshes, the crucial advantages of such new WENO scheme are its simplicity and compactness with the application of only six unequal size spatial stencils for reconstructing unequal degree polynomials in the WENO type spatial procedures, and easy choice of the positive linear weights without considering the topology of the meshes. The original innovation of such scheme is to use a quadratic polynomial defined on a big central spatial stencil for obtaining third order numerical approximation at any points inside the target tetrahedral cell in smooth region and switch to at least one of five linear polynomials defined on small biased/central spatial stencils for sustaining sharp shock transitions and keeping essentially non-oscillatory property simultaneously. By performing such new procedures in spatial reconstructions and adopting a third order TVD Runge-Kutta time discretization method for solving the ordinary differential equation (ODE), the new scheme's memory occupancy is decreased and the computing efficiency is increased. So it is suitable for large scale engineering requirements on tetrahedral meshes. Some numerical results are provided to illustrate the good performance of such scheme.

Constructive Development of the Solutions of Linear Equations in Introductory Ordinary Differential Equations

ERIC Educational Resources Information Center

Mallet, D. G.; McCue, S. W.

2009-01-01

The solution of linear ordinary differential equations (ODEs) is commonly taught in first-year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognizing what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to…

Modeling Individual Damped Linear Oscillator Processes with Differential Equations: Using Surrogate Data Analysis to Estimate the Smoothing Parameter

ERIC Educational Resources Information Center

Deboeck, Pascal R.; Boker, Steven M.; Bergeman, C. S.

2008-01-01

Among the many methods available for modeling intraindividual time series, differential equation modeling has several advantages that make it promising for applications to psychological data. One interesting differential equation model is that of the damped linear oscillator (DLO), which can be used to model variables that have a tendency to…

Chandrasekhar equations for infinite dimensional systems

NASA Technical Reports Server (NTRS)

Ito, K.; Powers, R.

1985-01-01

The existence of Chandrasekhar equations for linear time-invariant systems defined on Hilbert spaces is investigated. An important consequence is that the solution to the evolutional Riccati equation is strongly differentiable in time, and that a strong solution of the Riccati differential equation can be defined. A discussion of the linear-quadratic optimal-control problem for hereditary differential systems is also included.

WEAK GALERKIN METHODS FOR SECOND ORDER ELLIPTIC INTERFACE PROBLEMS

PubMed Central

MU, LIN; WANG, JUNPING; WEI, GUOWEI; YE, XIU; ZHAO, SHAN

2013-01-01

Weak Galerkin methods refer to general finite element methods for partial differential equations (PDEs) in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic PDEs with discontinuous coefficients and interfaces. Theoretically, it is proved that high order numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element. Extensive numerical experiments have been carried to validate the WG-FEM for solving second order elliptic interface problems. High order of convergence is numerically confirmed in both L2 and L∞ norms for the piecewise linear WG-FEM. Special attention is paid to solve many interface problems, in which the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design nearly second order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order O(h) to O(h1.5) for the solution itself in L∞ norm. It is demonstrated that the WG-FEM of the lowest order, i.e., the piecewise constant WG-FEM, is capable of delivering numerical approximations that are of order O(h1.75) to O(h2) in the L∞ norm for C1 or Lipschitz continuous interfaces associated with a C1 or H2 continuous solution. PMID:24072935

Theory of Metastable State Relaxation for Non-Critical Binary Systems with Non-Conserved Order Parameter

NASA Technical Reports Server (NTRS)

Izmailov, Alexander; Myerson, Allan S.

1993-01-01

A new mathematical ansatz for a solution of the time-dependent Ginzburg-Landau non-linear partial differential equation is developed for non-critical systems such as non-critical binary solutions (solute + solvent) described by the non-conserved scalar order parameter. It is demonstrated that in such systems metastability initiates heterogeneous solute redistribution which results in formation of the non-equilibrium singly-periodic spatial solute structure. It is found how the time-dependent period of this structure evolves in time. In addition, the critical radius r(sub c) for solute embryo of the new solute rich phase together with the metastable state lifetime t(sub c) are determined analytically and analyzed.

Aeroelastic System Development Using Proper Orthogonal Decomposition and Volterra Theory

NASA Technical Reports Server (NTRS)

Lucia, David J.; Beran, Philip S.; Silva, Walter A.

2003-01-01

This research combines Volterra theory and proper orthogonal decomposition (POD) into a hybrid methodology for reduced-order modeling of aeroelastic systems. The out-come of the method is a set of linear ordinary differential equations (ODEs) describing the modal amplitudes associated with both the structural modes and the POD basis functions for the uid. For this research, the structural modes are sine waves of varying frequency, and the Volterra-POD approach is applied to the fluid dynamics equations. The structural modes are treated as forcing terms which are impulsed as part of the uid model realization. Using this approach, structural and uid operators are coupled into a single aeroelastic operator. This coupling converts a free boundary uid problem into an initial value problem, while preserving the parameter (or parameters) of interest for sensitivity analysis. The approach is applied to an elastic panel in supersonic cross ow. The hybrid Volterra-POD approach provides a low-order uid model in state-space form. The linear uid model is tightly coupled with a nonlinear panel model using an implicit integration scheme. The resulting aeroelastic model provides correct limit-cycle oscillation prediction over a wide range of panel dynamic pressure values. Time integration of the reduced-order aeroelastic model is four orders of magnitude faster than the high-order solution procedure developed for this research using traditional uid and structural solvers.

Saving a Drug Poisoning Victim: A Kinetics Simulation

NASA Astrophysics Data System (ADS)

Selco, Jodye I.; Beery, Janet

2002-05-01

In this project, students, posing as hospital emergency room physicians, must save the life of a child who has accidentally overdosed on the asthma medication, theophylline. The progress of the drug through the child's body can be modeled as a chemical kinetics problem involving first-order consecutive reactions. Students begin by setting up a system of linear first-order differential equations describing the medication's absorption into and elimination from the child's bloodstream using half-lives obtained from the Physician's Desk Reference. By using a computer to solve the differential equations numerically, students discover that the child will almost certainly die if they, as physicians, do not intervene. The students then determine by how much they need to increase the drug's elimination rate in order to save the child. This dictates the appropriate medical action. Students discover that they need to use the more drastic treatment of extracorporeal filtering of the blood through charcoal, rather than simply administering oral doses of charcoal. We've found that this project appeals to a broad range of students; many students are interested in careers in the health professions and all are intrigued by the child's grave situation.

Electrocardiogram classification using delay differential equations

NASA Astrophysics Data System (ADS)

Lainscsek, Claudia; Sejnowski, Terrence J.

2013-06-01

Time series analysis with nonlinear delay differential equations (DDEs) reveals nonlinear as well as spectral properties of the underlying dynamical system. Here, global DDE models were used to analyze 5 min data segments of electrocardiographic (ECG) recordings in order to capture distinguishing features for different heart conditions such as normal heart beat, congestive heart failure, and atrial fibrillation. The number of terms and delays in the model as well as the order of nonlinearity of the model have to be selected that are the most discriminative. The DDE model form that best separates the three classes of data was chosen by exhaustive search up to third order polynomials. Such an approach can provide deep insight into the nature of the data since linear terms of a DDE correspond to the main time-scales in the signal and the nonlinear terms in the DDE are related to nonlinear couplings between the harmonic signal parts. The DDEs were able to detect atrial fibrillation with an accuracy of 72%, congestive heart failure with an accuracy of 88%, and normal heart beat with an accuracy of 97% from 5 min of ECG, a much shorter time interval than required to achieve comparable performance with other methods.

Bisimulation equivalence of differential-algebraic systems

NASA Astrophysics Data System (ADS)

Megawati, Noorma Yulia; Schaft, Arjan van der

2018-01-01

In this paper, the notion of bisimulation relation for linear input-state-output systems is extended to general linear differential-algebraic (DAE) systems. Geometric control theory is used to derive a linear-algebraic characterisation of bisimulation relations, and an algorithm for computing the maximal bisimulation relation between two linear DAE systems. The general definition is specialised to the case where the matrix pencil sE - A is regular. Furthermore, by developing a one-sided version of bisimulation, characterisations of simulation and abstraction are obtained.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Owen, Benjamin C.; Kenttamaa, Hilkka I.

The present disclosure provides a new tandem mass spectrometer and methods of using the same for analyzing charged particles. The differentially pumped dual linear quadrupole ion trap mass spectrometer of the present disclose includes a combination of two linear quadrupole (LQIT) mass spectrometers with differentially pumped vacuum chambers.

Computational method for analysis of polyethylene biodegradation

NASA Astrophysics Data System (ADS)

Watanabe, Masaji; Kawai, Fusako; Shibata, Masaru; Yokoyama, Shigeo; Sudate, Yasuhiro

2003-12-01

In a previous study concerning the biodegradation of polyethylene, we proposed a mathematical model based on two primary factors: the direct consumption or absorption of small molecules and the successive weight loss of large molecules due to β-oxidation. Our model is an initial value problem consisting of a differential equation whose independent variable is time. Its unknown variable represents the total weight of all the polyethylene molecules that belong to a molecular-weight class specified by a parameter. In this paper, we describe a numerical technique to introduce experimental results into analysis of our model. We first establish its mathematical foundation in order to guarantee its validity, by showing that the initial value problem associated with the differential equation has a unique solution. Our computational technique is based on a linear system of differential equations derived from the original problem. We introduce some numerical results to illustrate our technique as a practical application of the linear approximation. In particular, we show how to solve the inverse problem to determine the consumption rate and the β-oxidation rate numerically, and illustrate our numerical technique by analyzing the GPC patterns of polyethylene wax obtained before and after 5 weeks cultivation of a fungus, Aspergillus sp. AK-3. A numerical simulation based on these degradation rates confirms that the primary factors of the polyethylene biodegradation posed in modeling are indeed appropriate.

Optimal analytic method for the nonlinear Hasegawa-Mima equation

NASA Astrophysics Data System (ADS)

Baxter, Mathew; Van Gorder, Robert A.; Vajravelu, Kuppalapalle

2014-05-01

The Hasegawa-Mima equation is a nonlinear partial differential equation that describes the electric potential due to a drift wave in a plasma. In the present paper, we apply the method of homotopy analysis to a slightly more general Hasegawa-Mima equation, which accounts for hyper-viscous damping or viscous dissipation. First, we outline the method for the general initial/boundary value problem over a compact rectangular spatial domain. We use a two-stage method, where both the convergence control parameter and the auxiliary linear operator are optimally selected to minimize the residual error due to the approximation. To do the latter, we consider a family of operators parameterized by a constant which gives the decay rate of the solutions. After outlining the general method, we consider a number of concrete examples in order to demonstrate the utility of this approach. The results enable us to study properties of the initial/boundary value problem for the generalized Hasegawa-Mima equation. In several cases considered, we are able to obtain solutions with extremely small residual errors after relatively few iterations are computed (residual errors on the order of 10-15 are found in multiple cases after only three iterations). The results demonstrate that selecting a parameterized auxiliary linear operator can be extremely useful for minimizing residual errors when used concurrently with the optimal homotopy analysis method, suggesting that this approach can prove useful for a number of nonlinear partial differential equations arising in physics and nonlinear mechanics.

Chandrasekhar equations for infinite dimensional systems

NASA Technical Reports Server (NTRS)

Ito, K.; Powers, R. K.

1985-01-01

Chandrasekhar equations are derived for linear time invariant systems defined on Hilbert spaces using a functional analytic technique. An important consequence of this is that the solution to the evolutional Riccati equation is strongly differentiable in time and one can define a strong solution of the Riccati differential equation. A detailed discussion on the linear quadratic optimal control problem for hereditary differential systems is also included.

Adaptive learning in complex reproducing kernel Hilbert spaces employing Wirtinger's subgradients.

PubMed

Bouboulis, Pantelis; Slavakis, Konstantinos; Theodoridis, Sergios

2012-03-01

This paper presents a wide framework for non-linear online supervised learning tasks in the context of complex valued signal processing. The (complex) input data are mapped into a complex reproducing kernel Hilbert space (RKHS), where the learning phase is taking place. Both pure complex kernels and real kernels (via the complexification trick) can be employed. Moreover, any convex, continuous and not necessarily differentiable function can be used to measure the loss between the output of the specific system and the desired response. The only requirement is the subgradient of the adopted loss function to be available in an analytic form. In order to derive analytically the subgradients, the principles of the (recently developed) Wirtinger's calculus in complex RKHS are exploited. Furthermore, both linear and widely linear (in RKHS) estimation filters are considered. To cope with the problem of increasing memory requirements, which is present in almost all online schemes in RKHS, the sparsification scheme, based on projection onto closed balls, has been adopted. We demonstrate the effectiveness of the proposed framework in a non-linear channel identification task, a non-linear channel equalization problem and a quadrature phase shift keying equalization scheme, using both circular and non circular synthetic signal sources.

Implications of the Corotation Theorem on the MRI in Axial Symmetry

NASA Astrophysics Data System (ADS)

Montani, G.; Cianfrani, F.; Pugliese, D.

2016-08-01

We analyze the linear stability of an axially symmetric ideal plasma disk, embedded in a magnetic field and endowed with a differential rotation. This study is performed by adopting the magnetic flux function as the fundamental dynamical variable, in order to outline the role played by the corotation theorem on the linear mode structure. Using some specific assumptions (e.g., plasma incompressibility and propagation of the perturbations along the background magnetic field), we select the Alfvénic nature of the magnetorotational instability, and, in the geometric optics limit, we determine the dispersion relation describing the linear spectrum. We show how the implementation of the corotation theorem (valid for the background configuration) on the linear dynamics produces the cancellation of the vertical derivative of the disk angular velocity (we check such a feature also in the standard vector formalism to facilitate comparison with previous literature, in both the axisymmetric and three-dimensional cases). As a result, we clarify that the unstable modes have, for a stratified disk, the same morphology, proper of a thin-disk profile, and the z-dependence has a simple parametric role.

Non-linear hydrodynamical evolution of rotating relativistic stars: numerical methods and code tests

NASA Astrophysics Data System (ADS)

Font, José A.; Stergioulas, Nikolaos; Kokkotas, Kostas D.

2000-04-01

We present numerical hydrodynamical evolutions of rapidly rotating relativistic stars, using an axisymmetric, non-linear relativistic hydrodynamics code. We use four different high-resolution shock-capturing (HRSC) finite-difference schemes (based on approximate Riemann solvers) and compare their accuracy in preserving uniformly rotating stationary initial configurations in long-term evolutions. Among these four schemes, we find that the third-order piecewise parabolic method scheme is superior in maintaining the initial rotation law in long-term evolutions, especially near the surface of the star. It is further shown that HRSC schemes are suitable for the evolution of perturbed neutron stars and for the accurate identification (via Fourier transforms) of normal modes of oscillation. This is demonstrated for radial and quadrupolar pulsations in the non-rotating limit, where we find good agreement with frequencies obtained with a linear perturbation code. The code can be used for studying small-amplitude or non-linear pulsations of differentially rotating neutron stars, while our present results serve as testbed computations for three-dimensional general-relativistic evolution codes.

A powerful local shear instability in weakly magnetized disks. I - Linear analysis. II - Nonlinear evolution

NASA Technical Reports Server (NTRS)

Balbus, Steven A.; Hawley, John F.

1991-01-01

A broad class of astronomical accretion disks is presently shown to be dynamically unstable to axisymmetric disturbances in the presence of a weak magnetic field, an insight with consequently broad applicability to gaseous, differentially-rotating systems. In the first part of this work, a linear analysis is presented of the instability, which is local and extremely powerful; the maximum growth rate, which is of the order of the angular rotation velocity, is independent of the strength of the magnetic field. Fluid motions associated with the instability directly generate both poloidal and toroidal field components. In the second part of this investigation, the scaling relation between the instability's wavenumber and the Alfven velocity is demonstrated, and the independence of the maximum growth rate from magnetic field strength is confirmed.

Linear and nonlinear stability criteria for compressible MHD flows in a gravitational field

NASA Astrophysics Data System (ADS)

Moawad, S. M.; Moawad

2013-10-01

The equilibrium and stability properties of ideal magnetohydrodynamics (MHD) of compressible flow in a gravitational field with a translational symmetry are investigated. Variational principles for the steady-state equations are formulated. The MHD equilibrium equations are obtained as critical points of a conserved Lyapunov functional. This functional consists of the sum of the total energy, the mass, the circulation along field lines (cross helicity), the momentum, and the magnetic helicity. In the unperturbed case, the equilibrium states satisfy a nonlinear second-order partial differential equation (PDE) associated with hydrodynamic Bernoulli law. The PDE can be an elliptic or a parabolic equation depending on increasing the poloidal flow speed. Linear and nonlinear Lyapunov stability conditions under translational symmetric perturbations are established for the equilibrium states.

On Instability of Geostrophic Current with Linear Vertical Shear at Length Scales of Interleaving

NASA Astrophysics Data System (ADS)

Kuzmina, N. P.; Skorokhodov, S. L.; Zhurbas, N. V.; Lyzhkov, D. A.

2018-01-01

The instability of long-wave disturbances of a geostrophic current with linear velocity shear is studied with allowance for the diffusion of buoyancy. A detailed derivation of the model problem in dimensionless variables is presented, which is used for analyzing the dynamics of disturbances in a vertically bounded layer and for describing the formation of large-scale intrusions in the Arctic basin. The problem is solved numerically based on a high-precision method developed for solving fourth-order differential equations. It is established that there is an eigenvalue in the spectrum of eigenvalues that corresponds to unstable (growing with time) disturbances, which are characterized by a phase velocity exceeding the maximum velocity of the geostrophic flow. A discussion is presented to explain some features of the instability.

Method of fabricating an article with cavities. [with thin bottom walls

NASA Technical Reports Server (NTRS)

Dale, W. J.; Jurscaga, G. M. (Inventor)

1974-01-01

An article having a cavity with a thin bottom wall is provided by assembling a thin sheet, for example, a metal sheet, adjacent to the surface of a member having one or more apertures. A bonding adhesive is interposed between the thin sheet and the subadjacent member, and the thin sheet is subjected to a high fluid pressure. In order to prevent the differential pressure from being exerted against the thin sheet, the aperture is filled with a plug of solid material having a linear coefficient of thermal expansion higher than that of the member. When the assembly is subjected to pressure, the material is heated to a temperature such that its expansion exerts a pressure against the thin sheet thus reducing the differential pressure.

A differential equation for the asymptotic fitness distribution in the Bak-Sneppen model with five species.

PubMed

Schlemm, Eckhard

2015-09-01

The Bak-Sneppen model is an abstract representation of a biological system that evolves according to the Darwinian principles of random mutation and selection. The species in the system are characterized by a numerical fitness value between zero and one. We show that in the case of five species the steady-state fitness distribution can be obtained as a solution to a linear differential equation of order five with hypergeometric coefficients. Similar representations for the asymptotic fitness distribution in larger systems may help pave the way towards a resolution of the question of whether or not, in the limit of infinitely many species, the fitness is asymptotically uniformly distributed on the interval [fc, 1] with fc ≳ 2/3. Copyright © 2015 Elsevier Inc. All rights reserved.

On Polynomial Solutions of Linear Differential Equations with Polynomial Coefficients

ERIC Educational Resources Information Center

Si, Do Tan

1977-01-01

Demonstrates a method for solving linear differential equations with polynomial coefficients based on the fact that the operators z and D + d/dz are known to be Hermitian conjugates with respect to the Bargman and Louck-Galbraith scalar products. (MLH)

Nonlocal theory of curved rods. 2-D, high order, Timoshenko's and Euler-Bernoulli models

NASA Astrophysics Data System (ADS)

Zozulya, V. V.

2017-09-01

New models for plane curved rods based on linear nonlocal theory of elasticity have been developed. The 2-D theory is developed from general 2-D equations of linear nonlocal elasticity using a special curvilinear system of coordinates related to the middle line of the rod along with special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby, all equations of elasticity including nonlocal constitutive relations have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of local elasticity, a system of differential equations in terms of displacements for Fourier coefficients has been obtained. First and second order approximations have been considered in detail. Timoshenko's and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear nonlocal theory of elasticity which are considered in a special curvilinear system of coordinates related to the middle line of the rod. The obtained equations can be used to calculate stress-strain and to model thin walled structures in micro- and nanoscales when taking into account size dependent and nonlocal effects.

Analysis of Different Hyperspectral Variables for Diagnosing Leaf Nitrogen Accumulation in Wheat.

PubMed

Tan, Changwei; Du, Ying; Zhou, Jian; Wang, Dunliang; Luo, Ming; Zhang, Yongjian; Guo, Wenshan

2018-01-01

Hyperspectral remote sensing is a rapid non-destructive method for diagnosing nitrogen status in wheat crops. In this study, a quantitative correlation was associated with following parameters: leaf nitrogen accumulation (LNA), raw hyperspectral reflectance, first-order differential hyperspectra, and hyperspectral characteristics of wheat. In this study, integrated linear regression of LNA was obtained with raw hyperspectral reflectance (measurement wavelength = 790.4 nm). Furthermore, an exponential regression of LNA was obtained with first-order differential hyperspectra (measurement wavelength = 831.7 nm). Coefficients ( R 2 ) were 0.813 and 0.847; root mean squared errors (RMSE) were 2.02 g·m -2 and 1.72 g·m -2 ; and relative errors (RE) were 25.97% and 20.85%, respectively. Both the techniques were considered as optimal in the diagnoses of wheat LNA. Nevertheless, the better one was the new normalized variable (SD r - SD b )/(SD r + SD b ) , which was based on vegetation indices of R 2 = 0.935, RMSE = 0.98, and RE = 11.25%. In addition, (SD r - SD b )/(SD r + SD b ) was reliable in the application of a different cultivar or even wheat grown elsewhere. This indicated a superior fit and better performance for (SD r - SD b )/(SD r + SD b ) . For diagnosing LNA in wheat, the newly normalized variable (SD r - SD b )/(SD r + SD b ) was more effective than the previously reported data of raw hyperspectral reflectance, first-order differential hyperspectra, and red-edge parameters.

Comparison of finite-difference schemes for analysis of shells of revolution. [stress and free vibration analysis

NASA Technical Reports Server (NTRS)

Noor, A. K.; Stephens, W. B.

1973-01-01

Several finite difference schemes are applied to the stress and free vibration analysis of homogeneous isotropic and layered orthotropic shells of revolution. The study is based on a form of the Sanders-Budiansky first-approximation linear shell theory modified such that the effects of shear deformation and rotary inertia are included. A Fourier approach is used in which all the shell stress resultants and displacements are expanded in a Fourier series in the circumferential direction, and the governing equations reduce to ordinary differential equations in the meridional direction. While primary attention is given to finite difference schemes used in conjunction with first order differential equation formulation, comparison is made with finite difference schemes used with other formulations. These finite difference discretization models are compared with respect to simplicity of application, convergence characteristics, and computational efficiency. Numerical studies are presented for the effects of variations in shell geometry and lamination parameters on the accuracy and convergence of the solutions obtained by the different finite difference schemes. On the basis of the present study it is shown that the mixed finite difference scheme based on the first order differential equation formulation and two interlacing grids for the different fundamental unknowns combines a number of advantages over other finite difference schemes previously reported in the literature.

Mapping soil total nitrogen of cultivated land at county scale by using hyperspectral image

NASA Astrophysics Data System (ADS)

Gu, Xiaohe; Zhang, Li Yan; Shu, Meiyan; Yang, Guijun

2018-02-01

Monitoring total nitrogen content (TNC) in the soil of cultivated land quantitively and mastering its spatial distribution are helpful for crop growing, soil fertility adjustment and sustainable development of agriculture. The study aimed to develop a universal method to map total nitrogen content in soil of cultivated land by HSI image at county scale. Several mathematical transformations were used to improve the expression ability of HSI image. The correlations between soil TNC and the reflectivity and its mathematical transformations were analyzed. Then the susceptible bands and its transformations were screened to develop the optimizing model of map soil TNC in the Anping County based on the method of multiple linear regression. Results showed that the bands of 14th, 16th, 19th, 37th and 60th with different mathematical transformations were screened as susceptible bands. Differential transformation was helpful for reducing the noise interference to the diagnosis ability of the target spectrum. The determination coefficient of the first order differential of logarithmic transformation was biggest (0.505), while the RMSE was lowest. The study confirmed the first order differential of logarithm transformation as the optimal inversion model for soil TNC, which was used to map soil TNC of cultivated land in the study area.

Propagation of arbitrary initial wave packets in a quantum parametric oscillator: Instability zones for higher order moments

NASA Astrophysics Data System (ADS)

Biswas, Subhadip; Chattopadhyay, Rohitashwa; Bhattacharjee, Jayanta K.

2018-05-01

We consider the dynamics of a particle in a parametric oscillator with a view to exploring any quantum feature of the initial wave packet that shows divergent (in time) behaviour for parameter values where the classical motion dynamics of the mean position is bounded. We use Ehrenfest's theorem to explore the dynamics of nth order moment which reduces exactly to a linear non autonomous differential equation of order n + 1. It is found that while the width and skewness of the packet is unbounded exactly in the zones where the classical motion is unbounded, the kurtosis of an initially non-gaussian wave packet can become infinitely large in certain additional zones. This implies that the shape of the wave packet can change drastically with time in these zones.

Covariance and the hierarchy of frame bundles

NASA Technical Reports Server (NTRS)

Estabrook, Frank B.

1987-01-01

This is an essay on the general concept of covariance, and its connection with the structure of the nested set of higher frame bundles over a differentiable manifold. Examples of covariant geometric objects include not only linear tensor fields, densities and forms, but affinity fields, sectors and sector forms, higher order frame fields, etc., often having nonlinear transformation rules and Lie derivatives. The intrinsic, or invariant, sets of forms that arise on frame bundles satisfy the graded Cartan-Maurer structure equations of an infinite Lie algebra. Reduction of these gives invariant structure equations for Lie pseudogroups, and for G-structures of various orders. Some new results are introduced for prolongation of structure equations, and for treatment of Riemannian geometry with higher-order moving frames. The use of invariant form equations for nonlinear field physics is implicitly advocated.

Bayesian parameter estimation for nonlinear modelling of biological pathways.

PubMed

Ghasemi, Omid; Lindsey, Merry L; Yang, Tianyi; Nguyen, Nguyen; Huang, Yufei; Jin, Yu-Fang

2011-01-01

The availability of temporal measurements on biological experiments has significantly promoted research areas in systems biology. To gain insight into the interaction and regulation of biological systems, mathematical frameworks such as ordinary differential equations have been widely applied to model biological pathways and interpret the temporal data. Hill equations are the preferred formats to represent the reaction rate in differential equation frameworks, due to their simple structures and their capabilities for easy fitting to saturated experimental measurements. However, Hill equations are highly nonlinearly parameterized functions, and parameters in these functions cannot be measured easily. Additionally, because of its high nonlinearity, adaptive parameter estimation algorithms developed for linear parameterized differential equations cannot be applied. Therefore, parameter estimation in nonlinearly parameterized differential equation models for biological pathways is both challenging and rewarding. In this study, we propose a Bayesian parameter estimation algorithm to estimate parameters in nonlinear mathematical models for biological pathways using time series data. We used the Runge-Kutta method to transform differential equations to difference equations assuming a known structure of the differential equations. This transformation allowed us to generate predictions dependent on previous states and to apply a Bayesian approach, namely, the Markov chain Monte Carlo (MCMC) method. We applied this approach to the biological pathways involved in the left ventricle (LV) response to myocardial infarction (MI) and verified our algorithm by estimating two parameters in a Hill equation embedded in the nonlinear model. We further evaluated our estimation performance with different parameter settings and signal to noise ratios. Our results demonstrated the effectiveness of the algorithm for both linearly and nonlinearly parameterized dynamic systems. Our proposed Bayesian algorithm successfully estimated parameters in nonlinear mathematical models for biological pathways. This method can be further extended to high order systems and thus provides a useful tool to analyze biological dynamics and extract information using temporal data.

Optimal preview control for a linear continuous-time stochastic control system in finite-time horizon

NASA Astrophysics Data System (ADS)

Wu, Jiang; Liao, Fucheng; Tomizuka, Masayoshi

2017-01-01

This paper discusses the design of the optimal preview controller for a linear continuous-time stochastic control system in finite-time horizon, using the method of augmented error system. First, an assistant system is introduced for state shifting. Then, in order to overcome the difficulty of the state equation of the stochastic control system being unable to be differentiated because of Brownian motion, the integrator is introduced. Thus, the augmented error system which contains the integrator vector, control input, reference signal, error vector and state of the system is reconstructed. This leads to the tracking problem of the optimal preview control of the linear stochastic control system being transformed into the optimal output tracking problem of the augmented error system. With the method of dynamic programming in the theory of stochastic control, the optimal controller with previewable signals of the augmented error system being equal to the controller of the original system is obtained. Finally, numerical simulations show the effectiveness of the controller.

Turbulence closure for mixing length theories

NASA Astrophysics Data System (ADS)

Jermyn, Adam S.; Lesaffre, Pierre; Tout, Christopher A.; Chitre, Shashikumar M.

2018-05-01

We present an approach to turbulence closure based on mixing length theory with three-dimensional fluctuations against a two-dimensional background. This model is intended to be rapidly computable for implementation in stellar evolution software and to capture a wide range of relevant phenomena with just a single free parameter, namely the mixing length. We incorporate magnetic, rotational, baroclinic, and buoyancy effects exactly within the formalism of linear growth theories with non-linear decay. We treat differential rotation effects perturbatively in the corotating frame using a novel controlled approximation, which matches the time evolution of the reference frame to arbitrary order. We then implement this model in an efficient open source code and discuss the resulting turbulent stresses and transport coefficients. We demonstrate that this model exhibits convective, baroclinic, and shear instabilities as well as the magnetorotational instability. It also exhibits non-linear saturation behaviour, and we use this to extract the asymptotic scaling of various transport coefficients in physically interesting limits.

Numerical solution methods for viscoelastic orthotropic materials

NASA Technical Reports Server (NTRS)

Gramoll, K. C.; Dillard, D. A.; Brinson, H. F.

1988-01-01

Numerical solution methods for viscoelastic orthotropic materials, specifically fiber reinforced composite materials, are examined. The methods include classical lamination theory using time increments, direction solution of the Volterra Integral, Zienkiewicz's linear Prony series method, and a new method called Nonlinear Differential Equation Method (NDEM) which uses a nonlinear Prony series. The criteria used for comparison of the various methods include the stability of the solution technique, time step size stability, computer solution time length, and computer memory storage. The Volterra Integral allowed the implementation of higher order solution techniques but had difficulties solving singular and weakly singular compliance function. The Zienkiewicz solution technique, which requires the viscoelastic response to be modeled by a Prony series, works well for linear viscoelastic isotropic materials and small time steps. The new method, NDEM, uses a modified Prony series which allows nonlinear stress effects to be included and can be used with orthotropic nonlinear viscoelastic materials. The NDEM technique is shown to be accurate and stable for both linear and nonlinear conditions with minimal computer time.

A macroscopic plasma Lagrangian and its application to wave interactions and resonances

NASA Technical Reports Server (NTRS)

Peng, Y. K. M.

1974-01-01

The derivation of a macroscopic plasma Lagrangian is considered, along with its application to the description of nonlinear three-wave interaction in a homogeneous plasma and linear resonance oscillations in a inhomogeneous plasma. One approach to obtain the Lagrangian is via the inverse problem of the calculus of variations for arbitrary first and second order quasilinear partial differential systems. Necessary and sufficient conditions for the given equations to be Euler-Lagrange equations of a Lagrangian are obtained. These conditions are then used to determine the transformations that convert some classes of non-Euler-Lagrange equations to Euler-Lagrange equation form. The Lagrangians for a linear resistive transmission line and a linear warm collisional plasma are derived as examples. Using energy considerations, the correct macroscopic plasma Lagrangian is shown to differ from the velocity-integrated low Lagrangian by a macroscopic potential energy that equals twice the particle thermal kinetic energy plus the energy lost by heat conduction.

Chemical networks with inflows and outflows: a positive linear differential inclusions approach.

PubMed

Angeli, David; De Leenheer, Patrick; Sontag, Eduardo D

2009-01-01

Certain mass-action kinetics models of biochemical reaction networks, although described by nonlinear differential equations, may be partially viewed as state-dependent linear time-varying systems, which in turn may be modeled by convex compact valued positive linear differential inclusions. A result is provided on asymptotic stability of such inclusions, and applied to a ubiquitous biochemical reaction network with inflows and outflows, known as the futile cycle. We also provide a characterization of exponential stability of general homogeneous switched systems which is not only of interest in itself, but also plays a role in the analysis of the futile cycle. 2009 American Institute of Chemical Engineers

Approximate analytical solutions of a pair of coupled anharmonic oscillators

NASA Astrophysics Data System (ADS)

Alam, Nasir; Mandal, Swapan; Öhberg, Patrik

2015-02-01

The Hamiltonian and the corresponding equations of motion involving the field operators of two quartic anharmonic oscillators indirectly coupled via a linear oscillator are constructed. The approximate analytical solutions of the coupled differential equations involving the non-commuting field operators are solved up to the second order in the anharmonic coupling. In the absence of nonlinearity these solutions are used to calculate the second order variances and hence the squeezing in pure and in mixed modes. The higher order quadrature squeezing and the amplitude squared squeezing of various field modes are also investigated where the squeezing in pure and in mixed modes are found to be suppressed. Moreover, the absence of a nonlinearity prohibits the higher order quadrature and higher ordered amplitude squeezing of the input coherent states. It is established that the mere coupling of two oscillators through a third one is unable to produce any squeezing effects of input coherent light, but the presence of a nonlinear interaction may provide squeezed states and other nonclassical phenomena.

Magnetic Field Generation Processes Involving Gravity and Differential Rotation. Solitary Plasma Rings Formation around Black Holes

NASA Astrophysics Data System (ADS)

Coppi, Bruno

2012-10-01

A clear theoretical framework to describe how magnetic fields are generated and amplified is provided by the magneto-gravitational modes that involve both differential rotation and gravity and for which other factors such as temperature gradients can contribute to their excitation. These modes are shown to be important for the evolution of plasma disks surrounding black holes.footnotetextB. Coppi, Phys. Plasmas 18, 032901 (2011) Non-linear and axi-symmetric plasmas and associated field configurations are found under stationary conditions that do not involve the presence of a pre-existing ``seed'' magnetic field unlike other configurations found previously.footnotetextIbid. The relevant magnetic energy density is of the order of the gravitationally confined plasma pressure. The solitary plasma rings that characterize these configurations are localized radially over regions with vanishing differential rotation and can be envisioned as the saturated state of magneto-gravitational modes. The ``source'' of these configurations is the combination of the gravitational force and of the plasma density gradient orthogonal to it.

Quantity Competition in a Differentiated Duopoly

NASA Astrophysics Data System (ADS)

Ferreira, Fernanda A.; Ferreira, Flávio; Ferreira, Miguel; Pinto, Alberto A.

In this paper, we consider a Stackelberg duopoly competition with differentiated goods, linear and symmetric demand and with unknown costs. In our model, the two firms play a non-cooperative game with two stages: in a first stage, firm F 1 chooses the quantity, q 1, that is going to produce; in the second stage, firm F 2 observes the quantity q 1 produced by firm F 1 and chooses its own quantity q 2. Firms choose their output levels in order to maximise their profits. We suppose that each firm has two different technologies, and uses one of them following a certain probability distribution. The use of either one or the other technology affects the unitary production cost. We show that there is exactly one perfect Bayesian equilibrium for this game. We analyse the variations of the expected profits with the parameters of the model, namely with the parameters of the probability distributions, and with the parameters of the demand and differentiation.

NUCLEAR REACTOR AS THE OBJECT OF CONTROL. AUTOMATIC CONTROL OF AIRCRAFT ENGINES . B.S. Voronkev Collection of Articles

DOE Office of Scientific and Technical Information (OSTI.GOV)

None

BS> The dynamics of a power reactor is treated in some detail. Although the reactor is described by a nonlinear differential equation of the seventh order, a two-group approximstion with prompt neutrons and one averaged group of delayed neutrons may be used. When the reactor is in equilibrium, the reactor equation may be linearized in two ways. The effects of positive and negative coefficients of tins of the reactor are discussed. The nonlinear character of the control rods is trested. (D.L.C.)

Iterative algorithms for computing the feedback Nash equilibrium point for positive systems

NASA Astrophysics Data System (ADS)

Ivanov, I.; Imsland, Lars; Bogdanova, B.

2017-03-01

The paper studies N-player linear quadratic differential games on an infinite time horizon with deterministic feedback information structure. It introduces two iterative methods (the Newton method as well as its accelerated modification) in order to compute the stabilising solution of a set of generalised algebraic Riccati equations. The latter is related to the Nash equilibrium point of the considered game model. Moreover, we derive the sufficient conditions for convergence of the proposed methods. Finally, we discuss two numerical examples so as to illustrate the performance of both of the algorithms.

Mapping the Critical Gestational Age at Birth that Alters Brain Development in Preterm-born Infants using Multi-Modal MRI

PubMed Central

Wu, Dan; Chang, Linda; Akazawa, Kentaro; Oishi, Kumiko; Skranes, Jon; Ernst, Thomas; Oishi, Kenichi

2017-01-01

Preterm birth adversely affects postnatal brain development. In order to investigate the critical gestational age at birth (GAB) that alters the developmental trajectory of gray and white matter structures in the brain, we investigated diffusion tensor and quantitative T2 mapping data in 43 term-born and 43 preterm-born infants. A novel multivariate linear model—the change point model, was applied to detect change points in fractional anisotropy, mean diffusivity, and T2 relaxation time. Change points captured the “critical” GAB value associated with a change in the linear relation between GAB and MRI measures. The analysis was performed in 126 regions across the whole brain using an atlas-based image quantification approach to investigate the spatial pattern of the critical GAB. Our results demonstrate that the critical GABs are region- and modality-specific, generally following a central-to-peripheral and bottom-to-top order of structural development. This study may offer unique insights into the postnatal neurological development associated with differential degrees of preterm birth. PMID:28111189

Mapping the critical gestational age at birth that alters brain development in preterm-born infants using multi-modal MRI.

PubMed

Wu, Dan; Chang, Linda; Akazawa, Kentaro; Oishi, Kumiko; Skranes, Jon; Ernst, Thomas; Oishi, Kenichi

2017-04-01

Preterm birth adversely affects postnatal brain development. In order to investigate the critical gestational age at birth (GAB) that alters the developmental trajectory of gray and white matter structures in the brain, we investigated diffusion tensor and quantitative T2 mapping data in 43 term-born and 43 preterm-born infants. A novel multivariate linear model-the change point model, was applied to detect change points in fractional anisotropy, mean diffusivity, and T2 relaxation time. Change points captured the "critical" GAB value associated with a change in the linear relation between GAB and MRI measures. The analysis was performed in 126 regions across the whole brain using an atlas-based image quantification approach to investigate the spatial pattern of the critical GAB. Our results demonstrate that the critical GABs are region- and modality-specific, generally following a central-to-peripheral and bottom-to-top order of structural development. This study may offer unique insights into the postnatal neurological development associated with differential degrees of preterm birth. Copyright © 2017 The Authors. Published by Elsevier Inc. All rights reserved.

Global Mittag-Leffler stability and synchronization analysis of fractional-order quaternion-valued neural networks with linear threshold neurons.

PubMed

Yang, Xujun; Li, Chuandong; Song, Qiankun; Chen, Jiyang; Huang, Junjian

2018-05-04

This paper talks about the stability and synchronization problems of fractional-order quaternion-valued neural networks (FQVNNs) with linear threshold neurons. On account of the non-commutativity of quaternion multiplication resulting from Hamilton rules, the FQVNN models are separated into four real-valued neural network (RVNN) models. Consequently, the dynamic analysis of FQVNNs can be realized by investigating the real-valued ones. Based on the method of M-matrix, the existence and uniqueness of the equilibrium point of the FQVNNs are obtained without detailed proof. Afterwards, several sufficient criteria ensuring the global Mittag-Leffler stability for the unique equilibrium point of the FQVNNs are derived by applying the Lyapunov direct method, the theory of fractional differential equation, the theory of matrix eigenvalue, and some inequality techniques. In the meanwhile, global Mittag-Leffler synchronization for the drive-response models of the addressed FQVNNs are investigated explicitly. Finally, simulation examples are designed to verify the feasibility and availability of the theoretical results. Copyright © 2018 Elsevier Ltd. All rights reserved.

Second order accurate finite difference approximations for the transonic small disturbance equation and the full potential equation

NASA Technical Reports Server (NTRS)

Mostrel, M. M.

1988-01-01

New shock-capturing finite difference approximations for solving two scalar conservation law nonlinear partial differential equations describing inviscid, isentropic, compressible flows of aerodynamics at transonic speeds are presented. A global linear stability theorem is applied to these schemes in order to derive a necessary and sufficient condition for the finite element method. A technique is proposed to render the described approximations total variation-stable by applying the flux limiters to the nonlinear terms of the difference equation dimension by dimension. An entropy theorem applying to the approximations is proved, and an implicit, forward Euler-type time discretization of the approximation is presented. Results of some numerical experiments using the approximations are reported.

Finite-time synchronization for second-order nonlinear multi-agent system via pinning exponent sliding mode control.

PubMed

Hou, Huazhou; Zhang, Qingling

2016-11-01

In this paper we investigate the finite-time synchronization for second-order multi-agent system via pinning exponent sliding mode control. Firstly, for the nonlinear multi-agent system, differential mean value theorem is employed to transfer the nonlinear system into linear system, then, by pinning only one node in the system with novel exponent sliding mode control, we can achieve synchronization in finite time. Secondly, considering the 3-DOF helicopter system with nonlinear dynamics and disturbances, the novel exponent sliding mode control protocol is applied to only one node to achieve the synchronization. Finally, the simulation results show the effectiveness and the advantages of the proposed method. Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Symmetry groups of integro-differential equations for linear thermoviscoelastic materials with memory

NASA Astrophysics Data System (ADS)

Zhou, L.-Q.; Meleshko, S. V.

2017-07-01

The group analysis method is applied to a system of integro-differential equations corresponding to a linear thermoviscoelastic model. A recently developed approach for calculating the symmetry groups of such equations is used. The general solution of the determining equations for the system is obtained. Using subalgebras of the admitted Lie algebra, two classes of partially invariant solutions of the considered system of integro-differential equations are studied.

Stochastic differential equation model for linear growth birth and death processes with immigration and emigration

DOE Office of Scientific and Technical Information (OSTI.GOV)

Granita, E-mail: granitafc@gmail.com; Bahar, A.

This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found.

Risk management algorithm for rear-side collision avoidance using a combined steering torque overlay and differential braking

NASA Astrophysics Data System (ADS)

Lee, Junyung; Yi, Kyongsu; Yoo, Hyunjae; Chong, Hyokjin; Ko, Bongchul

2015-06-01

This paper describes a risk management algorithm for rear-side collision avoidance. The proposed risk management algorithm consists of a supervisor and a coordinator. The supervisor is designed to monitor collision risks between the subject vehicle and approaching vehicle in the adjacent lane. An appropriate criterion of intervention, which satisfies high acceptance to drivers through the consideration of a realistic traffic, has been determined based on the analysis of the kinematics of the vehicles in longitudinal and lateral directions. In order to assist the driver actively and increase driver's safety, a coordinator is designed to combine lateral control using a steering torque overlay by motor-driven power steering and differential braking by vehicle stability control. In order to prevent the collision while limiting actuator's control inputs and vehicle dynamics to safe values for the assurance of the driver's comfort, the Lyapunov theory and linear matrix inequalities based optimisation methods have been used. The proposed risk management algorithm has been evaluated via simulation using CarSim and MATLAB/Simulink.

Coupled bending-torsion steady-state response of pretwisted, nonuniform rotating beams using a transfer-matrix method

NASA Technical Reports Server (NTRS)

Gray, Carl E., Jr.

1988-01-01

Using the Newtonian method, the equations of motion are developed for the coupled bending-torsion steady-state response of beams rotating at constant angular velocity in a fixed plane. The resulting equations are valid to first order strain-displacement relationships for a long beam with all other nonlinear terms retained. In addition, the equations are valid for beams with the mass centroidal axis offset (eccentric) from the elastic axis, nonuniform mass and section properties, and variable twist. The solution of these coupled, nonlinear, nonhomogeneous, differential equations is obtained by modifying a Hunter linear second-order transfer-matrix solution procedure to solve the nonlinear differential equations and programming the solution for a desk-top personal computer. The modified transfer-matrix method was verified by comparing the solution for a rotating beam with a geometric, nonlinear, finite-element computer code solution; and for a simple rotating beam problem, the modified method demonstrated a significant advantage over the finite-element solution in accuracy, ease of solution, and actual computer processing time required to effect a solution.

The use of x-ray interferometry to investigate the linearity of the NPL Differential Plane Mirror Optical Interferometer

NASA Astrophysics Data System (ADS)

Yacoot, Andrew; Downs, Michael J.

2000-08-01

The x-ray interferometer from the combined optical and x-ray interferometer (COXI) facility at NPL has been used to investigate the performance of the NPL Jamin Differential Plane Mirror Interferometer when it is fitted with stabilized and unstabilized lasers. This Jamin interferometer employs a common path design using a double pass configuration and one fringe is realized by a displacement of 158 nm between its two plane mirror retroreflectors. Displacements over ranges of several optical fringes were measured simultaneously using the COXI x-ray interferometer and the Jamin interferometer and the results were compared. In order to realize the highest measurement accuracy from the Jamin interferometer, the air paths were shielded to prevent effects from air turbulence and electrical signals generated by the photodetectors were analysed and corrected using an optimizing routine in order to subdivide the optical fringes accurately. When an unstabilized laser was used the maximum peak-to-peak difference between the two interferometers was 80 pm, compared with 20 pm when the stabilized laser was used.

Mathematics for generative processes: Living and non-living systems

NASA Astrophysics Data System (ADS)

Giannantoni, Corrado

2006-05-01

The traditional Differential Calculus often shows its limits when describing living systems. These in fact present such a richness of characteristics that are, in the majority of cases, much wider than the description capabilities of the usual differential equations. Such an aspect became particularly evident during the research (completed in 2001) for an appropriate formulation of Odum's Maximum Em-Power Principle (proposed by the Author as a possible Fourth Thermodynamic Principle). In fact, in such a context, the particular non-conservative Algebra, adopted to account for both Quality and quantity of generative processes, suggested we introduce a faithfully corresponding concept of "derivative" (of both integer and fractional order) to describe dynamic conditions however variable. The new concept not only succeeded in pointing out the corresponding differential bases of all the rules of Emergy Algebra, but also represented the preferential guide in order to recognize the most profound physical nature of the basic processes which mostly characterize self-organizing Systems (co-production, co-injection, inter-action, feed-back, splits, etc.).From a mathematical point of view, the most important novelties introduced by such a new approach are: (i) the derivative of any integer or fractional order can be obtained independently from the evaluation of its lower order derivatives; (ii) the exponential function plays an extremely hinge role, much more marked than in the case of traditional differential equations; (iii) wide classes of differential equations, traditionally considered as being non-linear, become "intrinsically" linear when reconsidered in terms of "incipient" derivatives; (iv) their corresponding explicit solutions can be given in terms of new classes of functions (such as "binary" and "duet" functions); (v) every solution shows a sort of "persistence of form" when representing the product generated with respect to the agents of the generating process; (iv) and, at the same time, an intrinsic "genetic" ordinality which reflects the fact that any product "generated" is something more than the sum of the generating elements. Consequently all these properties enable us to follow the evolution of the "product" of any generative process from the very beginning, in its "rising", in its "incipient" act of being born. This is why the new "operator" introduced, specifically apt when describing the above-mentioned aspects, was termed as "incipient" (or "spring") derivative.In addition, even if the considered approach was suggested by the analysis of self-organizing living Systems, some specific examples of non-living Systems will also be mentioned. In fact, what is much more surprising is that such an approach is even more valid (than the traditional one) to describe non-living Systems too. In fact the resulting "drift" between traditional solutions and "incipient" solutions led us to reconsider the phenomenon of Mercury's precessions. The satisfactory agreement with the astronomical data suggested, as a consequential hypothesis, a different interpretation of its physical origin, substantially based on the Maximum Em-Power Principle.

Topographical scattering of gravity waves

NASA Astrophysics Data System (ADS)

Miles, J. W.; Chamberlain, P. G.

1998-04-01

A systematic hierarchy of partial differential equations for linear gravity waves in water of variable depth is developed through the expansion of the average Lagrangian in powers of [mid R:][nabla del, Hamilton operator][mid R:] (h=depth, [nabla del, Hamilton operator]h=slope). The first and second members of this hierarchy, the Helmholtz and conventional mild-slope equations, are second order. The third member is fourth order but may be approximated by Chamberlain & Porter's (1995) ‘modified mild-slope’ equation, which is second order and comprises terms in [nabla del, Hamilton operator]2h and ([nabla del, Hamilton operator]h)2 that are absent from the mild-slope equation. Approximate solutions of the mild-slope and modified mild-slope equations for topographical scattering are determined through an iterative sequence, starting from a geometrical-optics approximation (which neglects reflection), then a quasi-geometrical-optics approximation, and on to higher-order results. The resulting reflection coefficient for a ramp of uniform slope is compared with the results of numerical integrations of each of the mild-slope equation (Booij 1983), the modified mild-slope equation (Porter & Staziker 1995), and the full linear equations (Booij 1983). Also considered is a sequence of sinusoidal sandbars, for which Bragg resonance may yield rather strong reflection and for which the modified mild-slope approximation is in close agreement with Mei's (1985) asymptotic approximation.

Pseudo-second order models for the adsorption of safranin onto activated carbon: comparison of linear and non-linear regression methods.

PubMed

Kumar, K Vasanth

2007-04-02

Kinetic experiments were carried out for the sorption of safranin onto activated carbon particles. The kinetic data were fitted to pseudo-second order model of Ho, Sobkowsk and Czerwinski, Blanchard et al. and Ritchie by linear and non-linear regression methods. Non-linear method was found to be a better way of obtaining the parameters involved in the second order rate kinetic expressions. Both linear and non-linear regression showed that the Sobkowsk and Czerwinski and Ritchie's pseudo-second order models were the same. Non-linear regression analysis showed that both Blanchard et al. and Ho have similar ideas on the pseudo-second order model but with different assumptions. The best fit of experimental data in Ho's pseudo-second order expression by linear and non-linear regression method showed that Ho pseudo-second order model was a better kinetic expression when compared to other pseudo-second order kinetic expressions.

Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE's. [ordinary differential equations

NASA Technical Reports Server (NTRS)

Geddes, K. O.

1977-01-01

If a linear ordinary differential equation with polynomial coefficients is converted into integrated form then the formal substitution of a Chebyshev series leads to recurrence equations defining the Chebyshev coefficients of the solution function. An explicit formula is presented for the polynomial coefficients of the integrated form in terms of the polynomial coefficients of the differential form. The symmetries arising from multiplication and integration of Chebyshev polynomials are exploited in deriving a general recurrence equation from which can be derived all of the linear equations defining the Chebyshev coefficients. Procedures for deriving the general recurrence equation are specified in a precise algorithmic notation suitable for translation into any of the languages for symbolic computation. The method is algebraic and it can therefore be applied to differential equations containing indeterminates.

Application of higher-order cepstral techniques in problems of fetal heart signal extraction

NASA Astrophysics Data System (ADS)

Sabry-Rizk, Madiha; Zgallai, Walid; Hardiman, P.; O'Riordan, J.

1996-10-01

Recently, cepstral analysis based on second order statistics and homomorphic filtering techniques have been used in the adaptive decomposition of overlapping, or otherwise, and noise contaminated ECG complexes of mothers and fetals obtained by a transabdominal surface electrodes connected to a monitoring instrument, an interface card, and a PC. Differential time delays of fetal heart beats measured from a reference point located on the mother complex after transformation to cepstra domains are first obtained and this is followed by fetal heart rate variability computations. Homomorphic filtering in the complex cepstral domain and the subuent transformation to the time domain results in fetal complex recovery. However, three problems have been identified with second-order based cepstral techniques that needed rectification in this paper. These are (1) errors resulting from the phase unwrapping algorithms and leading to fetal complex perturbation, (2) the unavoidable conversion of noise statistics from Gaussianess to non-Gaussianess due to the highly non-linear nature of homomorphic transform does warrant stringent noise cancellation routines, (3) due to the aforementioned problems in (1) and (2), it is difficult to adaptively optimize windows to include all individual fetal complexes in the time domain based on amplitude thresholding routines in the complex cepstral domain (i.e. the task of `zooming' in on weak fetal complexes requires more processing time). The use of third-order based high resolution differential cepstrum technique results in recovery of the delay of the order of 120 milliseconds.

Semi-Analytic Reconstruction of Flux in Finite Volume Formulations

NASA Technical Reports Server (NTRS)

Gnoffo, Peter A.

2006-01-01

Semi-analytic reconstruction uses the analytic solution to a second-order, steady, ordinary differential equation (ODE) to simultaneously evaluate the convective and diffusive flux at all interfaces of a finite volume formulation. The second-order ODE is itself a linearized approximation to the governing first- and second- order partial differential equation conservation laws. Thus, semi-analytic reconstruction defines a family of formulations for finite volume interface fluxes using analytic solutions to approximating equations. Limiters are not applied in a conventional sense; rather, diffusivity is adjusted in the vicinity of changes in sign of eigenvalues in order to achieve a sufficiently small cell Reynolds number in the analytic formulation across critical points. Several approaches for application of semi-analytic reconstruction for the solution of one-dimensional scalar equations are introduced. Results are compared with exact analytic solutions to Burger s Equation as well as a conventional, upwind discretization using Roe s method. One approach, the end-point wave speed (EPWS) approximation, is further developed for more complex applications. One-dimensional vector equations are tested on a quasi one-dimensional nozzle application. The EPWS algorithm has a more compact difference stencil than Roe s algorithm but reconstruction time is approximately a factor of four larger than for Roe. Though both are second-order accurate schemes, Roe s method approaches a grid converged solution with fewer grid points. Reconstruction of flux in the context of multi-dimensional, vector conservation laws including effects of thermochemical nonequilibrium in the Navier-Stokes equations is developed.

A simple smoothness indicator for the WENO scheme with adaptive order

NASA Astrophysics Data System (ADS)

Huang, Cong; Chen, Li Li

2018-01-01

The fifth order WENO scheme with adaptive order is competent for solving hyperbolic conservation laws, its reconstruction is a convex combination of a fifth order linear reconstruction and three third order linear reconstructions. Note that, on uniform mesh, the computational cost of smoothness indicator for fifth order linear reconstruction is comparable with the sum of ones for three third order linear reconstructions, thus it is too heavy; on non-uniform mesh, the explicit form of smoothness indicator for fifth order linear reconstruction is difficult to be obtained, and its computational cost is much heavier than the one on uniform mesh. In order to overcome these problems, a simple smoothness indicator for fifth order linear reconstruction is proposed in this paper.

Modified Chebyshev Picard Iteration for Efficient Numerical Integration of Ordinary Differential Equations

NASA Astrophysics Data System (ADS)

Macomber, B.; Woollands, R. M.; Probe, A.; Younes, A.; Bai, X.; Junkins, J.

2013-09-01

Modified Chebyshev Picard Iteration (MCPI) is an iterative numerical method for approximating solutions of linear or non-linear Ordinary Differential Equations (ODEs) to obtain time histories of system state trajectories. Unlike other step-by-step differential equation solvers, the Runge-Kutta family of numerical integrators for example, MCPI approximates long arcs of the state trajectory with an iterative path approximation approach, and is ideally suited to parallel computation. Orthogonal Chebyshev Polynomials are used as basis functions during each path iteration; the integrations of the Picard iteration are then done analytically. Due to the orthogonality of the Chebyshev basis functions, the least square approximations are computed without matrix inversion; the coefficients are computed robustly from discrete inner products. As a consequence of discrete sampling and weighting adopted for the inner product definition, Runge phenomena errors are minimized near the ends of the approximation intervals. The MCPI algorithm utilizes a vector-matrix framework for computational efficiency. Additionally, all Chebyshev coefficients and integrand function evaluations are independent, meaning they can be simultaneously computed in parallel for further decreased computational cost. Over an order of magnitude speedup from traditional methods is achieved in serial processing, and an additional order of magnitude is achievable in parallel architectures. This paper presents a new MCPI library, a modular toolset designed to allow MCPI to be easily applied to a wide variety of ODE systems. Library users will not have to concern themselves with the underlying mathematics behind the MCPI method. Inputs are the boundary conditions of the dynamical system, the integrand function governing system behavior, and the desired time interval of integration, and the output is a time history of the system states over the interval of interest. Examples from the field of astrodynamics are presented to compare the output from the MCPI library to current state-of-practice numerical integration methods. It is shown that MCPI is capable of out-performing the state-of-practice in terms of computational cost and accuracy.

On Impedance Spectroscopy of Supercapacitors

NASA Astrophysics Data System (ADS)

Uchaikin, V. V.; Sibatov, R. T.; Ambrozevich, A. S.

2016-10-01

Supercapacitors are often characterized by responses measured by methods of impedance spectroscopy. In the frequency domain these responses have the form of power-law functions or their linear combinations. The inverse Fourier transform leads to relaxation equations with integro-differential operators of fractional order under assumption that the frequency response is independent of the working voltage. To compare long-term relaxation kinetics predicted by these equations with the observed one, charging-discharging of supercapacitors (with nominal capacitances of 0.22, 0.47, and 1.0 F) have been studied by means of registration of the current response to a step voltage signal. It is established that the reaction of devices under study to variations of the charging regime disagrees with the model of a homogeneous linear response. It is demonstrated that relaxation is well described by a fractional stretched exponent.

Spillover, nonlinearity, and flexible structures

NASA Technical Reports Server (NTRS)

Bass, Robert W.; Zes, Dean

1991-01-01

Many systems whose evolution in time is governed by Partial Differential Equations (PDEs) are linearized around a known equilibrium before Computer Aided Control Engineering (CACE) is considered. In this case, there are infinitely many independent vibrational modes, and it is intuitively evident on physical grounds that infinitely many actuators would be needed in order to control all modes. A more precise, general formulation of this grave difficulty (spillover problem) is due to A.V. Balakrishnan. A possible route to circumvention of this difficulty lies in leaving the PDE in its original nonlinear form, and adding the essentially finite dimensional control action prior to linearization. One possibly applicable technique is the Liapunov Schmidt rigorous reduction of singular infinite dimensional implicit function problems to finite dimensional implicit function problems. Omitting details of Banach space rigor, the formalities of this approach are given.

Classifying bilinear differential equations by linear superposition principle

NASA Astrophysics Data System (ADS)

Zhang, Lijun; Khalique, Chaudry Masood; Ma, Wen-Xiu

2016-09-01

In this paper, we investigate the linear superposition principle of exponential traveling waves to construct a sub-class of N-wave solutions of Hirota bilinear equations. A necessary and sufficient condition for Hirota bilinear equations possessing this specific sub-class of N-wave solutions is presented. We apply this result to find N-wave solutions to the (2+1)-dimensional KP equation, a (3+1)-dimensional generalized Kadomtsev-Petviashvili (KP) equation, a (3+1)-dimensional generalized BKP equation and the (2+1)-dimensional BKP equation. The inverse question, i.e., constructing Hirota Bilinear equations possessing N-wave solutions, is considered and a refined 3-step algorithm is proposed. As examples, we construct two very general kinds of Hirota bilinear equations of order 4 possessing N-wave solutions among which one satisfies dispersion relation and another does not satisfy dispersion relation.

LEGO - A Class Library for Accelerator Design and Simulation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cai, Yunhai

1998-11-19

An object-oriented class library of accelerator design and simulation is designed and implemented in a simple and modular fashion. All physics of single-particle dynamics is implemented based on the Hamiltonian in the local frame of the component. Symplectic integrators are used to approximate the integration of the Hamiltonian. A differential algebra class is introduced to extract a Taylor map up to arbitrary order. Analysis of optics is done in the same way both for the linear and non-linear cases. Recently, Monte Carlo simulation of synchrotron radiation has been added into the library. The code is used to design and simulatemore » the lattices of the PEP-II and SPEAR3. And it is also used for the commissioning of the PEP-II. Some examples of how to use the library will be given.« less

Mueller-matrix mapping of optically anisotropic fluorophores of biological tissues in the diagnosis of cancer

NASA Astrophysics Data System (ADS)

Ushenko, Yu A.; Sidor, M. I.; Bodnar, G. B.; Koval', G. D.

2014-08-01

We report the results of studying the polarisation manifestations of laser autofluorescence of optically anisotropic structures in biological tissues. A Mueller-matrix model is proposed to describe their complex anisotropy (linear and circular birefringence, linear and circular dichroism). The relationship is established between the mechanisms of optical anisotropy and polarisation manifestations of laser autofluorescence of histological sections of rectal tissue biopsy in different spectral regions. The ranges of changes in the statistical moments of the 1st-to-4th orders, which describe the distribution of the azimuth-invariant elements of Mueller matrices of rectal tissue autofluorescence, are found. Effectiveness of laser autofluorescence polarimetry is determined and the histological sections of biopsy of benign (polyp) and malignant (adenocarcinoma) tumours of the rectal wall are differentiated for the first time.

Ultra-thin nanocrystalline diamond membranes as pressure sensors for harsh environments

DOE Office of Scientific and Technical Information (OSTI.GOV)

Janssens, S. D., E-mail: stoffel.d.janssens@gmail.com; Haenen, K., E-mail: ken.haenen@uhasselt.be; IMOMEC, IMEC vzw, Wetenschapspark 1, B-3590 Diepenbeek

2014-02-17

Glass and diamond are suitable materials for harsh environments. Here, a procedure for fabricating ultra-thin nanocrystalline diamond membranes on glass, acting as an electrically insulating substrate, is presented. In order to investigate the pressure sensing properties of such membranes, a circular, highly conductive boron-doped nanocrystalline diamond membrane with a resistivity of 38 mΩ cm, a thickness of 150 nm, and a diameter of 555 μm is fabricated in the middle of a Hall bar structure. During the application of a positive differential pressure under the membrane (0–0.7 bar), four point piezoresistive effect measurements are performed. From these measurements, it can be concluded that the resistancemore » response of the membrane, as a function of differential pressure, is highly linear and sensitive.« less

The Exact Solution for Linear Thermoelastic Axisymmetric Deformations of Generally Laminated Circular Cylindrical Shells

NASA Technical Reports Server (NTRS)

Nemeth, Michael P.; Schultz, Marc R.

2012-01-01

A detailed exact solution is presented for laminated-composite circular cylinders with general wall construction and that undergo axisymmetric deformations. The overall solution is formulated in a general, systematic way and is based on the solution of a single fourth-order, nonhomogeneous ordinary differential equation with constant coefficients in which the radial displacement is the dependent variable. Moreover, the effects of general anisotropy are included and positive-definiteness of the strain energy is used to define uniquely the form of the basis functions spanning the solution space of the ordinary differential equation. Loading conditions are considered that include axisymmetric edge loads, surface tractions, and temperature fields. Likewise, all possible axisymmetric boundary conditions are considered. Results are presented for five examples that demonstrate a wide range of behavior for specially orthotropic and fully anisotropic cylinders.

Differential Resonant Ring YIG Tuned Oscillator

NASA Technical Reports Server (NTRS)

Parrott, Ronald A.

2010-01-01

A differential SiGe oscillator circuit uses a resonant ring-oscillator topology in order to electronically tune the oscillator over multi-octave bandwidths. The oscillator s tuning is extremely linear, because the oscillator s frequency depends on the magnetic tuning of a YIG sphere, whose resonant frequency is equal to a fundamental constant times the DC magnetic field. This extremely simple circuit topology uses two coupling loops connecting a differential pair of SiGe bipolar transistors into a feedback configuration using a YIG tuned filter creating a closed-loop ring oscillator. SiGe device technology is used for this oscillator in order to keep the transistor s 1/f noise to an absolute minimum in order to achieve minimum RF phase noise. The single-end resonant ring oscillator currently has an advantage in fewer parts, but when the oscillation frequency is greater than 16 GHz, the package s parasitic behavior couples energy to the sphere and causes holes and poor phase noise performance. This is because the coupling to the YIG is extremely low, so that the oscillator operates at near the unloaded Q. With the differential resonant ring oscillator, the oscillation currents are just in the YIG coupling mechanisms. The phase noise is even better, and the physical size can be reduced to permit monolithic microwave integrated circuit oscillators. This invention is a YIG tuned oscillator circuit making use of a differential topology to simultaneously achieve an extremely broadband electronic tuning range and ultra-low phase noise. As a natural result of its differential circuit topology, all reactive elements, such as tuning stubs, which limit tuning bandwidth by contributing excessive open loop phase shift, have been eliminated. The differential oscillator s open-loop phase shift is associated with completely non-dispersive circuit elements such as the physical angle of the coupling loops, a differential loop crossover, and the high-frequency phase shift of the n-p-n transistors. At the input of the oscillator s feedback loop is a pair of differentially connected n-p-n SiGe transistors that provides extremely high gain, and because they are bulk-effect devices, extremely low 1/f noise (leading to ultralow RF phase noise). The 1/f corner frequency for n-p-n SiGe transistors is approximately 500 Hz. The RF energy from the transistor s collector output is connected directly to the top-coupling loop (the excitation loop) of a single-sphere YIG tuned filter. A uniform magnetic field to bias the YIG must be at a right angle to any vector associated with an RF current in a coupling loop in order for the precession to interact with the RF currents.

Optimal moving grids for time-dependent partial differential equations

NASA Technical Reports Server (NTRS)

Wathen, A. J.

1989-01-01

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of partial differential equation solutions in the least squares norm.

Linear transformation and oscillation criteria for Hamiltonian systems

NASA Astrophysics Data System (ADS)

Zheng, Zhaowen

2007-08-01

Using a linear transformation similar to the Kummer transformation, some new oscillation criteria for linear Hamiltonian systems are established. These results generalize and improve the oscillation criteria due to I.S. Kumari and S. Umanaheswaram [I. Sowjaya Kumari, S. Umanaheswaram, Oscillation criteria for linear matrix Hamiltonian systems, J. Differential Equations 165 (2000) 174-198], Q. Yang et al. [Q. Yang, R. Mathsen, S. Zhu, Oscillation theorems for self-adjoint matrix Hamiltonian systems, J. Differential Equations 190 (2003) 306-329], and S. Chen and Z. Zheng [Shaozhu Chen, Zhaowen Zheng, Oscillation criteria of Yan type for linear Hamiltonian systems, Comput. Math. Appl. 46 (2003) 855-862]. These criteria also unify many of known criteria in literature and simplify the proofs.

A Solution Space for a System of Null-State Partial Differential Equations: Part 3

NASA Astrophysics Data System (ADS)

Flores, Steven M.; Kleban, Peter

2015-01-01

This article is the third of four that completely and rigorously characterize a solution space for a homogeneous system of 2 N + 3 linear partial differential equations (PDEs) in 2 N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE κ ). The system comprises 2 N null-state equations and three conformal Ward identities that govern CFT correlation functions of 2 N one-leg boundary operators. In the first two articles (Flores and Kleban, in Commun Math Phys, arXiv:1212.2301, 2012; Commun Math Phys, arXiv:1404.0035, 2014), we use methods of analysis and linear algebra to prove that dim , with C N the Nth Catalan number. Extending these results, we prove in this article that dim and entirely consists of (real-valued) solutions constructed with the CFT Coulomb gas (contour integral) formalism. In order to prove this claim, we show that a certain set of C N such solutions is linearly independent. Because the formulas for these solutions are complicated, we prove linear independence indirectly. We use the linear injective map of Lemma 15 in Flores and Kleban (Commun Math Phys, arXiv:1212.2301, 2012) to send each solution of the mentioned set to a vector in , whose components we find as inner products of elements in a Temperley-Lieb algebra. We gather these vectors together as columns of a symmetric matrix, with the form of a meander matrix. If the determinant of this matrix does not vanish, then the set of C N Coulomb gas solutions is linearly independent. And if this determinant does vanish, then we construct an alternative set of C N Coulomb gas solutions and follow a similar procedure to show that this set is linearly independent. The latter situation is closely related to CFT minimal models. We emphasize that, although the system of PDEs arises in CFT in away that is typically non-rigorous, our treatment of this system here and in Flores and Kleban (Commun Math Phys, arXiv:1212.2301, 2012; Commun Math Phys, arXiv:1404.0035, 2014; Commun Math Phys, arXiv:1405.2747, 2014) is completely rigorous.

Modelling and Inverse-Modelling: Experiences with O.D.E. Linear Systems in Engineering Courses

ERIC Educational Resources Information Center

Martinez-Luaces, Victor

2009-01-01

In engineering careers courses, differential equations are widely used to solve problems concerned with modelling. In particular, ordinary differential equations (O.D.E.) linear systems appear regularly in Chemical Engineering, Food Technology Engineering and Environmental Engineering courses, due to the usefulness in modelling chemical kinetics,…

Analysis of high-speed rotating flow inside gas centrifuge casing

NASA Astrophysics Data System (ADS)

Pradhan, Sahadev, , Dr.

2017-10-01

The generalized analytical model for the radial boundary layer inside the gas centrifuge casing in which the inner cylinder is rotating at a constant angular velocity Ω_i while the outer one is stationary, is formulated for studying the secondary gas flow field due to wall thermal forcing, inflow/outflow of light gas along the boundaries, as well as due to the combination of the above two external forcing. The analytical model includes the sixth order differential equation for the radial boundary layer at the cylindrical curved surface in terms of master potential (χ) , which is derived from the equations of motion in an axisymmetric (r - z) plane. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is a solid-body rotation. Additional approximations in the analytical model include constant temperature in the base state (isothermal compressible Couette flow), high aspect ratio (length is large compared to the annular gap), high Reynolds number, but there is no limitation on the Mach number. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order in the radial direction for the generalized analytical equation) are obtained. The solutions for the secondary flow is determined in terms of these eigenvalues and eigenfunctions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations and found excellent agreement (with a difference of less than 15%) between the predictions of the analytical model and the DSMC simulations, provided the boundary conditions in the analytical model are accurately specified.

Analysis of high-speed rotating flow inside gas centrifuge casing

NASA Astrophysics Data System (ADS)

Pradhan, Sahadev, , Dr.

2017-09-01

The generalized analytical model for the radial boundary layer inside the gas centrifuge casing in which the inner cylinder is rotating at a constant angular velocity Ωi while the outer one is stationary, is formulated for studying the secondary gas flow field due to wall thermal forcing, inflow/outflow of light gas along the boundaries, as well as due to the combination of the above two external forcing. The analytical model includes the sixth order differential equation for the radial boundary layer at the cylindrical curved surface in terms of master potential (χ) , which is derived from the equations of motion in an axisymmetric (r - z) plane. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is a solid-body rotation. Additional approximations in the analytical model include constant temperature in the base state (isothermal compressible Couette flow), high aspect ratio (length is large compared to the annular gap), high Reynolds number, but there is no limitation on the Mach number. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order in the radial direction for the generalized analytical equation) are obtained. The solutions for the secondary flow is determined in terms of these eigenvalues and eigenfunctions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations and found excellent agreement (with a difference of less than 15%) between the predictions of the analytical model and the DSMC simulations, provided the boundary conditions in the analytical model are accurately specified.

Analysis of high-speed rotating flow inside gas centrifuge casing

NASA Astrophysics Data System (ADS)

Pradhan, Sahadev

2017-11-01

The generalized analytical model for the radial boundary layer inside the gas centrifuge casing in which the inner cylinder is rotating at a constant angular velocity Ωi while the outer one is stationary, is formulated for studying the secondary gas flow field due to wall thermal forcing, inflow/outflow of light gas along the boundaries, as well as due to the combination of the above two external forcing. The analytical model includes the sixth order differential equation for the radial boundary layer at the cylindrical curved surface in terms of master potential (χ) , which is derived from the equations of motion in an axisymmetric (r - z) plane. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is a solid-body rotation. Additional approximations in the analytical model include constant temperature in the base state (isothermal compressible Couette flow), high aspect ratio (length is large compared to the annular gap), high Reynolds number, but there is no limitation on the Mach number. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order in the radial direction for the generalized analytical equation) are obtained. The solutions for the secondary flow is determined in terms of these eigenvalues and eigenfunctions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations and found excellent agreement (with a difference of less than 15%) between the predictions of the analytical model and the DSMC simulations, provided the boundary conditions in the analytical model are accurately specified.

Perturbations of linear delay differential equations at the verge of instability.

PubMed

Lingala, N; Namachchivaya, N Sri

2016-06-01

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. This paper considers linear DDEs that are on the verge of instability, i.e., a pair of roots of the characteristic equation lies on the imaginary axis of the complex plane and all other roots have negative real parts. It is shown that when small noise perturbations are present, the probability distribution of the dynamics can be approximated by the probability distribution of a certain one-dimensional stochastic differential equation (SDE) without delay. This is advantageous because equations without delay are easier to simulate and one-dimensional SDEs are analytically tractable. When the perturbations are also linear, it is shown that the stability depends on a specific complex number. The theory is applied to study oscillators with delayed feedback. Some errors in other articles that use multiscale approach are pointed out.

Quasi-linear theory via the cumulant expansion approach

NASA Technical Reports Server (NTRS)

Jones, F. C.; Birmingham, T. J.

1974-01-01

The cumulant expansion technique of Kubo was used to derive an intergro-differential equation for f , the average one particle distribution function for particles being accelerated by electric and magnetic fluctuations of a general nature. For a very restricted class of fluctuations, the f equation degenerates exactly to a differential equation of Fokker-Planck type. Quasi-linear theory, including the adiabatic assumption, is an exact theory for this limited class of fluctuations. For more physically realistic fluctuations, however, quasi-linear theory is at best approximate.

Pseudo second order kinetics and pseudo isotherms for malachite green onto activated carbon: comparison of linear and non-linear regression methods.

PubMed

Kumar, K Vasanth; Sivanesan, S

2006-08-25

Pseudo second order kinetic expressions of Ho, Sobkowsk and Czerwinski, Blanachard et al. and Ritchie were fitted to the experimental kinetic data of malachite green onto activated carbon by non-linear and linear method. Non-linear method was found to be a better way of obtaining the parameters involved in the second order rate kinetic expressions. Both linear and non-linear regression showed that the Sobkowsk and Czerwinski and Ritchie's pseudo second order model were the same. Non-linear regression analysis showed that both Blanachard et al. and Ho have similar ideas on the pseudo second order model but with different assumptions. The best fit of experimental data in Ho's pseudo second order expression by linear and non-linear regression method showed that Ho pseudo second order model was a better kinetic expression when compared to other pseudo second order kinetic expressions. The amount of dye adsorbed at equilibrium, q(e), was predicted from Ho pseudo second order expression and were fitted to the Langmuir, Freundlich and Redlich Peterson expressions by both linear and non-linear method to obtain the pseudo isotherms. The best fitting pseudo isotherm was found to be the Langmuir and Redlich Peterson isotherm. Redlich Peterson is a special case of Langmuir when the constant g equals unity.

Social inequality, lifestyles and health - a non-linear canonical correlation analysis based on the approach of Pierre Bourdieu.

PubMed

Grosse Frie, Kirstin; Janssen, Christian

2009-01-01

Based on the theoretical and empirical approach of Pierre Bourdieu, a multivariate non-linear method is introduced as an alternative way to analyse the complex relationships between social determinants and health. The analysis is based on face-to-face interviews with 695 randomly selected respondents aged 30 to 59. Variables regarding socio-economic status, life circumstances, lifestyles, health-related behaviour and health were chosen for the analysis. In order to determine whether the respondents can be differentiated and described based on these variables, a non-linear canonical correlation analysis (OVERALS) was performed. The results can be described on three dimensions; Eigenvalues add up to the fit of 1.444, which can be interpreted as approximately 50 % of explained variance. The three-dimensional space illustrates correspondences between variables and provides a framework for interpretation based on latent dimensions, which can be described by age, education, income and gender. Using non-linear canonical correlation analysis, health characteristics can be analysed in conjunction with socio-economic conditions and lifestyles. Based on Bourdieus theoretical approach, the complex correlations between these variables can be more substantially interpreted and presented.

Imaging fluorescence detected linear dichroism of plant cell walls in laser scanning confocal microscope.

PubMed

Steinbach, Gábor; Pomozi, István; Zsiros, Ottó; Páy, Anikó; Horváth, Gábor V; Garab, Gyozo

2008-03-01

Anisotropy carries important information on the molecular organization of biological samples. Its determination requires a combination of microscopy and polarization spectroscopy tools. The authors constructed differential polarization (DP) attachments to a laser scanning microscope in order to determine physical quantities related to the anisotropic distribution of molecules in microscopic samples; here the authors focus on fluorescence-detected linear dichroism (FDLD). By modulating the linear polarization of the laser beam between two orthogonally polarized states and by using a demodulation circuit, the authors determine the associated transmitted and fluorescence intensity-difference signals, which serve the basis for LD (linear dichroism) and FDLD, respectively. The authors demonstrate on sections of Convallaria majalis root tissue stained with Acridin Orange that while (nonconfocal) LD images remain smeared and weak, FDLD images recorded in confocal mode reveal strong anisotropy of the cell wall. FDLD imaging is suitable for mapping the anisotropic distribution of transition dipoles in 3 dimensions. A mathematical model is proposed to account for the fiber-laminate ultrastructure of the cell wall and for the intercalation of the dye molecules in complex, highly anisotropic architecture. Copyright 2007 International Society for Analytical Cytology.

Biomechanically based simulation of brain deformations for intraoperative image correction: coupling of elastic and fluid models

NASA Astrophysics Data System (ADS)

Hagemann, Alexander; Rohr, Karl; Stiehl, H. Siegfried

2000-06-01

In order to improve the accuracy of image-guided neurosurgery, different biomechanical models have been developed to correct preoperative images w.r.t. intraoperative changes like brain shift or tumor resection. All existing biomechanical models simulate different anatomical structures by using either appropriate boundary conditions or by spatially varying material parameter values, while assuming the same physical model for all anatomical structures. In general, this leads to physically implausible results, especially in the case of adjacent elastic and fluid structures. Therefore, we propose a new approach which allows to couple different physical models. In our case, we simulate rigid, elastic, and fluid regions by using the appropriate physical description for each material, namely either the Navier equation or the Stokes equation. To solve the resulting differential equations, we derive a linear matrix system for each region by applying the finite element method (FEM). Thereafter, the linear matrix systems are linked together, ending up with one overall linear matrix system. Our approach has been tested using synthetic as well as tomographic images. It turns out from experiments, that the integrated treatment of rigid, elastic, and fluid regions significantly improves the prediction results in comparison to a pure linear elastic model.

ASP: Automated symbolic computation of approximate symmetries of differential equations

NASA Astrophysics Data System (ADS)

Jefferson, G. F.; Carminati, J.

2013-03-01

A recent paper (Pakdemirli et al. (2004) [12]) compared three methods of determining approximate symmetries of differential equations. Two of these methods are well known and involve either a perturbation of the classical Lie symmetry generator of the differential system (Baikov, Gazizov and Ibragimov (1988) [7], Ibragimov (1996) [6]) or a perturbation of the dependent variable/s and subsequent determination of the classical Lie point symmetries of the resulting coupled system (Fushchych and Shtelen (1989) [11]), both up to a specified order in the perturbation parameter. The third method, proposed by Pakdemirli, Yürüsoy and Dolapçi (2004) [12], simplifies the calculations required by Fushchych and Shtelen's method through the assignment of arbitrary functions to the non-linear components prior to computing symmetries. All three methods have been implemented in the new MAPLE package ASP (Automated Symmetry Package) which is an add-on to the MAPLE symmetry package DESOLVII (Vu, Jefferson and Carminati (2012) [25]). To our knowledge, this is the first computer package to automate all three methods of determining approximate symmetries for differential systems. Extensions to the theory have also been suggested for the third method and which generalise the first method to systems of differential equations. Finally, a number of approximate symmetries and corresponding solutions are compared with results in the literature.

Constructing general partial differential equations using polynomial and neural networks.

PubMed

Zjavka, Ladislav; Pedrycz, Witold

2016-01-01

Sum fraction terms can approximate multi-variable functions on the basis of discrete observations, replacing a partial differential equation definition with polynomial elementary data relation descriptions. Artificial neural networks commonly transform the weighted sum of inputs to describe overall similarity relationships of trained and new testing input patterns. Differential polynomial neural networks form a new class of neural networks, which construct and solve an unknown general partial differential equation of a function of interest with selected substitution relative terms using non-linear multi-variable composite polynomials. The layers of the network generate simple and composite relative substitution terms whose convergent series combinations can describe partial dependent derivative changes of the input variables. This regression is based on trained generalized partial derivative data relations, decomposed into a multi-layer polynomial network structure. The sigmoidal function, commonly used as a nonlinear activation of artificial neurons, may transform some polynomial items together with the parameters with the aim to improve the polynomial derivative term series ability to approximate complicated periodic functions, as simple low order polynomials are not able to fully make up for the complete cycles. The similarity analysis facilitates substitutions for differential equations or can form dimensional units from data samples to describe real-world problems. Copyright © 2015 Elsevier Ltd. All rights reserved.

Couple stress theory of curved rods. 2-D, high order, Timoshenko's and Euler-Bernoulli models

NASA Astrophysics Data System (ADS)

Zozulya, V. V.

2017-01-01

New models for plane curved rods based on linear couple stress theory of elasticity have been developed.2-D theory is developed from general 2-D equations of linear couple stress elasticity using a special curvilinear system of coordinates related to the middle line of the rod as well as special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First, stress and strain tensors, vectors of displacements and rotation along with body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate.Thereby, all equations of elasticity including Hooke's law have been transformed to the corresponding equations for Fourier coefficients. Then, in the same way as in the theory of elasticity, a system of differential equations in terms of displacements and boundary conditions for Fourier coefficients have been obtained. Timoshenko's and Euler-Bernoulli theories are based on the classical hypothesis and the 2-D equations of linear couple stress theory of elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scales when taking into account couple stress and rotation effects.

Angular-Rate Estimation Using Star Tracker Measurements

NASA Technical Reports Server (NTRS)

Azor, R.; Bar-Itzhack, I.; Deutschmann, Julie K.; Harman, Richard R.

1999-01-01

This paper presents algorithms for estimating the angular-rate vector of satellites using quaternion measurements. Two approaches are compared, one that uses differentiated quatemion measurements to yield coarse rate measurements which are then fed into two different estimators. In the other approach the raw quatemion measurements themselves are fed directly into the two estimators. The two estimators rely on the ability to decompose the non-linear rate dependent part of the rotational dynamics equation of a rigid body into a product of an angular-rate dependent matrix and the angular-rate vector itself This decomposition, which is not unique, enables the treatment of the nonlinear spacecraft dynamics model as a linear one and, consequently, the application of a Pseudo-Linear Kalman Filter (PSELIKA). It also enables the application of a special Kalman filter which is based on the use of the solution of the State Dependent Algebraic Riccati Equation (SDARE) in order to compute the Kalman gain matrix and thus eliminates the need to propagate and update the filter covariance matrix. The replacement of the elaborate rotational dynamics by a simple first order Markov model is also examined. In this paper a special consideration is given to the problem of delayed quatemion measurements. Two solutions to this problem are suggested and tested. Real Rossi X-Ray Timing Explorer (RXTE) data is used to test these algorithms, and results of these tests are presented.

Angular-Rate Estimation using Star Tracker Measurements

NASA Technical Reports Server (NTRS)

Azor, R.; Bar-Itzhack, Itzhack Y.; Deutschmann, Julie K.; Harman, Richard R.

1999-01-01

This paper presents algorithms for estimating the angular-rate vector of satellites using quaternion measurements. Two approaches are compared, one that uses differentiated quaternion measurements to yield coarse rate measurements which are then fed into two different estimators. In the other approach the raw quaternion measurements themselves are fed directly into the two estimators. The two estimators rely on the ability to decompose the non-linear rate dependent part of the rotational dynamics equation of a rigid body into a product of an angular-rate dependent matrix and the angular-rate vector itself. This decomposition, which is not unique, enables the treatment of the nonlinear spacecraft dynamics model as a linear one and, consequently, the application of a Pseudo-Linear Kalman Filter (PSELIKA). It also enables the application of a special Kalman filter which is based on the use of the solution of the State Dependent Algebraic Riccati Equation (SDARE) in order to compute the Kalman gain matrix and thus eliminates the need to propagate and update the filter covariance matrix. The replacement of the elaborate rotational dynamics by a simple first order Markov model is also examined. In this paper a special consideration is given to the problem of delayed quaternion measurements. Two solutions to this problem are suggested and tested. Real Rossi X-Ray Timing Explorer (RXTE) data is used to test these algorithms, and results of these tests are presented.

State Estimation for Humanoid Robots

DTIC Science & Technology

2015-07-01

21 2.2.1 Linear Inverted Pendulum Model . . . . . . . . . . . . . . . . . . . 21 2.2.2 Planar Five-link Model...Linear Inverted Pendulum Model. LVDT Linear Variable Differential Transformers. MEMS Microelectromechanical Systems. MHE Moving Horizon Estimator. QP...

Utilization of the Discrete Differential Evolution for Optimization in Multidimensional Point Clouds.

PubMed

Uher, Vojtěch; Gajdoš, Petr; Radecký, Michal; Snášel, Václav

2016-01-01

The Differential Evolution (DE) is a widely used bioinspired optimization algorithm developed by Storn and Price. It is popular for its simplicity and robustness. This algorithm was primarily designed for real-valued problems and continuous functions, but several modified versions optimizing both integer and discrete-valued problems have been developed. The discrete-coded DE has been mostly used for combinatorial problems in a set of enumerative variants. However, the DE has a great potential in the spatial data analysis and pattern recognition. This paper formulates the problem as a search of a combination of distinct vertices which meet the specified conditions. It proposes a novel approach called the Multidimensional Discrete Differential Evolution (MDDE) applying the principle of the discrete-coded DE in discrete point clouds (PCs). The paper examines the local searching abilities of the MDDE and its convergence to the global optimum in the PCs. The multidimensional discrete vertices cannot be simply ordered to get a convenient course of the discrete data, which is crucial for good convergence of a population. A novel mutation operator utilizing linear ordering of spatial data based on the space filling curves is introduced. The algorithm is tested on several spatial datasets and optimization problems. The experiments show that the MDDE is an efficient and fast method for discrete optimizations in the multidimensional point clouds.

Utilization of the Discrete Differential Evolution for Optimization in Multidimensional Point Clouds

PubMed Central

Radecký, Michal; Snášel, Václav

2016-01-01

The Differential Evolution (DE) is a widely used bioinspired optimization algorithm developed by Storn and Price. It is popular for its simplicity and robustness. This algorithm was primarily designed for real-valued problems and continuous functions, but several modified versions optimizing both integer and discrete-valued problems have been developed. The discrete-coded DE has been mostly used for combinatorial problems in a set of enumerative variants. However, the DE has a great potential in the spatial data analysis and pattern recognition. This paper formulates the problem as a search of a combination of distinct vertices which meet the specified conditions. It proposes a novel approach called the Multidimensional Discrete Differential Evolution (MDDE) applying the principle of the discrete-coded DE in discrete point clouds (PCs). The paper examines the local searching abilities of the MDDE and its convergence to the global optimum in the PCs. The multidimensional discrete vertices cannot be simply ordered to get a convenient course of the discrete data, which is crucial for good convergence of a population. A novel mutation operator utilizing linear ordering of spatial data based on the space filling curves is introduced. The algorithm is tested on several spatial datasets and optimization problems. The experiments show that the MDDE is an efficient and fast method for discrete optimizations in the multidimensional point clouds. PMID:27974884

Improved Pedagogy for Linear Differential Equations by Reconsidering How We Measure the Size of Solutions

ERIC Educational Resources Information Center

Tisdell, Christopher C.

2017-01-01

For over 50 years, the learning of teaching of "a priori" bounds on solutions to linear differential equations has involved a Euclidean approach to measuring the size of a solution. While the Euclidean approach to "a priori" bounds on solutions is somewhat manageable in the learning and teaching of the proofs involving…

The Pendulum: A Paradigm for the Linear Oscillator

ERIC Educational Resources Information Center

Newburgh, Ronald

2004-01-01

The simple pendulum is a model for the linear oscillator. The usual mathematical treatment of the problem begins with a differential equation that one solves with the techniques of the differential calculus, a formal process that tends to obscure the physics. In this paper we begin with a kinematic description of the motion obtained by experiment…

A novel interval type-2 fractional order fuzzy PID controller: Design, performance evaluation, and its optimal time domain tuning.

PubMed

Kumar, Anupam; Kumar, Vijay

2017-05-01

In this paper, a novel concept of an interval type-2 fractional order fuzzy PID (IT2FO-FPID) controller, which requires fractional order integrator and fractional order differentiator, is proposed. The incorporation of Takagi-Sugeno-Kang (TSK) type interval type-2 fuzzy logic controller (IT2FLC) with fractional controller of PID-type is investigated for time response measure due to both unit step response and unit load disturbance. The resulting IT2FO-FPID controller is examined on different delayed linear and nonlinear benchmark plants followed by robustness analysis. In order to design this controller, fractional order integrator-differentiator operators are considered as design variables including input-output scaling factors. A new hybridized algorithm named as artificial bee colony-genetic algorithm (ABC-GA) is used to optimize the parameters of the controller while minimizing weighted sum of integral of time absolute error (ITAE) and integral of square of control output (ISCO). To assess the comparative performance of the IT2FO-FPID, authors compared it against existing controllers, i.e., interval type-2 fuzzy PID (IT2-FPID), type-1 fractional order fuzzy PID (T1FO-FPID), type-1 fuzzy PID (T1-FPID), and conventional PID controllers. Furthermore, to show the effectiveness of the proposed controller, the perturbed processes along with the larger dead time are tested. Moreover, the proposed controllers are also implemented on multi input multi output (MIMO), coupled, and highly complex nonlinear two-link robot manipulator system in presence of un-modeled dynamics. Finally, the simulation results explicitly indicate that the performance of the proposed IT2FO-FPID controller is superior to its conventional counterparts in most of the cases. Copyright © 2017 ISA. Published by Elsevier Ltd. All rights reserved.

Two Legendre-Dual-Petrov-Galerkin Algorithms for Solving the Integrated Forms of High Odd-Order Boundary Value Problems

PubMed Central

Abd-Elhameed, Waleed M.; Doha, Eid H.; Bassuony, Mahmoud A.

2014-01-01

Two numerical algorithms based on dual-Petrov-Galerkin method are developed for solving the integrated forms of high odd-order boundary value problems (BVPs) governed by homogeneous and nonhomogeneous boundary conditions. Two different choices of trial functions and test functions which satisfy the underlying boundary conditions of the differential equations and the dual boundary conditions are used for this purpose. These choices lead to linear systems with specially structured matrices that can be efficiently inverted, hence greatly reducing the cost. The various matrix systems resulting from these discretizations are carefully investigated, especially their complexities and their condition numbers. Numerical results are given to illustrate the efficiency of the proposed algorithms, and some comparisons with some other methods are made. PMID:24616620

A strictly Markovian expansion for plasma turbulence theory

NASA Technical Reports Server (NTRS)

Jones, F. C.

1978-01-01

The collision operator that appears in the equation of motion for a particle distribution function that has been averaged over an ensemble of random Hamiltonians is non-Markovian. It is non-Markovian in that it involves a propagated integral over the past history of the ensemble averaged distribution function. All formal expansions of this nonlinear collision operator to date preserve this non-Markovian character term by term yielding an integro-differential equation that must be converted to a diffusion equation by an additional approximation. In this note we derive an expansion of the collision operator that is strictly Markovian to any finite order and yields a diffusion equation as the lowest non-trivial order. The validity of this expansion is seen to be the same as that of the standard quasi-linear expansion.

Absolute Position Sensing Based on a Robust Differential Capacitive Sensor with a Grounded Shield Window

PubMed Central

Bai, Yang; Lu, Yunfeng; Hu, Pengcheng; Wang, Gang; Xu, Jinxin; Zeng, Tao; Li, Zhengkun; Zhang, Zhonghua; Tan, Jiubin

2016-01-01

A simple differential capacitive sensor is provided in this paper to measure the absolute positions of length measuring systems. By utilizing a shield window inside the differential capacitor, the measurement range and linearity range of the sensor can reach several millimeters. What is more interesting is that this differential capacitive sensor is only sensitive to one translational degree of freedom (DOF) movement, and immune to the vibration along the other two translational DOFs. In the experiment, we used a novel circuit based on an AC capacitance bridge to directly measure the differential capacitance value. The experimental result shows that this differential capacitive sensor has a sensitivity of 2 × 10−4 pF/μm with 0.08 μm resolution. The measurement range of this differential capacitive sensor is 6 mm, and the linearity error are less than 0.01% over the whole absolute position measurement range. PMID:27187393

Second-order kinetic model for the sorption of cadmium onto tree fern: a comparison of linear and non-linear methods.

PubMed

Ho, Yuh-Shan

2006-01-01

A comparison was made of the linear least-squares method and a trial-and-error non-linear method of the widely used pseudo-second-order kinetic model for the sorption of cadmium onto ground-up tree fern. Four pseudo-second-order kinetic linear equations are discussed. Kinetic parameters obtained from the four kinetic linear equations using the linear method differed but they were the same when using the non-linear method. A type 1 pseudo-second-order linear kinetic model has the highest coefficient of determination. Results show that the non-linear method may be a better way to obtain the desired parameters.

A problem with inverse time for a singularly perturbed integro-differential equation with diagonal degeneration of the kernel of high order

NASA Astrophysics Data System (ADS)

Bobodzhanov, A. A.; Safonov, V. F.

2016-04-01

We consider an algorithm for constructing asymptotic solutions regularized in the sense of Lomov (see [1], [2]). We show that such problems can be reduced to integro-differential equations with inverse time. But in contrast to known papers devoted to this topic (see, for example, [3]), in this paper we study a fundamentally new case, which is characterized by the absence, in the differential part, of a linear operator that isolates, in the asymptotics of the solution, constituents described by boundary functions and by the fact that the integral operator has kernel with diagonal degeneration of high order. Furthermore, the spectrum of the regularization operator A(t) (see below) may contain purely imaginary eigenvalues, which causes difficulties in the application of the methods of construction of asymptotic solutions proposed in the monograph [3]. Based on an analysis of the principal term of the asymptotics, we isolate a class of inhomogeneities and initial data for which the exact solution of the original problem tends to the limit solution (as \\varepsilon\\to+0) on the entire time interval under consideration, also including a boundary-layer zone (that is, we solve the so-called initialization problem). The paper is of a theoretical nature and is designed to lead to a greater understanding of the problems in the theory of singular perturbations. There may be applications in various applied areas where models described by integro-differential equations are used (for example, in elasticity theory, the theory of electrical circuits, and so on).

Equations for normal-mode statistics of sound scattering by a rough elastic boundary in an underwater waveguide, including backscattering.

PubMed

Morozov, Andrey K; Colosi, John A

2017-09-01

Underwater sound scattering by a rough sea surface, ice, or a rough elastic bottom is studied. The study includes both the scattering from the rough boundary and the elastic effects in the solid layer. A coupled mode matrix is approximated by a linear function of one random perturbation parameter such as the ice-thickness or a perturbation of the surface position. A full two-way coupled mode solution is used to derive the stochastic differential equation for the second order statistics in a Markov approximation.

A Novel Numerical Method for Fuzzy Boundary Value Problems

NASA Astrophysics Data System (ADS)

Can, E.; Bayrak, M. A.; Hicdurmaz

2016-05-01

In the present paper, a new numerical method is proposed for solving fuzzy differential equations which are utilized for the modeling problems in science and engineering. Fuzzy approach is selected due to its important applications on processing uncertainty or subjective information for mathematical models of physical problems. A second-order fuzzy linear boundary value problem is considered in particular due to its important applications in physics. Moreover, numerical experiments are presented to show the effectiveness of the proposed numerical method on specific physical problems such as heat conduction in an infinite plate and a fin.

Icing research tunnel rotating bar calibration measurement system

NASA Technical Reports Server (NTRS)

Gibson, Theresa L.; Dearmon, John M.

1993-01-01

In order to measure icing patterns across a test section of the Icing Research Tunnel, an automated rotating bar measurement system was developed at the NASA Lewis Research Center. In comparison with the previously used manual measurement system, this system provides a number of improvements: increased accuracy and repeatability, increased number of data points, reduced tunnel operating time, and improved documentation. The automated system uses a linear variable differential transformer (LVDT) to measure ice accretion. This instrument is driven along the bar by means of an intelligent stepper motor which also controls data recording. This paper describes the rotating bar calibration measurement system.

The Magnus problem in Rodrigues-Hamilton parameters

NASA Astrophysics Data System (ADS)

Koshliakov, V. N.

1984-04-01

The formalism of Rodrigues-Hamilton parameters is applied to the Magnus problem related to the systematic drift of a gimbal-mounted astatic gyroscope due to the nutational vibration of the main axis of the rotor. It is shown that the use of the above formalism makes it possible to limit the analysis to a consideration of a linear system of differential equations written in perturbed values of Rodrigues-Hamilton parameters. A refined formula for the drift of the main axis of the gyroscope rotor is obtained, and an estimation is made of the effect of the truncation of higher-order terms.

Study of self-focusing of Non Gaussian laser beam in a plasma with density variation using moment theory approach

NASA Astrophysics Data System (ADS)

Pathak, Nidhi; Kaur, Sukhdeep; Singh, Sukhmander

2018-05-01

In this paper, self-focusing/defocusing effects have been studied by taking into account the combined effect of ponder-motive and relativistic non linearity during the laser plasma interaction with density variation. The formulation is based on the numerical analysis of second order nonlinear differential equation for appropriate set of laser and plasma parameters by employing moment theory approach. We found that self-focusing increases with increasing the laser intensity and density variation. The results obtained are valuable in high harmonic generation, inertial confinement fusion and charge particle acceleration.

Simulation of noise involved in synthetic aperture radar

NASA Astrophysics Data System (ADS)

Grandchamp, Myriam; Cavassilas, Jean-Francois

1996-08-01

The synthetic aperture radr (SAR) returns from a linear distribution of scatterers are simulated and processed in order to estimate the reflectivity coefficients of the ground. An original expression of this estimate is given, which establishes the relation between the terms of signal and noise. Both are compared. One application of this formulation consists of detecting a surface ship wake on a complex SAR image. A smoothing is first accomplished on the complex image. The choice of the integration area is determined by the preceding mathematical formulation. Then a differential filter is applied, and results are shown for two parts of the wake.

A complete and partial integrability technique of the Lorenz system

NASA Astrophysics Data System (ADS)

Bougoffa, Lazhar; Al-Awfi, Saud; Bougouffa, Smail

2018-06-01

In this paper we deal with the well-known nonlinear Lorenz system that describes the deterministic chaos phenomenon. We consider an interesting problem with time-varying phenomena in quantum optics. Then we establish from the motion equations the passage to the Lorenz system. Furthermore, we show that the reduction to the third order non linear equation can be performed. Therefore, the obtained differential equation can be analytically solved in some special cases and transformed to Abel, Dufing, Painlevé and generalized Emden-Fowler equations. So, a motivating technique that permitted a complete and partial integrability of the Lorenz system is presented.

On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations

NASA Technical Reports Server (NTRS)

Gunzburger, M. D.; Nicolaides, R. A.

1986-01-01

Substructuring methods are in common use in mechanics problems where typically the associated linear systems of algebraic equations are positive definite. Here these methods are extended to problems which lead to nonpositive definite, nonsymmetric matrices. The extension is based on an algorithm which carries out the block Gauss elimination procedure without the need for interchanges even when a pivot matrix is singular. Examples are provided wherein the method is used in connection with finite element solutions of the stationary Stokes equations and the Helmholtz equation, and dual methods for second-order elliptic equations.

Effects of heat and mass transfer on unsteady boundary layer flow of a chemical reacting Casson fluid

NASA Astrophysics Data System (ADS)

Khan, Kashif Ali; Butt, Asma Rashid; Raza, Nauman

2018-03-01

In this study, an endeavor is to observe the unsteady two-dimensional boundary layer flow with heat and mass transfer behavior of Casson fluid past a stretching sheet in presence of wall mass transfer by ignoring the effects of viscous dissipation. Chemical reaction of linear order is also invoked here. Similarity transformation have been applied to reduce the governing equations of momentum, energy and mass into non-linear ordinary differential equations; then Homotopy analysis method (HAM) is applied to solve these equations. Numerical work is done carefully with a well-known software MATHEMATICA for the examination of non-dimensional velocity, temperature, and concentration profiles, and then results are presented graphically. The skin friction (viscous drag), local Nusselt number (rate of heat transfer) and Sherwood number (rate of mass transfer) are discussed and presented in tabular form for several factors which are monitoring the flow model.

Spatial-frequency Fourier polarimetry of the complex degree of mutual anisotropy of linear and circular birefringence in the diagnostics of oncological changes in morphological structure of biological tissues

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ushenko, Yu A; Gorskii, M P; Dubolazov, A V

2012-08-31

Theory of polarisation-correlation analysis of laser images of histological sections of biopsy material from cervix tissue based on spatial frequency selection of linear and circular birefringence mechanisms is formulated. Comparative results of measuring the coordinate distributions of the complex degree of mutual anisotropy (CDMA), produced by fibrillar networks formed by myosin and collagen fibres of cervix tissue in different pathological conditions, namely, pre-cancer (dysplasia) and cancer (adenocarcinoma), are presented. The values and variation ranges of statistical (moments of the first - fourth order), correlation (excess-autocorrelation functions), and fractal (slopes of approximating curves and dispersion of extrema of logarithmic dependences ofmore » power spectra) parameters of the CDMA coordinate distributions are studied. Objective criteria for pathology diagnostics and differentiation of its severity degree are determined. (image processing)« less

Laser autofluorescence polarimetry of optically anisotropic structures of biological tissues in cancer diagnostics

NASA Astrophysics Data System (ADS)

Ushenko, Yu. A.

2015-06-01

The results of a new physical study of polarization manifestations of laser autofluorescence of optically anisotropic structures in human female reproductive tissues are presented. A Mueller-matrix model of describing the complex anisotropy (linear and circular birefringence, linear and circular dichroism) of such biological layers is proposed. Interrelations between mechanisms of optical anisotropy and polarization manifestations of laser autofluorescence of histological layers of the uterine cervix tissue in different spectral regions are determined. Magnitudes and variation ranges of statistical moments from the first to the fourth order describing the distributions of azimuthally stable elements of Mueller matrices of autofluorescence in human female reproductive tissues in different physiological states are found. The informative value of the proposed method is determined and the differentiation of histological biopsy sections of benign (dysplasia) and malignant (adenocarcinoma) uterine cervix tumors is implemented for the first time.

Spatial-frequency Fourier polarimetry of the complex degree of mutual anisotropy of linear and circular birefringence in the diagnostics of oncological changes in morphological structure of biological tissues

NASA Astrophysics Data System (ADS)

Ushenko, Yu A.; Gorskii, M. P.; Dubolazov, A. V.; Motrich, A. V.; Ushenko, V. A.; Sidor, M. I.

2012-08-01

Theory of polarisation-correlation analysis of laser images of histological sections of biopsy material from cervix tissue based on spatial frequency selection of linear and circular birefringence mechanisms is formulated. Comparative results of measuring the coordinate distributions of the complex degree of mutual anisotropy (CDMA), produced by fibrillar networks formed by myosin and collagen fibres of cervix tissue in different pathological conditions, namely, pre-cancer (dysplasia) and cancer (adenocarcinoma), are presented. The values and variation ranges of statistical (moments of the first — fourth order), correlation (excess-autocorrelation functions), and fractal (slopes of approximating curves and dispersion of extrema of logarithmic dependences of power spectra) parameters of the CDMA coordinate distributions are studied. Objective criteria for pathology diagnostics and differentiation of its severity degree are determined.

Tuning the control system of a nonlinear inverted pendulum by means of the new method of Lyapunov exponents estimation

NASA Astrophysics Data System (ADS)

Balcerzak, Marek; Dąbrowski, Artur; Pikunov, Danylo

2018-01-01

This paper presents a practical application of a new, simplified method of Lyapunov exponents estimation. The method has been applied to optimization of a real, nonlinear inverted pendulum system. Authors presented how the algorithm of the Largest Lyapunov Exponent (LLE) estimation can be applied to evaluate control systems performance. The new LLE-based control performance index has been proposed. Equations of the inverted pendulum system of the fourth order have been found. The nonlinear friction of the regulation object has been identified by means of the nonlinear least squares method. Three different friction models have been tested: linear, cubic and Coulomb model. The Differential Evolution (DE) algorithm has been used to search for the best set of parameters of the general linear regulator. This work proves that proposed method is efficient and results in faster perturbation rejection, especially when disturbances are significant.

Concerted spatial-frequency and polarization-phase filtering of laser images of polycrystalline networks of blood plasma smears

NASA Astrophysics Data System (ADS)

Ushenko, Yu A.

2012-11-01

The complex technique of concerted polarization-phase and spatial-frequency filtering of blood plasma laser images is suggested. The possibility of obtaining the coordinate distributions of phases of linearly and circularly birefringent protein networks of blood plasma separately is presented. The statistical (moments of the first to fourth orders) and scale self-similar (logarithmic dependences of power spectra) structure of phase maps of different types of birefringence of blood plasma of two groups of patients-healthy people (donors) and those suffering from rectal cancer-is investigated. The diagnostically sensitive parameters of a pathological change of the birefringence of blood plasma polycrystalline networks are determined. The effectiveness of this technique for detecting change in birefringence in the smears of other biological fluids in diagnosing the appearance of cholelithiasis (bile), operative differentiation of the acute and gangrenous appendicitis (exudate), and differentiation of inflammatory diseases of joints (synovial fluid) is shown.

A parallel time integrator for noisy nonlinear oscillatory systems

NASA Astrophysics Data System (ADS)

Subber, Waad; Sarkar, Abhijit

2018-06-01

In this paper, we adapt a parallel time integration scheme to track the trajectories of noisy non-linear dynamical systems. Specifically, we formulate a parallel algorithm to generate the sample path of nonlinear oscillator defined by stochastic differential equations (SDEs) using the so-called parareal method for ordinary differential equations (ODEs). The presence of Wiener process in SDEs causes difficulties in the direct application of any numerical integration techniques of ODEs including the parareal algorithm. The parallel implementation of the algorithm involves two SDEs solvers, namely a fine-level scheme to integrate the system in parallel and a coarse-level scheme to generate and correct the required initial conditions to start the fine-level integrators. For the numerical illustration, a randomly excited Duffing oscillator is investigated in order to study the performance of the stochastic parallel algorithm with respect to a range of system parameters. The distributed implementation of the algorithm exploits Massage Passing Interface (MPI).

Parameter identification for nonlinear aerodynamic systems

NASA Technical Reports Server (NTRS)

Pearson, Allan E.

1990-01-01

Parameter identification for nonlinear aerodynamic systems is examined. It is presumed that the underlying model can be arranged into an input/output (I/O) differential operator equation of a generic form. The algorithm estimation is especially efficient since the equation error can be integrated exactly given any I/O pair to obtain an algebraic function of the parameters. The algorithm for parameter identification was extended to the order determination problem for linear differential system. The degeneracy in a least squares estimate caused by feedback was addressed. A method of frequency analysis for determining the transfer function G(j omega) from transient I/O data was formulated using complex valued Fourier based modulating functions in contrast with the trigonometric modulating functions for the parameter estimation problem. A simulation result of applying the algorithm is given under noise-free conditions for a system with a low pass transfer function.

Heat transfer in a one-dimensional harmonic crystal in a viscous environment subjected to an external heat supply

NASA Astrophysics Data System (ADS)

Gavrilov, S. N.; Krivtsov, A. M.; Tsvetkov, D. V.

2018-05-01

We consider unsteady heat transfer in a one-dimensional harmonic crystal surrounded by a viscous environment and subjected to an external heat supply. The basic equations for the crystal particles are stated in the form of a system of stochastic differential equations. We perform a continualization procedure and derive an infinite set of linear partial differential equations for covariance variables. An exact analytic solution describing unsteady ballistic heat transfer in the crystal is obtained. It is shown that the stationary spatial profile of the kinetic temperature caused by a point source of heat supply of constant intensity is described by the Macdonald function of zero order. A comparison with the results obtained in the framework of the classical heat equation is presented. We expect that the results obtained in the paper can be verified by experiments with laser excitation of low-dimensional nanostructures.

Stability and synchronization analysis of inertial memristive neural networks with time delays.

PubMed

Rakkiyappan, R; Premalatha, S; Chandrasekar, A; Cao, Jinde

2016-10-01

This paper is concerned with the problem of stability and pinning synchronization of a class of inertial memristive neural networks with time delay. In contrast to general inertial neural networks, inertial memristive neural networks is applied to exhibit the synchronization and stability behaviors due to the physical properties of memristors and the differential inclusion theory. By choosing an appropriate variable transmission, the original system can be transformed into first order differential equations. Then, several sufficient conditions for the stability of inertial memristive neural networks by using matrix measure and Halanay inequality are derived. These obtained criteria are capable of reducing computational burden in the theoretical part. In addition, the evaluation is done on pinning synchronization for an array of linearly coupled inertial memristive neural networks, to derive the condition using matrix measure strategy. Finally, the two numerical simulations are presented to show the effectiveness of acquired theoretical results.

Oblique transport of gyrotactic microorganisms and bioconvection nanoparticles with convective mass flux

NASA Astrophysics Data System (ADS)

Iqbal, Z.; Mehmood, Zaffar; Maraj, E. N.

2017-04-01

The present study deals with examination of steady two dimensional nanofluid containing both nanoparticles and gyrotactic microorganisms. Moreover the study comprises stagnation point flow of an obliquely striking nanofluid. The governing partial differential equations are complex and highly non-linear in nature. These are converted into system of ordinary differential equations using suitable transformations. The system is then solved numerically using shooting technique with Runge - Kutta Fehlberg method of order 5. Further, effect of different physical parameters on velocity f ‧ (η) , temperature θ (η) , density of motile microorganisms w (η) and concentration ϕ (η) along with skin friction coefficient Cf, local Nusselt Nux, Sherwood Shx and density of motile microorganism Nnx numbers have been discussed through graphs and tables. Results depict that temperature, concentration, density of motile microorganisms and local Nusselt number are increasing functions of thermophoresis parameter Nt. Whereas Nt contributes in lessening Sherwood and local density numbers.

The renormalization group and the implicit function theorem for amplitude equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kirkinis, Eleftherios

2008-07-15

This article lays down the foundations of the renormalization group (RG) approach for differential equations characterized by multiple scales. The renormalization of constants through an elimination process and the subsequent derivation of the amplitude equation [Chen et al., Phys. Rev. E 54, 376 (1996)] are given a rigorous but not abstract mathematical form whose justification is based on the implicit function theorem. Developing the theoretical framework that underlies the RG approach leads to a systematization of the renormalization process and to the derivation of explicit closed-form expressions for the amplitude equations that can be carried out with symbolic computation formore » both linear and nonlinear scalar differential equations and first order systems but independently of their particular forms. Certain nonlinear singular perturbation problems are considered that illustrate the formalism and recover well-known results from the literature as special cases.« less

Similarity transformation for equilibrium boundary layers, including effects of blowing and suction

NASA Astrophysics Data System (ADS)

Chen, Xi; Hussain, Fazle

2017-03-01

We present a similarity transformation for the mean velocity profiles in sink flow turbulent boundary layers, including effects of blowing and suction. It is based on symmetry analysis which transforms the governing partial differential equations (for mean mass and momentum) into an ordinary differential equation and yields a new result including an exact, linear relation between the mean normal (V ) and streamwise (U ) velocities. A characteristic length function is further introduced which, under a first order expansion (whose coefficient is η ) in wall blowing and suction velocity, leads to the similarity transformation for U with the value of η ≈-1 /9 . This transformation is shown to be a group invariant and maps different U profiles under different blowing and suction conditions into a (universal) profile for no blowing or suction. Its inverse transformation enables predictions of all mean quantities in the mean mass and momentum equations, in good agreement with DNS data.

Model and Comparative Study for Flow of Viscoelastic Nanofluids with Cattaneo-Christov Double Diffusion

PubMed Central

Hayat, Tasawar; Aziz, Arsalan; Muhammad, Taseer; Alsaedi, Ahmed

2017-01-01

Here two classes of viscoelastic fluids have been analyzed in the presence of Cattaneo-Christov double diffusion expressions of heat and mass transfer. A linearly stretched sheet has been used to create the flow. Thermal and concentration diffusions are characterized firstly by introducing Cattaneo-Christov fluxes. Novel features regarding Brownian motion and thermophoresis are retained. The conversion of nonlinear partial differential system to nonlinear ordinary differential system has been taken into place by using suitable transformations. The resulting nonlinear systems have been solved via convergent approach. Graphs have been sketched in order to investigate how the velocity, temperature and concentration profiles are affected by distinct physical flow parameters. Numerical values of skin friction coefficient and heat and mass transfer rates at the wall are also computed and discussed. Our observations demonstrate that the temperature and concentration fields are decreasing functions of thermal and concentration relaxation parameters. PMID:28046011

Anthropometric characteristics and motor skills in talent selection and development in indoor soccer.

PubMed

Ré, Alessandro H Nicolai; Corrêa, Umberto César; Böhme, Maria Tereza S

2010-06-01

Kick performance, anthropometric characteristics, slalom, and linear running were assessed in 49 (24 elite, 25 nonelite) postpubertal indoor soccer players in order to (a) verify whether anthropometric characteristics and physical and technical capacities can distinguish players of different competitive levels, (b) compare the kicking kinematics of these groups, with and without a defined target, and (c) compare results on the assessments and coaches' subjective rankings of the players. Thigh circumference and specific technical capacities differentiated the players by level of play; cluster analysis correctly classified 77.5% of the players. The correlation between players' standardized measures and the coaches' rankings was 0.29. Anthropometric characteristics and physical capacities do not necessarily differentiate players at post-pubertal stages and should not be overvalued during early development. Considering the coaches' rankings, performance measures outside the specific game conditions may not be useful in identification of talented players.

Searching fundamental information in ordinary differential equations. Nondimensionalization technique.

PubMed

Sánchez Pérez, J F; Conesa, M; Alhama, I; Alhama, F; Cánovas, M

2017-01-01

Classical dimensional analysis and nondimensionalization are assumed to be two similar approaches in the search for dimensionless groups. Both techniques, simplify the study of many problems. The first approach does not need to know the mathematical model, being sufficient a deep understanding of the physical phenomenon involved, while the second one begins with the governing equations and reduces them to their dimensionless form by simple mathematical manipulations. In this work, a formal protocol is proposed for applying the nondimensionalization process to ordinary differential equations, linear or not, leading to dimensionless normalized equations from which the resulting dimensionless groups have two inherent properties: In one hand, they are physically interpreted as balances between counteracting quantities in the problem, and on the other hand, they are of the order of magnitude unity. The solutions provided by nondimensionalization are more precise in every case than those from dimensional analysis, as it is illustrated by the applications studied in this work.

The determination of third order linear models from a seventh order nonlinear jet engine model

NASA Technical Reports Server (NTRS)

Lalonde, Rick J.; Hartley, Tom T.; De Abreu-Garcia, J. Alex

1989-01-01

Results are presented that demonstrate how good reduced-order models can be obtained directly by recursive parameter identification using input/output (I/O) data of high-order nonlinear systems. Three different methods of obtaining a third-order linear model from a seventh-order nonlinear turbojet engine model are compared. The first method is to obtain a linear model from the original model and then reduce the linear model by standard reduction techniques such as residualization and balancing. The second method is to identify directly a third-order linear model by recursive least-squares parameter estimation using I/O data of the original model. The third method is to obtain a reduced-order model from the original model and then linearize the reduced model. Frequency responses are used as the performance measure to evaluate the reduced models. The reduced-order models along with their Bode plots are presented for comparison purposes.

Bifurcations in two-image photometric stereo for orthogonal illuminations

NASA Astrophysics Data System (ADS)

Kozera, R.; Prokopenya, A.; Noakes, L.; Śluzek, A.

2017-07-01

This paper discusses the ambiguous shape recovery in two-image photometric stereo for a Lambertian surface. The current uniqueness analysis refers to linearly independent light-source directions p = (0, 0, -1) and q arbitrary. For this case necessary and sufficient condition determining ambiguous reconstruction is governed by a second-order linear partial differential equation with constant coefficients. In contrast, a general position of both non-colinear illumination directions p and q leads to a highly non-linear PDE which raises a number of technical difficulties. As recently shown, the latter can also be handled for another family of orthogonal illuminations parallel to the OXZ-plane. For the special case of p = (0, 0, -1) a potential ambiguity stems also from the possible bifurcations of sub-local solutions glued together along a curve defined by an algebraic equation in terms of the data. This paper discusses the occurrence of similar bifurcations for such configurations of orthogonal light-source directions. The discussion to follow is supplemented with examples based on continuous reflectance map model and generated synthetic images.

The method of Ritz applied to the equation of Hamilton. [for pendulum systems

NASA Technical Reports Server (NTRS)

Bailey, C. D.

1976-01-01

Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.

Diffusion maps for high-dimensional single-cell analysis of differentiation data.

PubMed

Haghverdi, Laleh; Buettner, Florian; Theis, Fabian J

2015-09-15

Single-cell technologies have recently gained popularity in cellular differentiation studies regarding their ability to resolve potential heterogeneities in cell populations. Analyzing such high-dimensional single-cell data has its own statistical and computational challenges. Popular multivariate approaches are based on data normalization, followed by dimension reduction and clustering to identify subgroups. However, in the case of cellular differentiation, we would not expect clear clusters to be present but instead expect the cells to follow continuous branching lineages. Here, we propose the use of diffusion maps to deal with the problem of defining differentiation trajectories. We adapt this method to single-cell data by adequate choice of kernel width and inclusion of uncertainties or missing measurement values, which enables the establishment of a pseudotemporal ordering of single cells in a high-dimensional gene expression space. We expect this output to reflect cell differentiation trajectories, where the data originates from intrinsic diffusion-like dynamics. Starting from a pluripotent stage, cells move smoothly within the transcriptional landscape towards more differentiated states with some stochasticity along their path. We demonstrate the robustness of our method with respect to extrinsic noise (e.g. measurement noise) and sampling density heterogeneities on simulated toy data as well as two single-cell quantitative polymerase chain reaction datasets (i.e. mouse haematopoietic stem cells and mouse embryonic stem cells) and an RNA-Seq data of human pre-implantation embryos. We show that diffusion maps perform considerably better than Principal Component Analysis and are advantageous over other techniques for non-linear dimension reduction such as t-distributed Stochastic Neighbour Embedding for preserving the global structures and pseudotemporal ordering of cells. The Matlab implementation of diffusion maps for single-cell data is available at https://www.helmholtz-muenchen.de/icb/single-cell-diffusion-map. fbuettner.phys@gmail.com, fabian.theis@helmholtz-muenchen.de Supplementary data are available at Bioinformatics online. © The Author 2015. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com.

Time-Domain Evaluation of Fractional Order Controllers’ Direct Discretization Methods

NASA Astrophysics Data System (ADS)

Ma, Chengbin; Hori, Yoichi

Fractional Order Control (FOC), in which the controlled systems and/or controllers are described by fractional order differential equations, has been applied to various control problems. Though it is not difficult to understand FOC’s theoretical superiority, realization issue keeps being somewhat problematic. Since the fractional order systems have an infinite dimension, proper approximation by finite difference equation is needed to realize the designed fractional order controllers. In this paper, the existing direct discretization methods are evaluated by their convergences and time-domain comparison with the baseline case. Proposed sampling time scaling property is used to calculate the baseline case with full memory length. This novel discretization method is based on the classical trapezoidal rule but with scaled sampling time. Comparative studies show good performance and simple algorithm make the Short Memory Principle method most practically superior. The FOC research is still at its primary stage. But its applications in modeling and robustness against non-linearities reveal the promising aspects. Parallel to the development of FOC theories, applying FOC to various control problems is also crucially important and one of top priority issues.

Chemoviscosity modeling for thermosetting resins - I

NASA Technical Reports Server (NTRS)

Hou, T. H.

1984-01-01

A new analytical model for chemoviscosity variation during cure of thermosetting resins was developed. This model is derived by modifying the widely used WLF (Williams-Landel-Ferry) Theory in polymer rheology. Major assumptions involved are that the rate of reaction is diffusion controlled and is linearly inversely proportional to the viscosity of the medium over the entire cure cycle. The resultant first order nonlinear differential equation is solved numerically, and the model predictions compare favorably with experimental data of EPON 828/Agent U obtained on a Rheometrics System 4 Rheometer. The model describes chemoviscosity up to a range of six orders of magnitude under isothermal curing conditions. The extremely non-linear chemoviscosity profile for a dynamic heating cure cycle is predicted as well. The model is also shown to predict changes of glass transition temperature for the thermosetting resin during cure. The physical significance of this prediction is unclear at the present time, however, and further research is required. From the chemoviscosity simulation point of view, the technique of establishing an analytical model as described here is easily applied to any thermosetting resin. The model thus obtained is used in real-time process controls for fabricating composite materials.

PAN AIR: A computer program for predicting subsonic or supersonic linear potential flows about arbitrary configurations using a higher order panel method. Volume 1: Theory document (version 3.0)

NASA Technical Reports Server (NTRS)

Epton, Michael A.; Magnus, Alfred E.

1990-01-01

An outline of the derivation of the differential equation governing linear subsonic and supersonic potential flow is given. The use of Green's Theorem to obtain an integral equation over the boundary surface is discussed. The engineering techniques incorporated in the Panel Aerodynamics (PAN AIR) program (a discretization method which solves the integral equation for arbitrary first order boundary conditions) are then discussed in detail. Items discussed include the construction of the compressibility transformation, splining techniques, imposition of the boundary conditions, influence coefficient computation (including the concept of the finite part of an integral), computation of pressure coefficients, and computation of forces and moments. Principal revisions to version 3.0 are the following: (1) appendices H and K more fully describe the Aerodynamic Influence Coefficient (AIC) construction; (2) appendix L now provides a complete description of the AIC solution process; (3) appendix P is new and discusses the theory for the new FDP module (which calculates streamlines and offbody points); and (4) numerous small corrections and revisions reflecting the MAG module rewrite.

Modelling and control issues of dynamically substructured systems: adaptive forward prediction taken as an example

PubMed Central

Tu, Jia-Ying; Hsiao, Wei-De; Chen, Chih-Ying

2014-01-01

Testing techniques of dynamically substructured systems dissects an entire engineering system into parts. Components can be tested via numerical simulation or physical experiments and run synchronously. Additional actuator systems, which interface numerical and physical parts, are required within the physical substructure. A high-quality controller, which is designed to cancel unwanted dynamics introduced by the actuators, is important in order to synchronize the numerical and physical outputs and ensure successful tests. An adaptive forward prediction (AFP) algorithm based on delay compensation concepts has been proposed to deal with substructuring control issues. Although the settling performance and numerical conditions of the AFP controller are improved using new direct-compensation and singular value decomposition methods, the experimental results show that a linear dynamics-based controller still outperforms the AFP controller. Based on experimental observations, the least-squares fitting technique, effectiveness of the AFP compensation and differences between delay and ordinary differential equations are discussed herein, in order to reflect the fundamental issues of actuator modelling in relevant literature and, more specifically, to show that the actuator and numerical substructure are heterogeneous dynamic components and should not be collectively modelled as a homogeneous delay differential equation. PMID:25104902

Anodic Oxidation of Etodolac and its Linear Sweep, Square Wave and Differential Pulse Voltammetric Determination in Pharmaceuticals

PubMed Central

Yilmaz, B.; Kaban, S.; Akcay, B. K.

2015-01-01

In this study, simple, fast and reliable cyclic voltammetry, linear sweep voltammetry, square wave voltammetry and differential pulse voltammetry methods were developed and validated for determination of etodolac in pharmaceutical preparations. The proposed methods were based on electrochemical oxidation of etodolac at platinum electrode in acetonitrile solution containing 0.1 M lithium perchlorate. The well-defined oxidation peak was observed at 1.03 V. The calibration curves were linear for etodolac at the concentration range of 2.5-50 μg/ml for linear sweep, square wave and differential pulse voltammetry methods, respectively. Intra- and inter-day precision values for etodolac were less than 4.69, and accuracy (relative error) was better than 2.00%. The mean recovery of etodolac was 100.6% for pharmaceutical preparations. No interference was found from three tablet excipients at the selected assay conditions. Developed methods in this study are accurate, precise and can be easily applied to Etol, Tadolak and Etodin tablets as pharmaceutical preparation. PMID:26664057

Non-linear feedback control of the p53 protein-mdm2 inhibitor system using the derivative-free non-linear Kalman filter.

PubMed

Rigatos, Gerasimos G

2016-06-01

It is proven that the model of the p53-mdm2 protein synthesis loop is a differentially flat one and using a diffeomorphism (change of state variables) that is proposed by differential flatness theory it is shown that the protein synthesis model can be transformed into the canonical (Brunovsky) form. This enables the design of a feedback control law that maintains the concentration of the p53 protein at the desirable levels. To estimate the non-measurable elements of the state vector describing the p53-mdm2 system dynamics, the derivative-free non-linear Kalman filter is used. Moreover, to compensate for modelling uncertainties and external disturbances that affect the p53-mdm2 system, the derivative-free non-linear Kalman filter is re-designed as a disturbance observer. The derivative-free non-linear Kalman filter consists of the Kalman filter recursion applied on the linearised equivalent of the protein synthesis model together with an inverse transformation based on differential flatness theory that enables to retrieve estimates for the state variables of the initial non-linear model. The proposed non-linear feedback control and perturbations compensation method for the p53-mdm2 system can result in more efficient chemotherapy schemes where the infusion of medication will be better administered.

Customized Steady-State Constraints for Parameter Estimation in Non-Linear Ordinary Differential Equation Models

PubMed Central

Rosenblatt, Marcus; Timmer, Jens; Kaschek, Daniel

2016-01-01

Ordinary differential equation models have become a wide-spread approach to analyze dynamical systems and understand underlying mechanisms. Model parameters are often unknown and have to be estimated from experimental data, e.g., by maximum-likelihood estimation. In particular, models of biological systems contain a large number of parameters. To reduce the dimensionality of the parameter space, steady-state information is incorporated in the parameter estimation process. For non-linear models, analytical steady-state calculation typically leads to higher-order polynomial equations for which no closed-form solutions can be obtained. This can be circumvented by solving the steady-state equations for kinetic parameters, which results in a linear equation system with comparatively simple solutions. At the same time multiplicity of steady-state solutions is avoided, which otherwise is problematic for optimization. When solved for kinetic parameters, however, steady-state constraints tend to become negative for particular model specifications, thus, generating new types of optimization problems. Here, we present an algorithm based on graph theory that derives non-negative, analytical steady-state expressions by stepwise removal of cyclic dependencies between dynamical variables. The algorithm avoids multiple steady-state solutions by construction. We show that our method is applicable to most common classes of biochemical reaction networks containing inhibition terms, mass-action and Hill-type kinetic equations. Comparing the performance of parameter estimation for different analytical and numerical methods of incorporating steady-state information, we show that our approach is especially well-tailored to guarantee a high success rate of optimization. PMID:27243005

Customized Steady-State Constraints for Parameter Estimation in Non-Linear Ordinary Differential Equation Models.

PubMed

Rosenblatt, Marcus; Timmer, Jens; Kaschek, Daniel

2016-01-01

Ordinary differential equation models have become a wide-spread approach to analyze dynamical systems and understand underlying mechanisms. Model parameters are often unknown and have to be estimated from experimental data, e.g., by maximum-likelihood estimation. In particular, models of biological systems contain a large number of parameters. To reduce the dimensionality of the parameter space, steady-state information is incorporated in the parameter estimation process. For non-linear models, analytical steady-state calculation typically leads to higher-order polynomial equations for which no closed-form solutions can be obtained. This can be circumvented by solving the steady-state equations for kinetic parameters, which results in a linear equation system with comparatively simple solutions. At the same time multiplicity of steady-state solutions is avoided, which otherwise is problematic for optimization. When solved for kinetic parameters, however, steady-state constraints tend to become negative for particular model specifications, thus, generating new types of optimization problems. Here, we present an algorithm based on graph theory that derives non-negative, analytical steady-state expressions by stepwise removal of cyclic dependencies between dynamical variables. The algorithm avoids multiple steady-state solutions by construction. We show that our method is applicable to most common classes of biochemical reaction networks containing inhibition terms, mass-action and Hill-type kinetic equations. Comparing the performance of parameter estimation for different analytical and numerical methods of incorporating steady-state information, we show that our approach is especially well-tailored to guarantee a high success rate of optimization.

A computational analysis subject to thermophysical aspects of Sisko fluid flow over a cylindrical surface

NASA Astrophysics Data System (ADS)

Awais, M.; Khalil-Ur-Rehman; Malik, M. Y.; Hussain, Arif; Salahuddin, T.

2017-09-01

The present analysis is devoted to probing the salient features of the mixed convection and non-linear thermal radiation effects on non-Newtonian Sisko fluid flow over a linearly stretching cylindrical surface. Properties of heat transfer are outlined via variable thermal conductivity and convective boundary conditions. The boundary layer approach is implemented to construct the mathematical model in the form of partial differential equations. Then, the requisite PDEs are transmuted into a complex ordinary differential system by invoking appropriate dimensionless variables. Solution of subsequent ODEs is obtained by utilizing the Runge-Kutta algorithm (fifth order) along with the shooting scheme. The graphical illustrations are presented to interpret the features of the involved pertinent flow parameters on concerning profiles. For a better description of the fluid flow, numerical variations in local skin friction coefficient and local Nusselt number are scrutinized in tables. From thorough analysis, it is inferred that the mixed convection parameter and the curvature parameter increase the velocity while temperature shows a different behavior. Additionally, both momentum and thermal distribution of fluid flow decrease with increasing values of the non-linearity index. Furthermore, variable thermal parameter and heat generation/absorption parameter amplify the temperature significantly. The skin friction is an increasing function of all momentum controlling parameters. The local Nusselt number also shows a similar behavior against heat radiation parameter and variable thermal conductivity parameter while it shows a dual nature for the heat generation/absorption parameter. Finally, the obtained results are validated by comparison with the existing literature and hence the correctness of the analysis is proved.

Effects of parabens on adipocyte differentiation.

PubMed

Hu, Pan; Chen, Xin; Whitener, Rick J; Boder, Eric T; Jones, Jeremy O; Porollo, Aleksey; Chen, Jiangang; Zhao, Ling

2013-01-01

Parabens are a group of alkyl esters of p-hydroxybenzoic acid that include methylparaben, ethylparaben, propylparaben, butylparaben, and benzylparaben. Paraben esters and their salts are widely used as preservatives in cosmetics, toiletries, food, and pharmaceuticals. Humans are exposed to parabens through the use of such products from dermal contact, ingestion, and inhalation. However, research on the effects of parabens on health is limited, and the effects of parabens on adipogenesis have not been systematically studied. Here, we report that (1) parabens promote adipogenesis (or adipocyte differentiation) in murine 3T3-L1 cells, as revealed by adipocyte morphology, lipid accumulation, and mRNA expression of adipocyte-specific markers; (2) the adipogenic potency of parabens is increased with increasing length of the linear alkyl chain in the following potency ranking order: methyl- < ethyl- < propyl- < butylparaben. The extension of the linear alkyl chain with an aromatic ring in benzylparaben further augments the adipogenic ability, whereas 4-hydroxybenzoic acid, the common metabolite of all parabens, and the structurally related benzoic acid (without the OH group) are inactive in promoting 3T3-L1 adipocyte differentiation; (3) parabens activate glucocorticoid receptor and/or peroxisome proliferator-activated receptor γ in 3T3-L1 preadipocytes; however, no direct binding to, or modulation of, the ligand binding domain of the glucocorticoid receptor by parabens was detected by glucocorticoid receptor competitor assays; and lastly, (4) parabens, butyl- and benzylparaben in particular, also promote adipose conversion of human adipose-derived multipotent stromal cells. Our results suggest that parabens may contribute to obesity epidemic, and the role of parabens in adipogenesis in vivo needs to be examined further.

An electroglottographical analysis-based discriminant function model differentiating multiple sclerosis patients from healthy controls.

PubMed

Vavougios, George D; Doskas, Triantafyllos; Konstantopoulos, Kostas

2018-05-01

Dysarthrophonia is a predominant symptom in many neurological diseases, affecting the quality of life of the patients. In this study, we produced a discriminant function equation that can differentiate MS patients from healthy controls, using electroglottographic variables not analyzed in a previous study. We applied stepwise linear discriminant function analysis in order to produce a function and score derived from electroglottographic variables extracted from a previous study. The derived discriminant function's statistical significance was determined via Wilk's λ test (and the associated p value). Finally, a 2 × 2 confusion matrix was used to determine the function's predictive accuracy, whereas the cross-validated predictive accuracy is estimated via the "leave-one-out" classification process. Discriminant function analysis (DFA) was used to create a linear function of continuous predictors. DFA produced the following model (Wilk's λ = 0.043, χ2 = 388.588, p < 0.0001, Tables 3 and 4): D (MS vs controls) = 0.728*DQx1 mean monologue + 0.325*CQx monologue + 0.298*DFx1 90% range monologue + 0.443*DQx1 90% range reading - 1.490*DQx1 90% range monologue. The derived discriminant score (S1) was used subsequently in order to form the coordinates of a ROC curve. Thus, a cutoff score of - 0.788 for S1 corresponded to a perfect classification (100% sensitivity and 100% specificity, p = 1.67e -22 ). Consistent with previous findings, electroglottographic evaluation represents an easy to implement and potentially important assessment in MS patients, achieving adequate classification accuracy. Further evaluation is needed to determine its use as a biomarker.

Bifurcation of rupture path by linear and cubic damping force

NASA Astrophysics Data System (ADS)

Dennis L. C., C.; Chew X., Y.; Lee Y., C.

2014-06-01

Bifurcation of rupture path is studied for the effect of linear and cubic damping. Momentum equation with Rayleigh factor was transformed into ordinary differential form. Bernoulli differential equation was obtained and solved by the separation of variables. Analytical or exact solutions yielded the bifurcation was visible at imaginary part when the wave was non dispersive. For the dispersive wave, bifurcation of rupture path was invisible.

Control problem for a system of linear loaded differential equations

NASA Astrophysics Data System (ADS)

Barseghyan, V. R.; Barseghyan, T. V.

2018-04-01

The problem of control and optimal control for a system of linear loaded differential equations is considered. Necessary and sufficient conditions for complete controllability and conditions for the existence of a program control and the corresponding motion are formulated. The explicit form of control action for the control problem is constructed and a method for solving the problem of optimal control is proposed.

The existence of solutions of q-difference-differential equations.

PubMed

Wang, Xin-Li; Wang, Hua; Xu, Hong-Yan

2016-01-01

By using the Nevanlinna theory of value distribution, we investigate the existence of solutions of some types of non-linear q-difference differential equations. In particular, we generalize the Rellich-Wittich-type theorem and Malmquist-type theorem about differential equations to the case of q-difference differential equations (system).

An efficient method for solving the steady Euler equations

NASA Technical Reports Server (NTRS)

Liou, M. S.

1986-01-01

An efficient numerical procedure for solving a set of nonlinear partial differential equations is given, specifically for the steady Euler equations. Solutions of the equations were obtained by Newton's linearization procedure, commonly used to solve the roots of nonlinear algebraic equations. In application of the same procedure for solving a set of differential equations we give a theorem showing that a quadratic convergence rate can be achieved. While the domain of quadratic convergence depends on the problems studied and is unknown a priori, we show that firstand second-order derivatives of flux vectors determine whether the condition for quadratic convergence is satisfied. The first derivatives enter as an implicit operator for yielding new iterates and the second derivatives indicates smoothness of the flows considered. Consequently flows involving shocks are expected to require larger number of iterations. First-order upwind discretization in conjunction with the Steger-Warming flux-vector splitting is employed on the implicit operator and a diagonal dominant matrix results. However the explicit operator is represented by first- and seond-order upwind differencings, using both Steger-Warming's and van Leer's splittings. We discuss treatment of boundary conditions and solution procedures for solving the resulting block matrix system. With a set of test problems for one- and two-dimensional flows, we show detailed study as to the efficiency, accuracy, and convergence of the present method.

Multi-scale Eulerian model within the new National Environmental Modeling System

NASA Astrophysics Data System (ADS)

Janjic, Zavisa; Janjic, Tijana; Vasic, Ratko

2010-05-01

The unified Non-hydrostatic Multi-scale Model on the Arakawa B grid (NMMB) is being developed at NCEP within the National Environmental Modeling System (NEMS). The finite-volume horizontal differencing employed in the model preserves important properties of differential operators and conserves a variety of basic and derived dynamical and quadratic quantities. Among these, conservation of energy and enstrophy improves the accuracy of nonlinear dynamics of the model. Within further model development, advection schemes of fourth order of formal accuracy have been developed. It is argued that higher order advection schemes should not be used in the thermodynamic equation in order to preserve consistency with the second order scheme used for computation of the pressure gradient force. Thus, the fourth order scheme is applied only to momentum advection. Three sophisticated second order schemes were considered for upgrade. Two of them, proposed in Janjic(1984), conserve energy and enstrophy, but with enstrophy calculated differently. One of them conserves enstrophy as computed by the most accurate second order Laplacian operating on stream function. The other scheme conserves enstrophy as computed from the B grid velocity. The third scheme (Arakawa 1972) is arithmetic mean of the former two. It does not conserve enstrophy strictly, but it conserves other quadratic quantities that control the nonlinear energy cascade. Linearization of all three schemes leads to the same second order linear advection scheme. The second order term of the truncation error of the linear advection scheme has a special form so that it can be eliminated by simply preconditioning the advected quantity. Tests with linear advection of a cone confirm the advantage of the fourth order scheme. However, if a localized, large amplitude and high wave-number pattern is present in initial conditions, the clear advantage of the fourth order scheme disappears. In real data runs, problems with noisy data may appear due to mountains. Thus, accuracy and formal accuracy may not be synonymous. The nonlinear fourth order schemes are quadratic conservative and reduce to the Arakawa Jacobian in case of non-divergent flow. In case of general flow the conservation properties of the new momentum advection schemes impose stricter constraint on the nonlinear cascade than the original second order schemes. However, for non-divergent flow, the conservation properties of the fourth order schemes cannot be proven in the same way as those of the original second order schemes. Therefore, nonlinear tests were carried out in order to check how well the fourth order schemes control the nonlinear energy cascade. In the tests nonlinear shallow water equations are solved in a rotating rectangular domain (Janjic, 1984). The domain is covered with only 17 x 17 grid points. A diagnostic quantity is used to monitor qualitative changes in the spectrum over 116 days of simulated time. All schemes maintained meaningful solutions throughout the test. Among the second order schemes, the best result was obtained with the scheme that conserved enstrophy as computed by the second order Laplacian of the stream function. It was closely followed by the Arakawa (1972) scheme, while the remaining scheme was distant third. The fourth order schemes ranked in the same order, and were competitive throughout the experiments with their second order counterparts in preventing accumulation of energy at small scales. Finally, the impact was examined of the fourth order momentum advection on global medium range forecasts. The 500 mb anomaly correlation coefficient is used as a measure of success of the forecasts. Arakawa, A., 1972: Design of the UCLA general circulation model. Tech. Report No. 7, Department of Meteorology, University of California, Los Angeles, 116 pp. Janjic, Z. I., 1984: Non-linear advection schemes and energy cascade on semi-staggered grids. Monthly Weather Review, 112, 1234-1245.

New algorithms for solving third- and fifth-order two point boundary value problems based on nonsymmetric generalized Jacobi Petrov–Galerkin method

PubMed Central

Doha, E.H.; Abd-Elhameed, W.M.; Youssri, Y.H.

2014-01-01

Two families of certain nonsymmetric generalized Jacobi polynomials with negative integer indexes are employed for solving third- and fifth-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions using a dual Petrov–Galerkin method. The idea behind our method is to use trial functions satisfying the underlying boundary conditions of the differential equations and the test functions satisfying the dual boundary conditions. The resulting linear systems from the application of our method are specially structured and they can be efficiently inverted. The use of generalized Jacobi polynomials simplify the theoretical and numerical analysis of the method and also leads to accurate and efficient numerical algorithms. The presented numerical results indicate that the proposed numerical algorithms are reliable and very efficient. PMID:26425358

Two-dimensional mesh embedding for Galerkin B-spline methods

NASA Technical Reports Server (NTRS)

Shariff, Karim; Moser, Robert D.

1995-01-01

A number of advantages result from using B-splines as basis functions in a Galerkin method for solving partial differential equations. Among them are arbitrary order of accuracy and high resolution similar to that of compact schemes but without the aliasing error. This work develops another property, namely, the ability to treat semi-structured embedded or zonal meshes for two-dimensional geometries. This can drastically reduce the number of grid points in many applications. Both integer and non-integer refinement ratios are allowed. The report begins by developing an algorithm for choosing basis functions that yield the desired mesh resolution. These functions are suitable products of one-dimensional B-splines. Finally, test cases for linear scalar equations such as the Poisson and advection equation are presented. The scheme is conservative and has uniformly high order of accuracy throughout the domain.

Non-linear analysis of wave progagation using transform methods and plates and shells using integral equations

NASA Astrophysics Data System (ADS)

Pipkins, Daniel Scott

Two diverse topics of relevance in modern computational mechanics are treated. The first involves the modeling of linear and non-linear wave propagation in flexible, lattice structures. The technique used combines the Laplace Transform with the Finite Element Method (FEM). The procedure is to transform the governing differential equations and boundary conditions into the transform domain where the FEM formulation is carried out. For linear problems, the transformed differential equations can be solved exactly, hence the method is exact. As a result, each member of the lattice structure is modeled using only one element. In the non-linear problem, the method is no longer exact. The approximation introduced is a spatial discretization of the transformed non-linear terms. The non-linear terms are represented in the transform domain by making use of the complex convolution theorem. A weak formulation of the resulting transformed non-linear equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solution of the linear part of the transformed governing differential equation. Numerical results are presented for both linear and non-linear systems. The linear systems modeled are longitudinal and torsional rods and Bernoulli-Euler and Timoshenko beams. For non-linear systems, a viscoelastic rod and Von Karman type beam are modeled. The second topic is the analysis of plates and shallow shells under-going finite deflections by the Field/Boundary Element Method. Numerical results are presented for two plate problems. The first is the bifurcation problem associated with a square plate having free boundaries which is loaded by four, self equilibrating corner forces. The results are compared to two existing numerical solutions of the problem which differ substantially.

Investigation of ODE integrators using interactive graphics. [Ordinary Differential Equations

NASA Technical Reports Server (NTRS)

Brown, R. L.

1978-01-01

Two FORTRAN programs using an interactive graphic terminal to generate accuracy and stability plots for given multistep ordinary differential equation (ODE) integrators are described. The first treats the fixed stepsize linear case with complex variable solutions, and generates plots to show accuracy and error response to step driving function of a numerical solution, as well as the linear stability region. The second generates an analog to the stability region for classes of non-linear ODE's as well as accuracy plots. Both systems can compute method coefficients from a simple specification of the method. Example plots are given.

A Semi-linear Backward Parabolic Cauchy Problem with Unbounded Coefficients of Hamilton–Jacobi–Bellman Type and Applications to Optimal Control

DOE Office of Scientific and Technical Information (OSTI.GOV)

Addona, Davide, E-mail: d.addona@campus.unimib.it

2015-08-15

We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton–Jacobi–Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified.

Description of a computer program and numerical techniques for developing linear perturbation models from nonlinear systems simulations

NASA Technical Reports Server (NTRS)

Dieudonne, J. E.

1978-01-01

A numerical technique was developed which generates linear perturbation models from nonlinear aircraft vehicle simulations. The technique is very general and can be applied to simulations of any system that is described by nonlinear differential equations. The computer program used to generate these models is discussed, with emphasis placed on generation of the Jacobian matrices, calculation of the coefficients needed for solving the perturbation model, and generation of the solution of the linear differential equations. An example application of the technique to a nonlinear model of the NASA terminal configured vehicle is included.

Optimal moving grids for time-dependent partial differential equations

NASA Technical Reports Server (NTRS)

Wathen, A. J.

1992-01-01

Various adaptive moving grid techniques for the numerical solution of time-dependent partial differential equations were proposed. The precise criterion for grid motion varies, but most techniques will attempt to give grids on which the solution of the partial differential equation can be well represented. Moving grids are investigated on which the solutions of the linear heat conduction and viscous Burgers' equation in one space dimension are optimally approximated. Precisely, the results of numerical calculations of optimal moving grids for piecewise linear finite element approximation of PDE solutions in the least-squares norm are reported.

Fidelity and enhanced sensitivity of differential transcription profiles following linear amplification of nanogram amounts of endothelial mRNA

NASA Technical Reports Server (NTRS)

Polacek, Denise C.; Passerini, Anthony G.; Shi, Congzhu; Francesco, Nadeene M.; Manduchi, Elisabetta; Grant, Gregory R.; Powell, Steven; Bischof, Helen; Winkler, Hans; Stoeckert, Christian J Jr;

Stabilization of the Rayleigh-Taylor instability in quantum magnetized plasmas

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wang, L. F.; Ye, W. H.; He, X. T.

2012-07-15

In this research, stabilization of the Rayleigh-Taylor instability (RTI) due to density gradients, magnetic fields, and quantum effects, in an ideal incompressible plasma, is studied analytically and numerically. A second-order ordinary differential equation (ODE) for the RTI including quantum corrections, with a continuous density profile, in a uniform external magnetic field, is obtained. Analytic expressions of the linear growth rate of the RTI, considering modifications of density gradients, magnetic fields, and quantum effects, are presented. Numerical approaches are performed to solve the second-order ODE. The analytical model proposed here agrees with the numerical calculation. It is found that the densitymore » gradients, the magnetic fields, and the quantum effects, respectively, have a stabilizing effect on the RTI (reduce the linear growth of the RTI). The RTI can be completely quenched by the magnetic field stabilization and/or the quantum effect stabilization in proper circumstances leading to a cutoff wavelength. The quantum effect stabilization plays a central role in systems with large Atwood number and small normalized density gradient scale length. The presence of external transverse magnetic fields beside the quantum effects will bring about more stability on the RTI. The stabilization of the linear growth of the RTI, for parameters closely related to inertial confinement fusion and white dwarfs, is discussed. Results could potentially be valuable for the RTI treatment to analyze the mixing in supernovas and other RTI-driven objects.« less

Improvements In Ball-Screw Linear Actuators

NASA Technical Reports Server (NTRS)

Iskenderian, Theodore; Joffe, Benjamin; Summers, Robert

1996-01-01

Report describes modifications of design of type of ball-screw linear actuator driven by dc motor, with linear-displacement feedback via linear variable-differential transformer (LVDT). Actuators used to position spacecraft engines to direct thrust. Modifications directed toward ensuring reliable and predictable operation during planned 12-year cruise and interval of hard use at end of cruise.

A Textbook for a First Course in Computational Fluid Dynamics

NASA Technical Reports Server (NTRS)

Zingg, D. W.; Pulliam, T. H.; Nixon, David (Technical Monitor)

1999-01-01

This paper describes and discusses the textbook, Fundamentals of Computational Fluid Dynamics by Lomax, Pulliam, and Zingg, which is intended for a graduate level first course in computational fluid dynamics. This textbook emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. Its underlying philosophy is that the theory of linear algebra and the attendant eigenanalysis of linear systems provides a mathematical framework to describe and unify most numerical methods in common use in the field of fluid dynamics. Two linear model equations, the linear convection and diffusion equations, are used to illustrate concepts throughout. Emphasis is on the semi-discrete approach, in which the governing partial differential equations (PDE's) are reduced to systems of ordinary differential equations (ODE's) through a discretization of the spatial derivatives. The ordinary differential equations are then reduced to ordinary difference equations (O(Delta)E's) using a time-marching method. This methodology, using the progression from PDE through ODE's to O(Delta)E's, together with the use of the eigensystems of tridiagonal matrices and the theory of O(Delta)E's, gives the book its distinctiveness and provides a sound basis for a deep understanding of fundamental concepts in computational fluid dynamics.

On new classes of solutions of nonlinear partial differential equations in the form of convergent special series

NASA Astrophysics Data System (ADS)

Filimonov, M. Yu.

2017-12-01

The method of special series with recursively calculated coefficients is used to solve nonlinear partial differential equations. The recurrence of finding the coefficients of the series is achieved due to a special choice of functions, in powers of which the solution is expanded in a series. We obtain a sequence of linear partial differential equations to find the coefficients of the series constructed. In many cases, one can deal with a sequence of linear ordinary differential equations. We construct classes of solutions in the form of convergent series for a certain class of nonlinear evolution equations. A new class of solutions of generalized Boussinesque equation with an arbitrary function in the form of a convergent series is constructed.

A new perspective for quintic B-spline based Crank-Nicolson-differential quadrature method algorithm for numerical solutions of the nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Başhan, Ali; Uçar, Yusuf; Murat Yağmurlu, N.; Esen, Alaattin

2018-01-01

In the present paper, a Crank-Nicolson-differential quadrature method (CN-DQM) based on utilizing quintic B-splines as a tool has been carried out to obtain the numerical solutions for the nonlinear Schrödinger (NLS) equation. For this purpose, first of all, the Schrödinger equation has been converted into coupled real value differential equations and then they have been discretized using both the forward difference formula and the Crank-Nicolson method. After that, Rubin and Graves linearization techniques have been utilized and the differential quadrature method has been applied to obtain an algebraic equation system. Next, in order to be able to test the efficiency of the newly applied method, the error norms, L2 and L_{∞}, as well as the two lowest invariants, I1 and I2, have been computed. Besides those, the relative changes in those invariants have been presented. Finally, the newly obtained numerical results have been compared with some of those available in the literature for similar parameters. This comparison clearly indicates that the currently utilized method, namely CN-DQM, is an effective and efficient numerical scheme and allows us to propose to solve a wide range of nonlinear equations.

Linear and non-linear regression analysis for the sorption kinetics of methylene blue onto activated carbon.

PubMed

Kumar, K Vasanth

2006-10-11

Batch kinetic experiments were carried out for the sorption of methylene blue onto activated carbon. The experimental kinetics were fitted to the pseudo first-order and pseudo second-order kinetics by linear and a non-linear method. The five different types of Ho pseudo second-order expression have been discussed. A comparison of linear least-squares method and a trial and error non-linear method of estimating the pseudo second-order rate kinetic parameters were examined. The sorption process was found to follow a both pseudo first-order kinetic and pseudo second-order kinetic model. Present investigation showed that it is inappropriate to use a type 1 and type pseudo second-order expressions as proposed by Ho and Blanachard et al. respectively for predicting the kinetic rate constants and the initial sorption rate for the studied system. Three correct possible alternate linear expressions (type 2 to type 4) to better predict the initial sorption rate and kinetic rate constants for the studied system (methylene blue/activated carbon) was proposed. Linear method was found to check only the hypothesis instead of verifying the kinetic model. Non-linear regression method was found to be the more appropriate method to determine the rate kinetic parameters.

Comment on "A note on generalized radial mesh generation for plasma electronic structure"

NASA Astrophysics Data System (ADS)

Pain, J.-Ch.

2011-12-01

In a recent note, B.G. Wilson and V. Sonnad [1] proposed a very useful closed form expression for the efficient generation of analytic log-linear radial meshes. The central point of the note is an implicit equation for the parameter h, involving Lambert's function W[x]. The authors mention that they are unaware of any direct proof of this equation (they obtained it by re-summing the Taylor expansion of h[α] using high-order coefficients obtained by analytic differentiation of the implicit definition using symbolic manipulation). In the present comment, we propose a direct proof of that equation.

Differential Si ring resonators for label-free biosensing

NASA Astrophysics Data System (ADS)

Taniguchi, Tomoya; Yokoyama, Shuhei; Amemiya, Yoshiteru; Ikeda, Takeshi; Kuroda, Akio; Yokoyama, Shin

2016-04-01

Differential Si ring optical resonator sensors have been fabricated. Their detection sensitivity was 10-3-10-2% for sucrose solution, which corresponds to a sensitivity of ˜1.0 ng/ml for prostate-specific antigen (PSA), which is satisfactory for practical use. In the differential sensing the input light is incident to two rings, and one of the outputs is connected to a π phase shifter then the two outputs are merged again. For the differential detection, not only is the common-mode noise canceled, resulting in high sensitivity, but also the temperature stability is much improved. A fluid channel is fabricated so that the detecting liquid flows to the detection ring and the reference liquid flows to the reference ring. We have proposed a method of obtaining a constant sensitivity for the integrated sensors even though the resonance wavelengths of the two rings of the differential sensor are slightly different. It was found that a region exists with a linear relationship between the differential output and the difference in the resonance wavelengths of the two rings. By intentionally differentiating the resonance wavelengths in this linear region, the sensors have a constant sensitivity. Many differential sensors with different ring spaces have been fabricated and the output scattering characteristics were statistically evaluated. As a result, a standard deviation of resonance wavelength σ = 8 × 10-3 nm was obtained for a ring space of 31 µm. From the width of the linear region and the standard deviation, it was estimated from the Gaussian distribution of the resonance wavelength that 93.8% of the devices have the same sensitivity.

A preconditioner for the finite element computation of incompressible, nonlinear elastic deformations

NASA Astrophysics Data System (ADS)

Whiteley, J. P.

2017-10-01

Large, incompressible elastic deformations are governed by a system of nonlinear partial differential equations. The finite element discretisation of these partial differential equations yields a system of nonlinear algebraic equations that are usually solved using Newton's method. On each iteration of Newton's method, a linear system must be solved. We exploit the structure of the Jacobian matrix to propose a preconditioner, comprising two steps. The first step is the solution of a relatively small, symmetric, positive definite linear system using the preconditioned conjugate gradient method. This is followed by a small number of multigrid V-cycles for a larger linear system. Through the use of exemplar elastic deformations, the preconditioner is demonstrated to facilitate the iterative solution of the linear systems arising. The number of GMRES iterations required has only a very weak dependence on the number of degrees of freedom of the linear systems.

Probe-level linear model fitting and mixture modeling results in high accuracy detection of differential gene expression.

PubMed

Lemieux, Sébastien

2006-08-25

The identification of differentially expressed genes (DEGs) from Affymetrix GeneChips arrays is currently done by first computing expression levels from the low-level probe intensities, then deriving significance by comparing these expression levels between conditions. The proposed PL-LM (Probe-Level Linear Model) method implements a linear model applied on the probe-level data to directly estimate the treatment effect. A finite mixture of Gaussian components is then used to identify DEGs using the coefficients estimated by the linear model. This approach can readily be applied to experimental design with or without replication. On a wholly defined dataset, the PL-LM method was able to identify 75% of the differentially expressed genes within 10% of false positives. This accuracy was achieved both using the three replicates per conditions available in the dataset and using only one replicate per condition. The method achieves, on this dataset, a higher accuracy than the best set of tools identified by the authors of the dataset, and does so using only one replicate per condition.

Hybrid finite element method for describing the electrical response of biological cells to applied fields.

PubMed

Ying, Wenjun; Henriquez, Craig S

2007-04-01

A novel hybrid finite element method (FEM) for modeling the response of passive and active biological membranes to external stimuli is presented. The method is based on the differential equations that describe the conservation of electric flux and membrane currents. By introducing the electric flux through the cell membrane as an additional variable, the algorithm decouples the linear partial differential equation part from the nonlinear ordinary differential equation part that defines the membrane dynamics of interest. This conveniently results in two subproblems: a linear interface problem and a nonlinear initial value problem. The linear interface problem is solved with a hybrid FEM. The initial value problem is integrated by a standard ordinary differential equation solver such as the Euler and Runge-Kutta methods. During time integration, these two subproblems are solved alternatively. The algorithm can be used to model the interaction of stimuli with multiple cells of almost arbitrary geometries and complex ion-channel gating at the plasma membrane. Numerical experiments are presented demonstrating the uses of the method for modeling field stimulation and action potential propagation.

Variationally consistent discretization schemes and numerical algorithms for contact problems

NASA Astrophysics Data System (ADS)

Wohlmuth, Barbara

We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-order a priori convergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal-dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-based a posteriori error estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.

Evaluating Differential Effects Using Regression Interactions and Regression Mixture Models

ERIC Educational Resources Information Center

Van Horn, M. Lee; Jaki, Thomas; Masyn, Katherine; Howe, George; Feaster, Daniel J.; Lamont, Andrea E.; George, Melissa R. W.; Kim, Minjung

2015-01-01

Research increasingly emphasizes understanding differential effects. This article focuses on understanding regression mixture models, which are relatively new statistical methods for assessing differential effects by comparing results to using an interactive term in linear regression. The research questions which each model answers, their…

Gene selection for the reconstruction of stem cell differentiation trees: a linear programming approach.

PubMed

Ghadie, Mohamed A; Japkowicz, Nathalie; Perkins, Theodore J

2015-08-15

Stem cell differentiation is largely guided by master transcriptional regulators, but it also depends on the expression of other types of genes, such as cell cycle genes, signaling genes, metabolic genes, trafficking genes, etc. Traditional approaches to understanding gene expression patterns across multiple conditions, such as principal components analysis or K-means clustering, can group cell types based on gene expression, but they do so without knowledge of the differentiation hierarchy. Hierarchical clustering can organize cell types into a tree, but in general this tree is different from the differentiation hierarchy itself. Given the differentiation hierarchy and gene expression data at each node, we construct a weighted Euclidean distance metric such that the minimum spanning tree with respect to that metric is precisely the given differentiation hierarchy. We provide a set of linear constraints that are provably sufficient for the desired construction and a linear programming approach to identify sparse sets of weights, effectively identifying genes that are most relevant for discriminating different parts of the tree. We apply our method to microarray gene expression data describing 38 cell types in the hematopoiesis hierarchy, constructing a weighted Euclidean metric that uses just 175 genes. However, we find that there are many alternative sets of weights that satisfy the linear constraints. Thus, in the style of random-forest training, we also construct metrics based on random subsets of the genes and compare them to the metric of 175 genes. We then report on the selected genes and their biological functions. Our approach offers a new way to identify genes that may have important roles in stem cell differentiation. tperkins@ohri.ca Supplementary data are available at Bioinformatics online. © The Author 2015. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oup.com.

Piecewise linear emulator of the nonlinear Schrödinger equation and the resulting analytic solutions for Bose-Einstein condensates.

PubMed

Theodorakis, Stavros

2003-06-01

We emulate the cubic term Psi(3) in the nonlinear Schrödinger equation by a piecewise linear term, thus reducing the problem to a set of uncoupled linear inhomogeneous differential equations. The resulting analytic expressions constitute an excellent approximation to the exact solutions, as is explicitly shown in the case of the kink, the vortex, and a delta function trap. Such a piecewise linear emulation can be used for any differential equation where the only nonlinearity is a Psi(3) one. In particular, it can be used for the nonlinear Schrödinger equation in the presence of harmonic traps, giving analytic Bose-Einstein condensate solutions that reproduce very accurately the numerically calculated ones in one, two, and three dimensions.

Functional differentiability in time-dependent quantum mechanics.

PubMed

Penz, Markus; Ruggenthaler, Michael

2015-03-28

In this work, we investigate the functional differentiability of the time-dependent many-body wave function and of derived quantities with respect to time-dependent potentials. For properly chosen Banach spaces of potentials and wave functions, Fréchet differentiability is proven. From this follows an estimate for the difference of two solutions to the time-dependent Schrödinger equation that evolve under the influence of different potentials. Such results can be applied directly to the one-particle density and to bounded operators, and present a rigorous formulation of non-equilibrium linear-response theory where the usual Lehmann representation of the linear-response kernel is not valid. Further, the Fréchet differentiability of the wave function provides a new route towards proving basic properties of time-dependent density-functional theory.

Some Theoretical Aspects of Nonzero Sum Differential Games and Applications to Combat Problems

DTIC Science & Technology

1971-06-01

the Equilibrium Solution . 7 Hamilton-Jacobi-Bellman Partial Differential Equations ............. .............. 9 Influence Function Differential...Linearly .......... ............ 18 Problem Statement .......... ............ 18 Formulation of LJB Equations, Influence Function Equations and the TPBVP...19 Control Lawe . . .. ...... ........... 21 Conditions for Influence Function Continuity along Singular Surfaces

HgCdTe Avalanche Photodiode Detectors for Airborne and Spaceborne Lidar at Infrared Wavelengths

NASA Technical Reports Server (NTRS)

Sun, Xiaoli; Abshire, James B.; Beck, Jeffrey D.; Mitra, Pradip; Reiff, Kirk; Yang, Guangning

2017-01-01

We report results from characterizing the HgCdTe avalanche photodiode (APD) sensorchip assemblies (SCA) developed for lidar at infrared wavelength using the high density vertically integrated photodiodes (HDVIP) technique. These devices demonstrated high quantum efficiency, typically greater than 90 between 0.8 micrometers and the cut-off wavelength, greater than 600 APD gain, near unity excess noise factor, 6-10 MHz electrical bandwidth and less than 0.5 fW/Hz(exp.1/2) noise equivalent power (NEP). The detectors provide linear analog output with a dynamic range of 2-3 orders of magnitude at a fixed APD gain without averaging, and over 5 orders of magnitude by adjusting the APD and preamplifier gain settings. They have been successfully used in airborne CO2 and CH4 integrated path differential absorption (IPDA) lidar as a precursor for space lidar applications.

Intrinsic character of Stokes matrices

NASA Astrophysics Data System (ADS)

Gagnon, Jean-François; Rousseau, Christiane

2017-02-01

Two germs of linear analytic differential systems x k + 1Y‧ = A (x) Y with a non-resonant irregular singularity are analytically equivalent if and only if they have the same eigenvalues and equivalent collections of Stokes matrices. The Stokes matrices are the transition matrices between sectors on which the system is analytically equivalent to its formal normal form. Each sector contains exactly one separating ray for each pair of eigenvalues. A rotation in S allows supposing that R+ lies in the intersection of two sectors. Reordering of the coordinates of Y allows ordering the real parts of the eigenvalues, thus yielding triangular Stokes matrices. However, the choice of the rotation in x is not canonical. In this paper we establish how the collection of Stokes matrices depends on this rotation, and hence on a chosen order of the projection of the eigenvalues on a line through the origin.

Least-squares finite element methods for compressible Euler equations

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Carey, G. F.

1990-01-01

A method based on backward finite differencing in time and a least-squares finite element scheme for first-order systems of partial differential equations in space is applied to the Euler equations for gas dynamics. The scheme minimizes the L-sq-norm of the residual within each time step. The method naturally generates numerical dissipation proportional to the time step size. An implicit method employing linear elements has been implemented and proves robust. For high-order elements, computed solutions based on the L-sq method may have oscillations for calculations at similar time step sizes. To overcome this difficulty, a scheme which minimizes the weighted H1-norm of the residual is proposed and leads to a successful scheme with high-degree elements. Finally, a conservative least-squares finite element method is also developed. Numerical results for two-dimensional problems are given to demonstrate the shock resolution of the methods and compare different approaches.

A least-squares finite element method for 3D incompressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Lin, T. L.; Hou, Lin-Jun; Povinelli, Louis A.

1993-01-01

The least-squares finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in symmetric, positive definite algebraic system. An additional compatibility equation, i.e., the divergence of vorticity vector should be zero, is included to make the first-order system elliptic. The Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. The flow in a half of 3D cubic cavity is calculated at Re = 100, 400, and 1,000 with 50 x 52 x 25 trilinear elements. The Taylor-Gortler-like vortices are observed at Re = 1,000.

Dealing with Uncertainties in Initial Orbit Determination

NASA Technical Reports Server (NTRS)

Armellin, Roberto; Di Lizia, Pierluigi; Zanetti, Renato

2015-01-01

A method to deal with uncertainties in initial orbit determination (IOD) is presented. This is based on the use of Taylor differential algebra (DA) to nonlinearly map the observation uncertainties from the observation space to the state space. When a minimum set of observations is available DA is used to expand the solution of the IOD problem in Taylor series with respect to measurement errors. When more observations are available high order inversion tools are exploited to obtain full state pseudo-observations at a common epoch. The mean and covariance of these pseudo-observations are nonlinearly computed by evaluating the expectation of high order Taylor polynomials. Finally, a linear scheme is employed to update the current knowledge of the orbit. Angles-only observations are considered and simplified Keplerian dynamics adopted to ease the explanation. Three test cases of orbit determination of artificial satellites in different orbital regimes are presented to discuss the feature and performances of the proposed methodology.

Synchronisation, electronic circuit implementation, and fractional-order analysis of 5D ordinary differential equations with hidden hyperchaotic attractors

NASA Astrophysics Data System (ADS)

Wei, Zhouchao; Rajagopal, Karthikeyan; Zhang, Wei; Kingni, Sifeu Takougang; Akgül, Akif

2018-04-01

Hidden hyperchaotic attractors can be generated with three positive Lyapunov exponents in the proposed 5D hyperchaotic Burke-Shaw system with only one stable equilibrium. To the best of our knowledge, this feature has rarely been previously reported in any other higher-dimensional systems. Unidirectional linear error feedback coupling scheme is used to achieve hyperchaos synchronisation, which will be estimated by using two indicators: the normalised average root-mean squared synchronisation error and the maximum cross-correlation coefficient. The 5D hyperchaotic system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integration. In addition, fractional-order hidden hyperchaotic system will be considered from the following three aspects: stability, bifurcation analysis and FPGA implementation. Such implementations in real time represent hidden hyperchaotic attractors with important consequences for engineering applications.

System theory as applied differential geometry. [linear system

NASA Technical Reports Server (NTRS)

Hermann, R.

1979-01-01

The invariants of input-output systems under the action of the feedback group was examined. The approach used the theory of Lie groups and concepts of modern differential geometry, and illustrated how the latter provides a basis for the discussion of the analytic structure of systems. Finite dimensional linear systems in a single independent variable are considered. Lessons of more general situations (e.g., distributed parameter and multidimensional systems) which are increasingly encountered as technology advances are presented.

Solving ordinary differential equations by electrical analogy: a multidisciplinary teaching tool

NASA Astrophysics Data System (ADS)

Sanchez Perez, J. F.; Conesa, M.; Alhama, I.

2016-11-01

Ordinary differential equations are the mathematical formulation for a great variety of problems in science and engineering, and frequently, two different problems are equivalent from a mathematical point of view when they are formulated by the same equations. Students acquire the knowledge of how to solve these equations (at least some types of them) using protocols and strict algorithms of mathematical calculation without thinking about the meaning of the equation. The aim of this work is that students learn to design network models or circuits in this way; with simple knowledge of them, students can establish the association of electric circuits and differential equations and their equivalences, from a formal point of view, that allows them to associate knowledge of two disciplines and promote the use of this interdisciplinary approach to address complex problems. Therefore, they learn to use a multidisciplinary tool that allows them to solve these kinds of equations, even students of first course of engineering, whatever the order, grade or type of non-linearity. This methodology has been implemented in numerous final degree projects in engineering and science, e.g., chemical engineering, building engineering, industrial engineering, mechanical engineering, architecture, etc. Applications are presented to illustrate the subject of this manuscript.

On the nonlinear stability of the unsteady, viscous flow of an incompressible fluid in a curved pipe

NASA Technical Reports Server (NTRS)

Shortis, Trudi A.; Hall, Philip

1995-01-01

The stability of the flow of an incompressible, viscous fluid through a pipe of circular cross-section curved about a central axis is investigated in a weakly nonlinear regime. A sinusoidal pressure gradient with zero mean is imposed, acting along the pipe. A WKBJ perturbation solution is constructed, taking into account the need for an inner solution in the vicinity of the outer bend, which is obtained by identifying the saddle point of the Taylor number in the complex plane of the cross-sectional angle co-ordinate. The equation governing the nonlinear evolution of the leading order vortex amplitude is thus determined. The stability analysis of this flow to periodic disturbances leads to a partial differential system dependent on three variables, and since the differential operators in this system are periodic in time, Floquet theory may be applied to reduce this system to a coupled infinite system of ordinary differential equations, together with homogeneous uncoupled boundary conditions. The eigenvalues of this system are calculated numerically to predict a critical Taylor number consistent with the analysis of Papageorgiou. A discussion of how nonlinear effects alter the linear stability analysis is also given, and the nature of the instability determined.

Differentiation of tea varieties using UV-Vis spectra and pattern recognition techniques

NASA Astrophysics Data System (ADS)

Palacios-Morillo, Ana; Alcázar, Ángela.; de Pablos, Fernando; Jurado, José Marcos

2013-02-01

Tea, one of the most consumed beverages all over the world, is of great importance in the economies of a number of countries. Several methods have been developed to classify tea varieties or origins based in pattern recognition techniques applied to chemical data, such as metal profile, amino acids, catechins and volatile compounds. Some of these analytical methods become tedious and expensive to be applied in routine works. The use of UV-Vis spectral data as discriminant variables, highly influenced by the chemical composition, can be an alternative to these methods. UV-Vis spectra of methanol-water extracts of tea have been obtained in the interval 250-800 nm. Absorbances have been used as input variables. Principal component analysis was used to reduce the number of variables and several pattern recognition methods, such as linear discriminant analysis, support vector machines and artificial neural networks, have been applied in order to differentiate the most common tea varieties. A successful classification model was built by combining principal component analysis and multilayer perceptron artificial neural networks, allowing the differentiation between tea varieties. This rapid and simple methodology can be applied to solve classification problems in food industry saving economic resources.

Micropolar curved rods. 2-D, high order, Timoshenko's and Euler-Bernoulli models

NASA Astrophysics Data System (ADS)

Zozulya, V. V.

2017-01-01

New models for micropolar plane curved rods have been developed. 2-D theory is developed from general 2-D equations of linear micropolar elasticity using a special curvilinear system of coordinates related to the middle line of the rod and special hypothesis based on assumptions that take into account the fact that the rod is thin.High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First stress and strain tensors,vectors of displacements and rotation and body force shave been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate.Thereby all equations of elasticity including Hooke's law have been transformed to the corresponding equations for Fourier coefficients. Then in the same way as in the theory of elasticity, system of differential equations in term of displacements and boundary conditions for Fourier coefficients have been obtained. The Timoshenko's and Euler-Bernoulli theories are based on the classical hypothesis and 2-D equations of linear micropolar elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scale when taking in to account micropolar couple stress and rotation effects.

Enhancing the stabilization of aircraft pitch motion control via intelligent and classical method

NASA Astrophysics Data System (ADS)

Lukman, H.; Munawwarah, S.; Azizan, A.; Yakub, F.; Zaki, S. A.; Rasid, Z. A.

2017-12-01

The pitching movement of an aircraft is very important to ensure passengers are intrinsically safe and the aircraft achieve its maximum stability. The equations governing the motion of an aircraft are a complex set of six nonlinear coupled differential equations. Under certain assumptions, it can be decoupled and linearized into longitudinal and lateral equations. Pitch control is a longitudinal problem and thus, only the longitudinal dynamics equations are involved in this system. It is a third order nonlinear system, which is linearized about the operating point. The system is also inherently unstable due to the presence of a free integrator. Because of this, a feedback controller is added in order to solve this problem and enhance the system performance. This study uses two approaches in designing controller: a conventional controller and an intelligent controller. The pitch control scheme consists of proportional, integral and derivatives (PID) for conventional controller and fuzzy logic control (FLC) for intelligent controller. Throughout the paper, the performance of the presented controllers are investigated and compared based on the common criteria of step response. Simulation results have been obtained and analysed by using Matlab and Simulink software. The study shows that FLC controller has higher ability to control and stabilize the aircraft's pitch angle as compared to PID controller.

Analysis of nonlinear axial vibration of single-walled carbon nanotubes using Homotopy perturbation method

NASA Astrophysics Data System (ADS)

Fatahi-Vajari, A.; Azimzadeh, Z.

2018-05-01

This paper investigates the nonlinear axial vibration of single-walled carbon nanotubes (SWCNTs) based on Homotopy perturbation method (HPM). A second order partial differential equation that governs the nonlinear axial vibration for such nanotubes is derived using doublet mechanics (DM) theory. To obtain the nonlinear natural frequency in axial vibration mode, this nonlinear equation is solved using HPM. The influences of some commonly used boundary conditions, amplitude of vibration, changes in vibration modes and variations of the nanotubes geometrical parameters on the nonlinear axial vibration characteristics of SWCNTs are discussed. It was shown that unlike the linear one, the nonlinear natural frequency is dependent to maximum vibration amplitude. Increasing the maximum vibration amplitude decreases the natural frequency of vibration compared to the predictions of the linear models. However, with increase in tube length, the effect of the amplitude on the natural frequency decreases. It was also shown that the amount and variation of nonlinear natural frequency is more apparent in higher mode vibration and two clamped boundary conditions. To show the accuracy and capability of this method, the results obtained herein were compared with the fourth order Runge-Kuta numerical results and good agreement was observed. It is notable that the results generated herein are new and can be served as a benchmark for future works.

Assessment of the non-Gaussianity and non-linearity levels of simulated sEMG signals on stationary segments.

PubMed

Messaoudi, Noureddine; Bekka, Raïs El'hadi; Ravier, Philippe; Harba, Rachid

2017-02-01

The purpose of this paper was to evaluate the effects of the longitudinal single differential (LSD), the longitudinal double differential (LDD) and the normal double differential (NDD) spatial filters, the electrode shape, the inter-electrode distance (IED) on non-Gaussianity and non-linearity levels of simulated surface EMG (sEMG) signals when the maximum voluntary contraction (MVC) varied from 10% to 100% by a step of 10%. The effects of recruitment range thresholds (RR), the firing rate (FR) strategy and the peak firing rate (PFR) of motor units were also considered. A cylindrical multilayer model of the volume conductor and a model of motor unit (MU) recruitment and firing rate were used to simulate sEMG signals in a pool of 120 MUs for 5s. Firstly, the stationarity of sEMG signals was tested by the runs, the reverse arrangements (RA) and the modified reverse arrangements (MRA) tests. Then the non-Gaussianity was characterised with bicoherence and kurtosis, and non-linearity levels was evaluated with linearity test. The kurtosis analysis showed that the sEMG signals detected by the LSD filter were the most Gaussian and those detected by the NDD filter were the least Gaussian. In addition, the sEMG signals detected by the LSD filter were the most linear. For a given filter, the sEMG signals detected by using rectangular electrodes were more Gaussian and more linear than that detected with circular electrodes. Moreover, the sEMG signals are less non-Gaussian and more linear with reverse onion-skin firing rate strategy than those with onion-skin strategy. The levels of sEMG signal Gaussianity and linearity increased with the increase of the IED, RR and PFR. Copyright © 2016 Elsevier Ltd. All rights reserved.

Artificial Intelligence Methodologies in Flight Related Differential Game, Control and Optimization Problems

DTIC Science & Technology

1993-01-31

28 Controllability and Observability ............................. .32 ’ Separation of Learning and Control ... ... 37 Linearization via... Linearization via Transformation of Coordinates and Nonlinear Fedlback . .1 Main Result ......... .............................. 13 Discussion...9 2.1 Basic Structure of a NLM........................ .󈧟 2.2 General Structure of NNLM .......................... .28 2.3 Linear System

Effects of arbuscular mycorrhizal colonization and phosphorus application on nuclear ploidy in Allium porrum plants.

PubMed

Fusconi, Anna; Lingua, Guido; Trotta, Antonio; Berta, Graziella

2005-07-01

Arbuscular mycorrhizal (AM) colonization can strongly affect the plant cell nucleus, causing displacement from the periphery to the center of the cell, hypertrophy and polyploidization. The hypertrophy response has been shown in a variety of AM plants whilst polyploidization has been reported only in Lycopersicon esculentum, a multiploid species with a small genome. In order to determine whether polyploidization is a general plant response to AM colonization, analyses were performed on Allium porrum, a plant with a large genome, which is much less subject to polyploidization than L. esculentum. The ploidy status of leaves, complete root systems and four zones of the adventitious roots was investigated in relation to phosphorus content, AM colonization and root differentiation in A. porrum plants grown under two different regimes of phosphate nutrition in order to distinguish direct effects of the fungus from those of improved nutrition. Results showed the presence of two nuclear populations (2C and 4C) in all treatments and samples. Linear regression analyses suggested a general negative correlation between phosphorus content and the proportion of 2C nuclei. The percentage of 2C nuclei (and consequently that of 4C nuclei), was also influenced by AM colonization, differentiation and ageing of the root cells, which resulted in earlier occurrence, in time and space, of polyploid nuclei.

Nonlinear stability and control study of highly maneuverable high performance aircraft, phase 2

NASA Technical Reports Server (NTRS)

Mohler, R. R.

1992-01-01

This research should lead to the development of new nonlinear methodologies for the adaptive control and stability analysis of high angle-of-attack aircraft such as the F18 (HARV). The emphasis has been on nonlinear adaptive control, but associated model development, system identification, stability analysis and simulation is performed in some detail as well. Various models under investigation for different purposes are summarized in tabular form. Models and simulation for the longitudinal dynamics have been developed for all types except the nonlinear ordinary differential equation model. Briefly, studies completed indicate that nonlinear adaptive control can outperform linear adaptive control for rapid maneuvers with large changes in alpha. The transient responses are compared where the desired alpha varies from 5 degrees to 60 degrees to 30 degrees and back to 5 degrees in all about 16 sec. Here, the horizontal stabilator is the only control used with an assumed first-order linear actuator with a 1/30 sec time constant.

Polarization-correlation optical microscopy of anisotropic biological layers

NASA Astrophysics Data System (ADS)

Ushenko, A. G.; Dubolazov, A. V.; Ushenko, V. A.; Ushenko, Yu. A.; Sakhnovskiy, M. Y.; Balazyuk, V. N.; Khukhlina, O.; Viligorska, K.; Bykov, A.; Doronin, A.; Meglinski, I.

2016-09-01

The theoretical background of azimuthally stable method of Jones-matrix mapping of histological sections of biopsy of myocardium tissue on the basis of spatial frequency selection of the mechanisms of linear and circular birefringence is presented. The diagnostic application of a new correlation parameter - complex degree of mutual anisotropy - is analytically substantiated. The method of measuring coordinate distributions of complex degree of mutual anisotropy with further spatial filtration of their high- and low-frequency components is developed. The interconnections of such distributions with parameters of linear and circular birefringence of myocardium tissue histological sections are found. The comparative results of measuring the coordinate distributions of complex degree of mutual anisotropy formed by fibrillar networks of myosin fibrils of myocardium tissue of different necrotic states - dead due to coronary heart disease and acute coronary insufficiency are shown. The values and ranges of change of the statistical (moments of the 1st - 4th order) parameters of complex degree of mutual anisotropy coordinate distributions are studied. The objective criteria of differentiation of cause of death are determined.

Nonlinear recurrent neural networks for finite-time solution of general time-varying linear matrix equations.

PubMed

Xiao, Lin; Liao, Bolin; Li, Shuai; Chen, Ke

2018-02-01

In order to solve general time-varying linear matrix equations (LMEs) more efficiently, this paper proposes two nonlinear recurrent neural networks based on two nonlinear activation functions. According to Lyapunov theory, such two nonlinear recurrent neural networks are proved to be convergent within finite-time. Besides, by solving differential equation, the upper bounds of the finite convergence time are determined analytically. Compared with existing recurrent neural networks, the proposed two nonlinear recurrent neural networks have a better convergence property (i.e., the upper bound is lower), and thus the accurate solutions of general time-varying LMEs can be obtained with less time. At last, various different situations have been considered by setting different coefficient matrices of general time-varying LMEs and a great variety of computer simulations (including the application to robot manipulators) have been conducted to validate the better finite-time convergence of the proposed two nonlinear recurrent neural networks. Copyright © 2017 Elsevier Ltd. All rights reserved.

Two-dimensional computer simulation of EMVJ and grating solar cells under AMO illumination

NASA Technical Reports Server (NTRS)

Gray, J. L.; Schwartz, R. J.

1984-01-01

A computer program, SCAP2D (Solar Cell Analysis Program in 2-Dimensions), is used to evaluate the Etched Multiple Vertical Junction (EMVJ) and grating solar cells. The aim is to demonstrate how SCAP2D can be used to evaluate cell designs. The cell designs studied are by no means optimal designs. The SCAP2D program solves the three coupled, nonlinear partial differential equations, Poisson's Equation and the hole and electron continuity equations, simultaneously in two-dimensions using finite differences to discretize the equations and Newton's Method to linearize them. The variables solved for are the electrostatic potential and the hole and electron concentrations. Each linear system of equations is solved directly by Gaussian Elimination. Convergence of the Newton Iteration is assumed when the largest correction to the electrostatic potential or hole or electron quasi-potential is less than some predetermined error. A typical problem involves 2000 nodes with a Jacobi matrix of order 6000 and a bandwidth of 243.

Computational manipulation of a radiative MHD flow with Hall current and chemical reaction in the presence of rotating fluid

NASA Astrophysics Data System (ADS)

Alias Suba, Subbu; Muthucumaraswamy, R.

2018-04-01

A numerical analysis of transient radiative MHD(MagnetoHydroDynamic) natural convective flow of a viscous, incompressible, electrically conducting and rotating fluid along a semi-infinite isothermal vertical plate is carried out taking into consideration Hall current, rotation and first order chemical reaction.The coupled non-linear partial differential equations are expressed in difference form using implicit finite difference scheme. The difference equations are then reduced to a system of linear algebraic equations with a tri-diagonal structure which is solved by Thomas Algorithm. The primary and secondary velocity profiles, temperature profile, concentration profile, skin friction, Nusselt number and Sherwood Number are depicted graphically for a range of values of rotation parameter, Hall parameter,magnetic parameter, chemical reaction parameter, radiation parameter, Prandtl number and Schmidt number.It is recognized that rate of heat transfer and rate of mass transfer decrease with increase in time but they increase with increasing values of radiation parameter and Schmidt number respectively.

Influence of two-stream relativistic electron beam parameters on the space-charge wave with broad frequency spectrum formation

NASA Astrophysics Data System (ADS)

Alexander, LYSENKO; Iurii, VOLK

2018-03-01

We developed a cubic non-linear theory describing the dynamics of the multiharmonic space-charge wave (SCW), with harmonics frequencies smaller than the two-stream instability critical frequency, with different relativistic electron beam (REB) parameters. The self-consistent differential equation system for multiharmonic SCW harmonic amplitudes was elaborated in a cubic non-linear approximation. This system considers plural three-wave parametric resonant interactions between wave harmonics and the two-stream instability effect. Different REB parameters such as the input angle with respect to focusing magnetic field, the average relativistic factor value, difference of partial relativistic factors, and plasma frequency of partial beams were investigated regarding their influence on the frequency spectrum width and multiharmonic SCW saturation levels. We suggested ways in which the multiharmonic SCW frequency spectrum widths could be increased in order to use them in multiharmonic two-stream superheterodyne free-electron lasers, with the main purpose of forming a powerful multiharmonic electromagnetic wave.

Fit Point-Wise AB Initio Calculation Potential Energies to a Multi-Dimension Long-Range Model

NASA Astrophysics Data System (ADS)

Zhai, Yu; Li, Hui; Le Roy, Robert J.

2016-06-01

A potential energy surface (PES) is a fundamental tool and source of understanding for theoretical spectroscopy and for dynamical simulations. Making correct assignments for high-resolution rovibrational spectra of floppy polyatomic and van der Waals molecules often relies heavily on predictions generated from a high quality ab initio potential energy surface. Moreover, having an effective analytic model to represent such surfaces can be as important as the ab initio results themselves. For the one-dimensional potentials of diatomic molecules, the most successful such model to date is arguably the ``Morse/Long-Range'' (MLR) function developed by R. J. Le Roy and coworkers. It is very flexible, is everywhere differentiable to all orders. It incorporates correct predicted long-range behaviour, extrapolates sensibly at both large and small distances, and two of its defining parameters are always the physically meaningful well depth {D}_e and equilibrium distance r_e. Extensions of this model, called the Multi-Dimension Morse/Long-Range (MD-MLR) function, linear molecule-linear molecule systems and atom-non-linear molecule system. have been applied successfully to atom-plus-linear molecule, linear molecule-linear molecule and atom-non-linear molecule systems. However, there are several technical challenges faced in modelling the interactions of general molecule-molecule systems, such as the absence of radial minima for some relative alignments, difficulties in fitting short-range potential energies, and challenges in determining relative-orientation dependent long-range coefficients. This talk will illustrate some of these challenges and describe our ongoing work in addressing them. Mol. Phys. 105, 663 (2007); J. Chem. Phys. 131, 204309 (2009); Mol. Phys. 109, 435 (2011). Phys. Chem. Chem. Phys. 10, 4128 (2008); J. Chem. Phys. 130, 144305 (2009) J. Chem. Phys. 132, 214309 (2010) J. Chem. Phys. 140, 214309 (2010)

A Unified Introduction to Ordinary Differential Equations

ERIC Educational Resources Information Center

Lutzer, Carl V.

2006-01-01

This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

NASA Technical Reports Server (NTRS)

Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Gray, Hugh R. (Technical Monitor)

2002-01-01

We discuss an Adams-type predictor-corrector method for the numerical solution of fractional differential equations. The method may be used both for linear and for nonlinear problems, and it may be extended to multi-term equations (involving more than one differential operator) too.

Nonlinear core deflection in injection molding

NASA Astrophysics Data System (ADS)

Poungthong, P.; Giacomin, A. J.; Saengow, C.; Kolitawong, C.; Liao, H.-C.; Tseng, S.-C.

2018-05-01

Injection molding of thin slender parts is often complicated by core deflection. This deflection is caused by molten plastics race tracking through the slit between the core and the rigid cavity wall. The pressure of this liquid exerts a lateral force of the slender core causing the core to bend, and this bending is governed by a nonlinear fifth order ordinary differential equation for the deflection that is not directly in the position along the core. Here we subject this differential equation to 6 sets of boundary conditions, corresponding to 6 commercial core constraints. For each such set of boundary conditions, we develop an explicit approximate analytical solution, including both a linear term and a nonlinear term. By comparison with finite difference solutions, we find our new analytical solutions to be accurate. We then use these solutions to derive explicit analytical approximations for maximum deflections and for the core position of these maximum deflections. Our experiments on the base-gated free-tip boundary condition agree closely with our new explicit approximate analytical solution.

Generalized Lagrangian Jacobi Gauss collocation method for solving unsteady isothermal gas through a micro-nano porous medium

NASA Astrophysics Data System (ADS)

Parand, Kourosh; Latifi, Sobhan; Delkhosh, Mehdi; Moayeri, Mohammad M.

2018-01-01

In the present paper, a new method based on the Generalized Lagrangian Jacobi Gauss (GLJG) collocation method is proposed. The nonlinear Kidder equation, which explains unsteady isothermal gas through a micro-nano porous medium, is a second-order two-point boundary value ordinary differential equation on the unbounded interval [0, ∞). Firstly, using the quasilinearization method, the equation is converted to a sequence of linear ordinary differential equations. Then, by using the GLJG collocation method, the problem is reduced to solving a system of algebraic equations. It must be mentioned that this equation is solved without domain truncation and variable changing. A comparison with some numerical solutions made and the obtained results indicate that the presented solution is highly accurate. The important value of the initial slope, y'(0), is obtained as -1.191790649719421734122828603800159364 for η = 0.5. Comparing to the best result obtained so far, it is accurate up to 36 decimal places.

Elemental profiling and geographical differentiation of Ethiopian coffee samples through inductively coupled plasma-optical emission spectroscopy (ICP-OES), ICP-mass spectrometry (ICP-MS) and direct mercury analyzer (DMA).

PubMed

Habte, Girum; Hwang, In Min; Kim, Jae Sung; Hong, Joon Ho; Hong, Young Sin; Choi, Ji Yeon; Nho, Eun Yeong; Jamila, Nargis; Khan, Naeem; Kim, Kyong Su

2016-12-01

This study was aimed to establish the elemental profiling and provenance of coffee samples collected from eleven major coffee producing regions of Ethiopia. A total of 129 samples were analyzed for forty-five elements using inductively coupled plasma (ICP)-optical emission spectroscopy (OES), ICP-mass spectrometry (MS) and direct mercury analyzer (DMA). Among the macro elements, K showed the highest levels whereas Fe was found to have the lowest concentration values. In all the samples, Ca, K, Mg, P and S contents were statistically significant (p<0.05). Micro elements showed the concentrations order of: Mn>Cu>Sr>Zn>Rb>Ni>B. Contents of the trace elements were lower than the permissible standard values. Inter-regions differentiation by cluster analysis (CA), linear discriminant analysis (LDA) and principal component analysis (PCA) showed that micro and trace elements are the best chemical descriptors of the analyzed coffee samples. Copyright © 2016 Elsevier Ltd. All rights reserved.

On multiple solutions of non-Newtonian Carreau fluid flow over an inclined shrinking sheet

NASA Astrophysics Data System (ADS)

Khan, Masood; Sardar, Humara; Gulzar, M. Mudassar; Alshomrani, Ali Saleh

2018-03-01

This paper presents the multiple solutions of a non-Newtonian Carreau fluid flow over a nonlinear inclined shrinking surface in presence of infinite shear rate viscosity. The governing boundary layer equations are derived for the Carreau fluid with infinite shear rate viscosity. The suitable transformations are employed to alter the leading partial differential equations to a set of ordinary differential equations. The consequential non-linear ODEs are solved numerically by an active numerical approach namely Runge-Kutta Fehlberg fourth-fifth order method accompanied by shooting technique. Multiple solutions are presented graphically and results are shown for various physical parameters. It is important to state that the velocity and momentum boundary layer thickness reduce with increasing viscosity ratio parameter in shear thickening fluid while opposite trend is observed for shear thinning fluid. Another important observation is that the wall shear stress is significantly decreased by the viscosity ratio parameter β∗ for the first solution and opposite trend is observed for the second solution.

An LFMCW detector with new structure and FRFT based differential distance estimation method.

PubMed

Yue, Kai; Hao, Xinhong; Li, Ping

2016-01-01

This paper describes a linear frequency modulated continuous wave (LFMCW) detector which is designed for a collision avoidance radar. This detector can estimate distance between the detector and pedestrians or vehicles, thereby it will help to reduce the likelihood of traffic accidents. The detector consists of a transceiver and a signal processor. A novel structure based on the intermediate frequency signal (IFS) is designed for the transceiver which is different from the traditional LFMCW transceiver using the beat frequency signal (BFS) based structure. In the signal processor, a novel fractional Fourier transform (FRFT) based differential distance estimation (DDE) method is used to detect the distance. The new IFS based structure is beneficial for the FRFT based DDE method to reduce the computation complexity, because it does not need the scan of the optimal FRFT order. Low computation complexity ensures the feasibility of practical applications. Simulations are carried out and results demonstrate the efficiency of the detector designed in this paper.

An approach to get thermodynamic properties from speed of sound

NASA Astrophysics Data System (ADS)

Núñez, M. A.; Medina, L. A.

2017-01-01

An approach for estimating thermodynamic properties of gases from the speed of sound u, is proposed. The square u2, the compression factor Z and the molar heat capacity at constant volume C V are connected by two coupled nonlinear partial differential equations. Previous approaches to solving this system differ in the conditions used on the range of temperature values [Tmin,Tmax]. In this work we propose the use of Dirichlet boundary conditions at Tmin, Tmax. The virial series of the compression factor Z = 1+Bρ+Cρ2+… and other properties leads the problem to the solution of a recursive set of linear ordinary differential equations for the B, C. Analytic solutions of the B equation for Argon are used to study the stability of our approach and previous ones under perturbation errors of the input data. The results show that the approach yields B with a relative error bounded basically by that of the boundary values and the error of other approaches can be some orders of magnitude lager.

Exact solutions for an oscillator with anti-symmetric quadratic nonlinearity

NASA Astrophysics Data System (ADS)

Beléndez, A.; Martínez, F. J.; Beléndez, T.; Pascual, C.; Alvarez, M. L.; Gimeno, E.; Arribas, E.

2018-04-01

Closed-form exact solutions for an oscillator with anti-symmetric quadratic nonlinearity are derived from the first integral of the nonlinear differential equation governing the behaviour of this oscillator. The mathematical model is an ordinary second order differential equation in which the sign of the quadratic nonlinear term changes. Two parameters characterize this oscillator: the coefficient of the linear term and the coefficient of the quadratic term. Not only the common case in which both coefficients are positive but also all possible combinations of positive and negative signs of these coefficients which provide periodic motions are considered, giving rise to four different cases. Three different periods and solutions are obtained, since the same result is valid in two of these cases. An interesting feature is that oscillatory motions whose equilibrium points are not at x = 0 are also considered. The periods are given in terms of an incomplete or complete elliptic integral of the first kind, and the exact solutions are expressed as functions including Jacobi elliptic cosine or sine functions.

Aeroelastic Analysis of Helicopter Rotor Blades Incorporating Anisotropic Piezoelectric Twist Actuation

NASA Technical Reports Server (NTRS)

Wilkie, W. Keats; Belvin, W. Keith; Park, K. C.

1996-01-01

A simple aeroelastic analysis of a helicopter rotor blade incorporating embedded piezoelectric fiber composite, interdigitated electrode blade twist actuators is described. The analysis consists of a linear torsion and flapwise bending model coupled with a nonlinear ONERA based unsteady aerodynamics model. A modified Galerkin procedure is performed upon the rotor blade partial differential equations of motion to develop a system of ordinary differential equations suitable for dynamics simulation using numerical integration. The twist actuation responses for three conceptual fullscale blade designs with realistic constraints on blade mass are numerically evaluated using the analysis. Numerical results indicate that useful amplitudes of nonresonant elastic twist, on the order of one to two degrees, are achievable under one-g hovering flight conditions for interdigitated electrode poling configurations. Twist actuation for the interdigitated electrode blades is also compared with the twist actuation of a conventionally poled piezoelectric fiber composite blade. Elastic twist produced using the interdigitated electrode actuators was found to be four to five times larger than that obtained with the conventionally poled actuators.

An aeroelastic analysis of helicopter rotor blades incorporating piezoelectric fiber composite twist actuation

NASA Technical Reports Server (NTRS)

Wilkie, W. Keats; Park, K. C.

1996-01-01

A simple aeroelastic analysis of a helicopter rotor blade incorporating embedded piezoelectric fiber composite, interdigitated electrode blade twist actuators is described. The analysis consist of a linear torsion and flapwise bending model coupled with a nonlinear ONERA based unsteady aerodynamics model. A modified Galerkin procedure is performed upon the rotor blade partial differential equations of motion to develop a system of ordinary differential equations suitable for numerical integration. The twist actuation responses for three conceptual full-scale blade designs with realistic constraints on blade mass are numerically evaluated using the analysis. Numerical results indicate that useful amplitudes of nonresonant elastic twist, on the order of one to two degrees, are achievable under one-g hovering flight conditions for interdigitated electrode poling configurations. Twist actuation for the interdigitated electrode blades is also compared with the twist actuation of a conventionally poled piezoelectric fiber composite blade. Elastic twist produced using the interdigitated electrode actuators was found to be four to five times larger than that obtained with the conventionally poled actuators.

Transformation elastodynamics and cloaking for flexural waves

NASA Astrophysics Data System (ADS)

Colquitt, D. J.; Brun, M.; Gei, M.; Movchan, A. B.; Movchan, N. V.; Jones, I. S.

2014-12-01

The paper addresses an important issue of cloaking transformations for fourth-order partial differential equations representing flexural waves in thin elastic plates. It is shown that, in contrast with the Helmholtz equation, the general form of the partial differential equation is not invariant with respect to the cloaking transformation. The significant result of this paper is the analysis of the transformed equation and its interpretation in the framework of the linear theory of pre-stressed plates. The paper provides a formal framework for transformation elastodynamics as applied to elastic plates. Furthermore, an algorithm is proposed for designing a broadband square cloak for flexural waves, which employs a regularised push-out transformation. Illustrative numerical examples show high accuracy and efficiency of the proposed cloaking algorithm. In particular, a physical configuration involving a perturbation of an interference pattern generated by two coherent sources is presented. It is demonstrated that the perturbation produced by a cloaked defect is negligibly small even for such a delicate interference pattern.

Stability Analysis of Finite Difference Schemes for Hyperbolic Systems, and Problems in Applied and Computational Linear Algebra.

DTIC Science & Technology

FINITE DIFFERENCE THEORY, * LINEAR ALGEBRA , APPLIED MATHEMATICS, APPROXIMATION(MATHEMATICS), BOUNDARY VALUE PROBLEMS, COMPUTATIONS, HYPERBOLAS, MATHEMATICAL MODELS, NUMERICAL ANALYSIS, PARTIAL DIFFERENTIAL EQUATIONS, STABILITY.

A conformal mapping based fractional order approach for sub-optimal tuning of PID controllers with guaranteed dominant pole placement

NASA Astrophysics Data System (ADS)

Saha, Suman; Das, Saptarshi; Das, Shantanu; Gupta, Amitava

2012-09-01

A novel conformal mapping based fractional order (FO) methodology is developed in this paper for tuning existing classical (Integer Order) Proportional Integral Derivative (PID) controllers especially for sluggish and oscillatory second order systems. The conventional pole placement tuning via Linear Quadratic Regulator (LQR) method is extended for open loop oscillatory systems as well. The locations of the open loop zeros of a fractional order PID (FOPID or PIλDμ) controller have been approximated in this paper vis-à-vis a LQR tuned conventional integer order PID controller, to achieve equivalent integer order PID control system. This approach eases the implementation of analog/digital realization of a FOPID controller with its integer order counterpart along with the advantages of fractional order controller preserved. It is shown here in the paper that decrease in the integro-differential operators of the FOPID/PIλDμ controller pushes the open loop zeros of the equivalent PID controller towards greater damping regions which gives a trajectory of the controller zeros and dominant closed loop poles. This trajectory is termed as "M-curve". This phenomena is used to design a two-stage tuning algorithm which reduces the existing PID controller's effort in a significant manner compared to that with a single stage LQR based pole placement method at a desired closed loop damping and frequency.

Where Does the Ordered Line Come From? Evidence From a Culture of Papua New Guinea.

PubMed

Cooperrider, Kensy; Marghetis, Tyler; Núñez, Rafael

2017-05-01

Number lines, calendars, and measuring sticks all represent order along some dimension (e.g., magnitude) as position on a line. In high-literacy, industrialized societies, this principle of spatial organization- linear order-is a fixture of visual culture and everyday cognition. But what are the principle's origins, and how did it become such a fixture? Three studies investigated intuitions about linear order in the Yupno, members of a culture of Papua New Guinea that lacks conventional representations involving ordered lines, and in U.S. undergraduates. Presented with cards representing differing sizes and numerosities, both groups arranged them using linear order or sometimes spatial grouping, a competing principle. But whereas the U.S. participants produced ordered lines in all tasks, strongly favoring a left-to-right format, the Yupno produced them less consistently, and with variable orientations. Conventional linear representations are thus not necessary to spark the intuition of linear order-which may have other experiential sources-but they nonetheless regiment when and how the principle is used.

Delay differential equations via the matrix Lambert W function and bifurcation analysis: application to machine tool chatter.

PubMed

Yi, Sun; Nelson, Patrick W; Ulsoy, A Galip

2007-04-01

In a turning process modeled using delay differential equations (DDEs), we investigate the stability of the regenerative machine tool chatter problem. An approach using the matrix Lambert W function for the analytical solution to systems of delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert W function, known to be useful for solving scalar first-order DDEs, has recently been extended to a matrix Lambert W function approach to solve systems of DDEs. The essential advantages of the matrix Lambert W approach are not only the similarity to the concept of the state transition matrix in lin ear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay, which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert W function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy and certain other advantages, when compared to traditional graphical, computational and approximate methods.

Water waves generated by impulsively moving obstacle

NASA Astrophysics Data System (ADS)

Makarenko, Nikolay; Kostikov, Vasily

2017-04-01

There are several mechanisms of tsunami-type wave formation such as piston displacement of the ocean floor due to a submarine earthquake, landslides, etc. We consider simplified mathematical formulation which involves non-stationary Euler equations of infinitely deep ideal fluid with submerged compact wave-maker. We apply semi-analytical method [1] based on the reduction of fully nonlinear water wave problem to the integral-differential system for the wave elevation together with normal and tangential fluid velocities at the free surface. Recently, small-time asymptotic solutions were constructed by this method for submerged piston modeled by thin elliptic cylinder which starts with constant acceleration from rest [2,3]. By that, the leading-order solution terms describe several regimes of non-stationary free surface flow such as formation of inertial fluid layer, splash jets and diverging waves over the obstacle. Now we construct asymptotic solution taking into account higher-order nonlinear terms in the case of submerged circular cylinder. The role of non-linearity in the formation mechanism of surface waves is clarified in comparison with linear approximations. This work was supported by RFBR (grant No 15-01-03942). References [1] Makarenko N.I. Nonlinear interaction of submerged cylinder with free surface, JOMAE Trans. ASME, 2003, 125(1), 75-78. [2] Makarenko N.I., Kostikov V.K. Unsteady motion of an elliptic cylinder under a free surface, J. Appl. Mech. Techn. Phys., 2013, 54(3), 367-376. [3] Makarenko N.I., Kostikov V.K. Non-linear water waves generated by impulsive motion of submerged obstacle, NHESS, 2014, 14(4), 751-756.

Differential Geometry and Lie Groups for Physicists

NASA Astrophysics Data System (ADS)

Fecko, Marián.

2006-10-01

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

Differential Geometry and Lie Groups for Physicists

NASA Astrophysics Data System (ADS)

Fecko, Marián.

2011-03-01

Introduction; 1. The concept of a manifold; 2. Vector and tensor fields; 3. Mappings of tensors induced by mappings of manifolds; 4. Lie derivative; 5. Exterior algebra; 6. Differential calculus of forms; 7. Integral calculus of forms; 8. Particular cases and applications of Stoke's Theorem; 9. Poincaré Lemma and cohomologies; 10. Lie Groups - basic facts; 11. Differential geometry of Lie Groups; 12. Representations of Lie Groups and Lie Algebras; 13. Actions of Lie Groups and Lie Algebras on manifolds; 14. Hamiltonian mechanics and symplectic manifolds; 15. Parallel transport and linear connection on M; 16. Field theory and the language of forms; 17. Differential geometry on TM and T*M; 18. Hamiltonian and Lagrangian equations; 19. Linear connection and the frame bundle; 20. Connection on a principal G-bundle; 21. Gauge theories and connections; 22. Spinor fields and Dirac operator; Appendices; Bibliography; Index.

Stabilisation of time-varying linear systems via Lyapunov differential equations

NASA Astrophysics Data System (ADS)

Zhou, Bin; Cai, Guang-Bin; Duan, Guang-Ren

2013-02-01

This article studies stabilisation problem for time-varying linear systems via state feedback. Two types of controllers are designed by utilising solutions to Lyapunov differential equations. The first type of feedback controllers involves the unique positive-definite solution to a parametric Lyapunov differential equation, which can be solved when either the state transition matrix of the open-loop system is exactly known, or the future information of the system matrices are accessible in advance. Different from the first class of controllers which may be difficult to implement in practice, the second type of controllers can be easily implemented by solving a state-dependent Lyapunov differential equation with a given positive-definite initial condition. In both cases, explicit conditions are obtained to guarantee the exponentially asymptotic stability of the associated closed-loop systems. Numerical examples show the effectiveness of the proposed approaches.

Mueller-matrix mapping of biological tissues in differential diagnosis of optical anisotropy mechanisms of protein networks

NASA Astrophysics Data System (ADS)

Ushenko, V. A.; Sidor, M. I.; Marchuk, Yu F.; Pashkovskaya, N. V.; Andreichuk, D. R.

2015-03-01

We report a model of Mueller-matrix description of optical anisotropy of protein networks in biological tissues with allowance for the linear birefringence and dichroism. The model is used to construct the reconstruction algorithms of coordinate distributions of phase shifts and the linear dichroism coefficient. In the statistical analysis of such distributions, we have found the objective criteria of differentiation between benign and malignant tissues of the female reproductive system. From the standpoint of evidence-based medicine, we have determined the operating characteristics (sensitivity, specificity and accuracy) of the Mueller-matrix reconstruction method of optical anisotropy parameters and demonstrated its effectiveness in the differentiation of benign and malignant tumours.

Nonferromagnetic linear variable differential transformer

DOEpatents

Ellis, James F.; Walstrom, Peter L.

1977-06-14

A nonferromagnetic linear variable differential transformer for accurately measuring mechanical displacements in the presence of high magnetic fields is provided. The device utilizes a movable primary coil inside a fixed secondary coil that consists of two series-opposed windings. Operation is such that the secondary output voltage is maintained in phase (depending on polarity) with the primary voltage. The transducer is well-suited to long cable runs and is useful for measuring small displacements in the presence of high or alternating magnetic fields.

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets

DTIC Science & Technology

2014-11-01

linear hybrid systems by linear algebraic methods. In SAS, volume 6337 of LNCS, pages 373–389. Springer, 2010. [19] E. W. Mayr. Membership in polynomial...383–394, 2009. [31] A. Tarski. A decision method for elementary algebra and geometry. Bull. Amer. Math. Soc., 59, 1951. [32] A. Tiwari. Abstractions...A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets Khalil Ghorbal1 Andrew Sogokon2 André Platzer1 November 2014 CMU

Time domain convergence properties of Lyapunov stable penalty methods

NASA Technical Reports Server (NTRS)

Kurdila, A. J.; Sunkel, John

1991-01-01

Linear hyperbolic partial differential equations are analyzed using standard techniques to show that a sequence of solutions generated by the Liapunov stable penalty equations approaches the solution of the differential-algebraic equations governing the dynamics of multibody problems arising in linear vibrations. The analysis does not require that the system be conservative and does not impose any specific integration scheme. Variational statements are derived which bound the error in approximation by the norm of the constraint violation obtained in the approximate solutions.

Conservation properties of numerical integration methods for systems of ordinary differential equations

NASA Technical Reports Server (NTRS)

Rosenbaum, J. S.

1976-01-01

If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.

[Multiple time scales analysis of spatial differentiation characteristics of non-point source nitrogen loss within watershed].

PubMed

Liu, Mei-bing; Chen, Xing-wei; Chen, Ying

2015-07-01

Identification of the critical source areas of non-point source pollution is an important means to control the non-point source pollution within the watershed. In order to further reveal the impact of multiple time scales on the spatial differentiation characteristics of non-point source nitrogen loss, a SWAT model of Shanmei Reservoir watershed was developed. Based on the simulation of total nitrogen (TN) loss intensity of all 38 subbasins, spatial distribution characteristics of nitrogen loss and critical source areas were analyzed at three time scales of yearly average, monthly average and rainstorms flood process, respectively. Furthermore, multiple linear correlation analysis was conducted to analyze the contribution of natural environment and anthropogenic disturbance on nitrogen loss. The results showed that there were significant spatial differences of TN loss in Shanmei Reservoir watershed at different time scales, and the spatial differentiation degree of nitrogen loss was in the order of monthly average > yearly average > rainstorms flood process. TN loss load mainly came from upland Taoxi subbasin, which was identified as the critical source area. At different time scales, land use types (such as farmland and forest) were always the dominant factor affecting the spatial distribution of nitrogen loss, while the effect of precipitation and runoff on the nitrogen loss was only taken in no fertilization month and several processes of storm flood at no fertilization date. This was mainly due to the significant spatial variation of land use and fertilization, as well as the low spatial variability of precipitation and runoff.

Second-order discrete Kalman filtering equations for control-structure interaction simulations

NASA Technical Reports Server (NTRS)

Park, K. C.; Belvin, W. Keith; Alvin, Kenneth F.

1991-01-01

A general form for the first-order representation of the continuous, second-order linear structural dynamics equations is introduced in order to derive a corresponding form of first-order Kalman filtering equations (KFE). Time integration of the resulting first-order KFE is carried out via a set of linear multistep integration formulas. It is shown that a judicious combined selection of computational paths and the undetermined matrices introduced in the general form of the first-order linear structural systems leads to a class of second-order discrete KFE involving only symmetric, N x N solution matrix.

System of multifunctional laser polarimetry of phase and amplitude anisotropy in the diagnosis of endometriosis

NASA Astrophysics Data System (ADS)

Ushenko, Yu. O.; Dubolazov, O. V.; Olar, O. V.

2015-11-01

The theoretical background of azimuthally stable method Jones matrix mapping of histological sections of biopsy of uterine neck on the basis of spatial-frequency selection of the mechanisms of linear and circular birefringence is presented. The comparative results of measuring the coordinate distributions of complex degree of mutual anisotropy formed by polycristalline networks of blood plasma layers of donors (group 1) and patients with endometriosis (group 2). The values and ranges of change of the statistical (moments of the 1st - 4th order) parameters of complex degree of mutual anisotropy coordinate distributions are studied. The objective criteria of diagnostics of the pathology and differentiation of its severity degree are determined.

Multifunctional polarization tomography of optical anisotropy of biological layers in diagnosis of endometriosis

NASA Astrophysics Data System (ADS)

Ushenko, O. G.; Koval, L. D.; Dubolazov, O. V.; Ushenko, Yu. O.; Savich, V. O.; Sidor, M. I.; Marchuk, Yu. F.

2015-09-01

The theoretical background of azimuthally stable method Jones matrix mapping of histological sections of biopsy of uterine neck on the basis of spatial-frequency selection of the mechanisms of linear and circular birefringence is presented. The comparative results of measuring the coordinate distributions of complex degree of mutual anisotropy formed by polycristalline networks of blood plasma layers of donors (group 1) and patients with endometriosis (group 2). The values and ranges of change of the statistical (moments of the 1st - 4th order) parameters of complex degree of mutual anisotropy coordinate distributions are studied. The objective criteria of diagnostics of the pathology and differentiation of its severity degree are determined.

Construction of normal-regular decisions of Bessel typed special system

NASA Astrophysics Data System (ADS)

Tasmambetov, Zhaksylyk N.; Talipova, Meiramgul Zh.

2017-09-01

Studying a special system of differential equations in the separate production of the second order is solved by the degenerate hypergeometric function reducing to the Bessel functions of two variables. To construct a solution of this system near regular and irregular singularities, we use the method of Frobenius-Latysheva applying the concepts of rank and antirank. There is proved the basic theorem that establishes the existence of four linearly independent solutions of studying system type of Bessel. To prove the existence of normal-regular solutions we establish necessary conditions for the existence of such solutions. The existence and convergence of a normally regular solution are shown using the notion of rank and antirank.

Use of Green's functions in the numerical solution of two-point boundary value problems

NASA Technical Reports Server (NTRS)

Gallaher, L. J.; Perlin, I. E.

1974-01-01

This study investigates the use of Green's functions in the numerical solution of the two-point boundary value problem. The first part deals with the role of the Green's function in solving both linear and nonlinear second order ordinary differential equations with boundary conditions and systems of such equations. The second part describes procedures for numerical construction of Green's functions and considers briefly the conditions for their existence. Finally, there is a description of some numerical experiments using nonlinear problems for which the known existence, uniqueness or convergence theorems do not apply. Examples here include some problems in finding rendezvous orbits of the restricted three body system.