A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations

PubMed Central

Güner, Özkan; Cevikel, Adem C.

2014-01-01

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions. PMID:24737972

General existence principles for Stieltjes differential equations with applications to mathematical biology

NASA Astrophysics Data System (ADS)

López Pouso, Rodrigo; Márquez Albés, Ignacio

2018-04-01

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper.

[Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (2)].

PubMed

Murase, Kenya

2015-01-01

In this issue, symbolic methods for solving differential equations were firstly introduced. Of the symbolic methods, Laplace transform method was also introduced together with some examples, in which this method was applied to solving the differential equations derived from a two-compartment kinetic model and an equivalent circuit model for membrane potential. Second, series expansion methods for solving differential equations were introduced together with some examples, in which these methods were used to solve Bessel's and Legendre's differential equations. In the next issue, simultaneous differential equations and various methods for solving these differential equations will be introduced together with some examples in medical physics.

Illness-death model: statistical perspective and differential equations.

PubMed

Brinks, Ralph; Hoyer, Annika

2018-01-27

The aim of this work is to relate the theory of stochastic processes with the differential equations associated with multistate (compartment) models. We show that the Kolmogorov Forward Differential Equations can be used to derive a relation between the prevalence and the transition rates in the illness-death model. Then, we prove mathematical well-definedness and epidemiological meaningfulness of the prevalence of the disease. As an application, we derive the incidence of diabetes from a series of cross-sections.

Construction and accuracy of partial differential equation approximations to the chemical master equation.

PubMed

Grima, Ramon

2011-11-01

The mesoscopic description of chemical kinetics, the chemical master equation, can be exactly solved in only a few simple cases. The analytical intractability stems from the discrete character of the equation, and hence considerable effort has been invested in the development of Fokker-Planck equations, second-order partial differential equation approximations to the master equation. We here consider two different types of higher-order partial differential approximations, one derived from the system-size expansion and the other from the Kramers-Moyal expansion, and derive the accuracy of their predictions for chemical reactive networks composed of arbitrary numbers of unimolecular and bimolecular reactions. In particular, we show that the partial differential equation approximation of order Q from the Kramers-Moyal expansion leads to estimates of the mean number of molecules accurate to order Ω(-(2Q-3)/2), of the variance of the fluctuations in the number of molecules accurate to order Ω(-(2Q-5)/2), and of skewness accurate to order Ω(-(Q-2)). We also show that for large Q, the accuracy in the estimates can be matched only by a partial differential equation approximation from the system-size expansion of approximate order 2Q. Hence, we conclude that partial differential approximations based on the Kramers-Moyal expansion generally lead to considerably more accurate estimates in the mean, variance, and skewness than approximations of the same order derived from the system-size expansion.

Estimation of Ordinary Differential Equation Parameters Using Constrained Local Polynomial Regression.

PubMed

Ding, A Adam; Wu, Hulin

2014-10-01

We propose a new method to use a constrained local polynomial regression to estimate the unknown parameters in ordinary differential equation models with a goal of improving the smoothing-based two-stage pseudo-least squares estimate. The equation constraints are derived from the differential equation model and are incorporated into the local polynomial regression in order to estimate the unknown parameters in the differential equation model. We also derive the asymptotic bias and variance of the proposed estimator. Our simulation studies show that our new estimator is clearly better than the pseudo-least squares estimator in estimation accuracy with a small price of computational cost. An application example on immune cell kinetics and trafficking for influenza infection further illustrates the benefits of the proposed new method.

Estimation of Ordinary Differential Equation Parameters Using Constrained Local Polynomial Regression

PubMed Central

Ding, A. Adam; Wu, Hulin

2015-01-01

We propose a new method to use a constrained local polynomial regression to estimate the unknown parameters in ordinary differential equation models with a goal of improving the smoothing-based two-stage pseudo-least squares estimate. The equation constraints are derived from the differential equation model and are incorporated into the local polynomial regression in order to estimate the unknown parameters in the differential equation model. We also derive the asymptotic bias and variance of the proposed estimator. Our simulation studies show that our new estimator is clearly better than the pseudo-least squares estimator in estimation accuracy with a small price of computational cost. An application example on immune cell kinetics and trafficking for influenza infection further illustrates the benefits of the proposed new method. PMID:26401093

A new approach to Catalan numbers using differential equations

NASA Astrophysics Data System (ADS)

Kim, D. S.; Kim, T.

2017-10-01

In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in a certain sense. From these differential equations, we obtain some new and explicit identities for Catalan and higher-order Catalan numbers. In addition, by other means than differential equations, we also derive some interesting identities involving Catalan numbers which are of arithmetic and combinatorial nature.

Factorization and the synthesis of optimal feedback kernels for differential-delay systems

NASA Technical Reports Server (NTRS)

Milman, Mark M.; Scheid, Robert E.

1987-01-01

A combination of ideas from the theories of operator Riccati equations and Volterra factorizations leads to the derivation of a novel, relatively simple set of hyperbolic equations which characterize the optimal feedback kernel for the finite-time regulator problem for autonomous differential-delay systems. Analysis of these equations elucidates the underlying structure of the feedback kernel and leads to the development of fast and accurate numerical methods for its computation. Unlike traditional formulations based on the operator Riccati equation, the gain is characterized by means of classical solutions of the derived set of equations. This leads to the development of approximation schemes which are analogous to what has been accomplished for systems of ordinary differential equations with given initial conditions.

Correcting the initialization of models with fractional derivatives via history-dependent conditions

NASA Astrophysics Data System (ADS)

Du, Maolin; Wang, Zaihua

2016-04-01

Fractional differential equations are more and more used in modeling memory (history-dependent, non-local, or hereditary) phenomena. Conventional initial values of fractional differential equations are defined at a point, while recent works define initial conditions over histories. We prove that the conventional initialization of fractional differential equations with a Riemann-Liouville derivative is wrong with a simple counter-example. The initial values were assumed to be arbitrarily given for a typical fractional differential equation, but we find one of these values can only be zero. We show that fractional differential equations are of infinite dimensions, and the initial conditions, initial histories, are defined as functions over intervals. We obtain the equivalent integral equation for Caputo case. With a simple fractional model of materials, we illustrate that the recovery behavior is correct with the initial creep history, but is wrong with initial values at the starting point of the recovery. We demonstrate the application of initial history by solving a forced fractional Lorenz system numerically.

Finite-difference models of ordinary differential equations - Influence of denominator functions

NASA Technical Reports Server (NTRS)

Mickens, Ronald E.; Smith, Arthur

1990-01-01

This paper discusses the influence on the solutions of finite-difference schemes of using a variety of denominator functions in the discrete modeling of the derivative for any ordinary differential equation. The results obtained are a consequence of using a generalized definition of the first derivative. A particular example of the linear decay equation is used to illustrate in detail the various solution possibilities that can occur.

Analytical solution of the nonlinear diffusion equation

NASA Astrophysics Data System (ADS)

Shanker Dubey, Ravi; Goswami, Pranay

2018-05-01

In the present paper, we derive the solution of the nonlinear fractional partial differential equations using an efficient approach based on the q -homotopy analysis transform method ( q -HATM). The fractional diffusion equations derivatives are considered in Caputo sense. The derived results are graphically demonstrated as well.

Analysis of nonlocal neural fields for both general and gamma-distributed connectivities

NASA Astrophysics Data System (ADS)

Hutt, Axel; Atay, Fatihcan M.

2005-04-01

This work studies the stability of equilibria in spatially extended neuronal ensembles. We first derive the model equation from statistical properties of the neuron population. The obtained integro-differential equation includes synaptic and space-dependent transmission delay for both general and gamma-distributed synaptic connectivities. The latter connectivity type reveals infinite, finite, and vanishing self-connectivities. The work derives conditions for stationary and nonstationary instabilities for both kernel types. In addition, a nonlinear analysis for general kernels yields the order parameter equation of the Turing instability. To compare the results to findings for partial differential equations (PDEs), two typical PDE-types are derived from the examined model equation, namely the general reaction-diffusion equation and the Swift-Hohenberg equation. Hence, the discussed integro-differential equation generalizes these PDEs. In the case of the gamma-distributed kernels, the stability conditions are formulated in terms of the mean excitatory and inhibitory interaction ranges. As a novel finding, we obtain Turing instabilities in fields with local inhibition-lateral excitation, while wave instabilities occur in fields with local excitation and lateral inhibition. Numerical simulations support the analytical results.

Differential equations for loop integrals in Baikov representation

NASA Astrophysics Data System (ADS)

Bosma, Jorrit; Larsen, Kasper J.; Zhang, Yang

2018-05-01

We present a proof that differential equations for Feynman loop integrals can always be derived in Baikov representation without involving dimension-shift identities. We moreover show that in a large class of two- and three-loop diagrams it is possible to avoid squared propagators in the intermediate steps of setting up the differential equations.

Remarks on the Non-Linear Differential Equation the Second Derivative of Theta Plus A Sine Theta Equals 0.

ERIC Educational Resources Information Center

Fay, Temple H.; O'Neal, Elizabeth A.

1985-01-01

The authors draw together a variety of facts concerning a nonlinear differential equation and compare the exact solution with approximate solutions. Then they provide an expository introduction to the elliptic sine function suitable for presentation in undergraduate courses on differential equations. (MNS)

A Nonlinear Programming Perspective on Sensitivity Calculations for Systems Governed by State Equations

NASA Technical Reports Server (NTRS)

Lewis, Robert Michael

1997-01-01

This paper discusses the calculation of sensitivities. or derivatives, for optimization problems involving systems governed by differential equations and other state relations. The subject is examined from the point of view of nonlinear programming, beginning with the analytical structure of the first and second derivatives associated with such problems and the relation of these derivatives to implicit differentiation and equality constrained optimization. We also outline an error analysis of the analytical formulae and compare the results with similar results for finite-difference estimates of derivatives. We then attend to an investigation of the nature of the adjoint method and the adjoint equations and their relation to directions of steepest descent. We illustrate the points discussed with an optimization problem in which the variables are the coefficients in a differential operator.

Solving Ordinary Differential Equations

NASA Technical Reports Server (NTRS)

Krogh, F. T.

1987-01-01

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

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.

The Pendulum and the Calculus.

ERIC Educational Resources Information Center

Sworder, Steven C.

A pair of experiments, appropriate for the lower division fourth semester calculus or differential equations course, are presented. The second order differential equation representing the equation of motion of a simple pendulum is derived. The period of oscillation for a particular pendulum can be predicted from the solution to this equation. As a…

W-transform for exponential stability of second order delay differential equations without damping terms.

PubMed

Domoshnitsky, Alexander; Maghakyan, Abraham; Berezansky, Leonid

2017-01-01

In this paper a method for studying stability of the equation [Formula: see text] not including explicitly the first derivative is proposed. We demonstrate that although the corresponding ordinary differential equation [Formula: see text] is not exponentially stable, the delay equation can be exponentially stable.

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.

Legendre-Tau approximation for functional differential equations. Part 3: Eigenvalue approximations and uniform stability

NASA Technical Reports Server (NTRS)

Ito, K.

1984-01-01

The stability and convergence properties of the Legendre-tau approximation for hereditary differential systems are analyzed. A charactristic equation is derived for the eigenvalues of the resulting approximate system. As a result of this derivation the uniform exponential stability of the solution semigroup is preserved under approximation. It is the key to obtaining the convergence of approximate solutions of the algebraic Riccati equation in trace norm.

An accessible four-dimensional treatment of Maxwell's equations in terms of differential forms

NASA Astrophysics Data System (ADS)

Sá, Lucas

2017-03-01

Maxwell’s equations are derived in terms of differential forms in the four-dimensional Minkowski representation, starting from the three-dimensional vector calculus differential version of these equations. Introducing all the mathematical and physical concepts needed (including the tool of differential forms), using only knowledge of elementary vector calculus and the local vector version of Maxwell’s equations, the equations are reduced to a simple and elegant set of two equations for a unified quantity, the electromagnetic field. The treatment should be accessible for students taking a first course on electromagnetism.

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.

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 generalized Langevin equations and generalized Fokker-Planck equations.

PubMed

Darve, Eric; Solomon, Jose; Kia, Amirali

2009-07-07

The Mori-Zwanzig formalism is an effective tool to derive differential equations describing the evolution of a small number of resolved variables. In this paper we present its application to the derivation of generalized Langevin equations and generalized non-Markovian Fokker-Planck equations. We show how long time scales rates and metastable basins can be extracted from these equations. Numerical algorithms are proposed to discretize these equations. An important aspect is the numerical solution of the orthogonal dynamics equation which is a partial differential equation in a high dimensional space. We propose efficient numerical methods to solve this orthogonal dynamics equation. In addition, we present a projection formalism of the Mori-Zwanzig type that is applicable to discrete maps. Numerical applications are presented from the field of Hamiltonian systems.

Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (3).

PubMed

Murase, Kenya

2016-01-01

In this issue, simultaneous differential equations were introduced. These differential equations are often used in the field of medical physics. The methods for solving them were also introduced, which include Laplace transform and matrix methods. Some examples were also introduced, in which Laplace transform and matrix methods were applied to solving simultaneous differential equations derived from a three-compartment kinetic model for analyzing the glucose metabolism in tissues and Bloch equations for describing the behavior of the macroscopic magnetization in magnetic resonance imaging.In the next (final) issue, partial differential equations and various methods for solving them will be introduced together with some examples in medical physics.

Measurement-based perturbation theory and differential equation parameter estimation with applications to satellite gravimetry

NASA Astrophysics Data System (ADS)

Xu, Peiliang

2018-06-01

The numerical integration method has been routinely used by major institutions worldwide, for example, NASA Goddard Space Flight Center and German Research Center for Geosciences (GFZ), to produce global gravitational models from satellite tracking measurements of CHAMP and/or GRACE types. Such Earth's gravitational products have found widest possible multidisciplinary applications in Earth Sciences. The method is essentially implemented by solving the differential equations of the partial derivatives of the orbit of a satellite with respect to the unknown harmonic coefficients under the conditions of zero initial values. From the mathematical and statistical point of view, satellite gravimetry from satellite tracking is essentially the problem of estimating unknown parameters in the Newton's nonlinear differential equations from satellite tracking measurements. We prove that zero initial values for the partial derivatives are incorrect mathematically and not permitted physically. The numerical integration method, as currently implemented and used in mathematics and statistics, chemistry and physics, and satellite gravimetry, is groundless, mathematically and physically. Given the Newton's nonlinear governing differential equations of satellite motion with unknown equation parameters and unknown initial conditions, we develop three methods to derive new local solutions around a nominal reference orbit, which are linked to measurements to estimate the unknown corrections to approximate values of the unknown parameters and the unknown initial conditions. Bearing in mind that satellite orbits can now be tracked almost continuously at unprecedented accuracy, we propose the measurement-based perturbation theory and derive global uniformly convergent solutions to the Newton's nonlinear governing differential equations of satellite motion for the next generation of global gravitational models. Since the solutions are global uniformly convergent, theoretically speaking, they are able to extract smallest possible gravitational signals from modern and future satellite tracking measurements, leading to the production of global high-precision, high-resolution gravitational models. By directly turning the nonlinear differential equations of satellite motion into the nonlinear integral equations, and recognizing the fact that satellite orbits are measured with random errors, we further reformulate the links between satellite tracking measurements and the global uniformly convergent solutions to the Newton's governing differential equations as a condition adjustment model with unknown parameters, or equivalently, the weighted least squares estimation of unknown differential equation parameters with equality constraints, for the reconstruction of global high-precision, high-resolution gravitational models from modern (and future) satellite tracking measurements.

From differential to difference equations for first order ODEs

NASA Technical Reports Server (NTRS)

Freed, Alan D.; Walker, Kevin P.

1991-01-01

When constructing an algorithm for the numerical integration of a differential equation, one should first convert the known ordinary differential equation (ODE) into an ordinary difference equation. Given this difference equation, one can develop an appropriate numerical algorithm. This technical note describes the derivation of two such ordinary difference equations applicable to a first order ODE. The implicit ordinary difference equation has the same asymptotic expansion as the ODE itself, whereas the explicit ordinary difference equation has an asymptotic that is similar in structure but different in value when compared with that of the ODE.

Dynamics of the Pin Pallet Runaway Escapement

DTIC Science & Technology

1978-06-01

for Continued Work 29 References 32 I Appendixes A Kinematics of Coupled Motion 34 B Differential Equation of Coupled Motion 38 f C Moment Arms 42 D...Expressions for these quantities are derived in appendix D. The differential equations for the free motion of the pallet and the escape-wheel are...Coupled Motion (location 100) To solve the differential equation of coupled motion (see equation .B (-10) of appendix B)- the main program calls on

Nonlinear Equations of Motion for Cantilever Rotor Blades in Hover with Pitch Link Flexibility, Twist, Precone, Droop, Sweep, Torque Offset, and Blade Root Offset

NASA Technical Reports Server (NTRS)

Hodges, D. H.

1976-01-01

Nonlinear equations of motion for a cantilever rotor blade are derived for the hovering flight condition. The blade is assumed to have twist, precone, droop, sweep, torque offset and blade root offset, and the elastic axis and the axes of center of mass, tension, and aerodynamic center coincident at the quarter chord. The blade is cantilevered in bending, but has a torsional root spring to simulate pitch link flexibility. Aerodynamic forces acting on the blade are derived from strip theory based on quasi-steady two-dimensional airfoil theory. The equations are hybrid, consisting of one integro-differential equation for root torsion and three integro-partial differential equations for flatwise and chordwise bending and elastic torsion. The equations are specialized for a uniform blade and reduced to nonlinear ordinary differential equations by Galerkin's method. They are linearized for small perturbation motions about the equilibrium operating condition. Modal analysis leads to formulation of a standard eigenvalue problem where the elements of the stability matrix depend on the solution of the equilibrium equations. Two different forms of the root torsion equation are derived that yield virtually identical numerical results. This provides a reasonable check for the accuracy of the equations.

Distribution theory for Schrödinger’s integral equation

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

Lange, Rutger-Jan, E-mail: rutger-jan.lange@cantab.net

2015-12-15

Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger’s equation. This paper, in contrast, investigates the integral form of Schrödinger’s equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger’s integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger’s differential equation. This hints at a possible deeper connection between bothmore » forms of the equation. We also sketch a generalisation of Kurasov’s [J. Math. Anal. Appl. 201(1), 297–323 (1996)] result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger’s integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger’s differential equation. Third, we derive boundary conditions for “super-singular” potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger’s integral equation is a viable tool for studying singular interactions in quantum mechanics.« less

A New Factorisation of a General Second Order Differential Equation

ERIC Educational Resources Information Center

Clegg, Janet

2006-01-01

A factorisation of a general second order ordinary differential equation is introduced from which the full solution to the equation can be obtained by performing two integrations. The method is compared with traditional methods for solving these type of equations. It is shown how the Green's function can be derived directly from the factorisation…

Finding the Equation for a Vibrating Car Antenna.

ERIC Educational Resources Information Center

Newburgh, Ronald; Newburgh, G. Alexander

2000-01-01

Presents the physical assumptions and mathematical expressions necessary to derive a fourth-order differential equation that describes the vibration of a particular car antenna. Contends that while students may not be able to derive or use the equation, they should be able to appreciate a guided derivation as an example of how physics is done.…

Numerical analysis for distributed-order differential equations

NASA Astrophysics Data System (ADS)

Diethelm, Kai; Ford, Neville J.

2009-03-01

In this paper we present and analyse a numerical method for the solution of a distributed-order differential equation of the general form where m is a positive real number and where the derivative is taken to be a fractional derivative of Caputo type of order r. We give a convergence theory for our method and conclude with some numerical examples.

A Posteriori Finite Element Bounds for Sensitivity Derivatives of Partial-Differential-Equation Outputs. Revised

NASA Technical Reports Server (NTRS)

Lewis, Robert Michael; Patera, Anthony T.; Peraire, Jaume

1998-01-01

We present a Neumann-subproblem a posteriori finite element procedure for the efficient and accurate calculation of rigorous, 'constant-free' upper and lower bounds for sensitivity derivatives of functionals of the solutions of partial differential equations. The design motivation for sensitivity derivative error control is discussed; the a posteriori finite element procedure is described; the asymptotic bounding properties and computational complexity of the method are summarized; and illustrative numerical results are presented.

Analysis of stability for stochastic delay integro-differential equations.

PubMed

Zhang, Yu; Li, Longsuo

2018-01-01

In this paper, we concern stability of numerical methods applied to stochastic delay integro-differential equations. For linear stochastic delay integro-differential equations, it is shown that the mean-square stability is derived by the split-step backward Euler method without any restriction on step-size, while the Euler-Maruyama method could reproduce the mean-square stability under a step-size constraint. We also confirm the mean-square stability of the split-step backward Euler method for nonlinear stochastic delay integro-differential equations. The numerical experiments further verify the theoretical results.

Differential Equation Models for Sharp Threshold Dynamics

DTIC Science & Technology

2012-08-01

dynamics, and the Lanchester model of armed conflict, where the loss of a key capability drastically changes dynamics. We derive and demonstrate a step...dynamics using differential equations. 15. SUBJECT TERMS Differential Equations, Markov Population Process, S-I-R Epidemic, Lanchester Model 16...infection, where a detection event drastically changes dynamics, and the Lanchester model of armed conflict, where the loss of a key capability

Algebraic and geometric structures of analytic partial differential equations

NASA Astrophysics Data System (ADS)

Kaptsov, O. V.

2016-11-01

We study the problem of the compatibility of nonlinear partial differential equations. We introduce the algebra of convergent power series, the module of derivations of this algebra, and the module of Pfaffian forms. Systems of differential equations are given by power series in the space of infinite jets. We develop a technique for studying the compatibility of differential systems analogous to the Gröbner bases. Using certain assumptions, we prove that compatible systems generate infinite manifolds.

Cellular Automata for Spatiotemporal Pattern Formation from Reaction-Diffusion Partial Differential Equations

NASA Astrophysics Data System (ADS)

Ohmori, Shousuke; Yamazaki, Yoshihiro

2016-01-01

Ultradiscrete equations are derived from a set of reaction-diffusion partial differential equations, and cellular automaton rules are obtained on the basis of the ultradiscrete equations. Some rules reproduce the dynamical properties of the original reaction-diffusion equations, namely, bistability and pulse annihilation. Furthermore, other rules bring about soliton-like preservation and periodic pulse generation with a pacemaker, which are not obtained from the original reaction-diffusion equations.

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.

Dynamically orthogonal field equations for stochastic flows and particle dynamics

DTIC Science & Technology

2011-02-01

where uncertainty ‘lives’ as well as a system of Stochastic Di erential Equations that de nes how the uncertainty evolves in the time varying stochastic ... stochastic dynamical component that are both time and space dependent, we derive a system of field equations consisting of a Partial Differential Equation...a system of Stochastic Differential Equations that defines how the stochasticity evolves in the time varying stochastic subspace. These new

Sparse dynamics for partial differential equations

PubMed Central

Schaeffer, Hayden; Caflisch, Russel; Hauck, Cory D.; Osher, Stanley

2013-01-01

We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms. PMID:23533273

Exact solutions for STO and (3+1)-dimensional KdV-ZK equations using (G‧/G2) -expansion method

NASA Astrophysics Data System (ADS)

Bibi, Sadaf; Mohyud-Din, Syed Tauseef; Ullah, Rahmat; Ahmed, Naveed; Khan, Umar

This article deals with finding some exact solutions of nonlinear fractional differential equations (NLFDEs) by applying a relatively new method known as (G‧/G2) -expansion method. Solutions of space-time fractional Sharma-Tasso-Olever (STO) equation of fractional order and (3+1)-dimensional KdV-Zakharov Kuznetsov (KdV-ZK) equation of fractional order are reckoned to demonstrate the validity of this method. The fractional derivative version of modified Riemann-Liouville, linked with Fractional complex transform is employed to transform fractional differential equations into the corresponding ordinary differential equations.

Sparse dynamics for partial differential equations.

PubMed

Schaeffer, Hayden; Caflisch, Russel; Hauck, Cory D; Osher, Stanley

2013-04-23

We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms.

Symmetries of the Gas Dynamics Equations using the Differential Form Method

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

Ramsey, Scott D.; Baty, Roy S.

Here, a brief review of the theory of exterior differential systems and isovector symmetry analysis methods is presented in the context of the one-dimensional inviscid compressible flow equations. These equations are formulated as an exterior differential system with equation of state (EOS) closure provided in terms of an adiabatic bulk modulus. The scaling symmetry generators—and corresponding EOS constraints—otherwise appearing in the existing literature are recovered through the application and invariance under Lie derivative dragging operations.

Symmetries of the Gas Dynamics Equations using the Differential Form Method

DOE PAGES

Ramsey, Scott D.; Baty, Roy S.

2017-11-21

Here, a brief review of the theory of exterior differential systems and isovector symmetry analysis methods is presented in the context of the one-dimensional inviscid compressible flow equations. These equations are formulated as an exterior differential system with equation of state (EOS) closure provided in terms of an adiabatic bulk modulus. The scaling symmetry generators—and corresponding EOS constraints—otherwise appearing in the existing literature are recovered through the application and invariance under Lie derivative dragging operations.

A result on differential inequalities and its application to higher order trajectory derivatives

NASA Technical Reports Server (NTRS)

Gunderson, R. W.

1973-01-01

A result on differential inequalities is obtained by considering the adjoint differential equation of the variational equation of the right side of the inequality. The main theorem is proved using basic results on differentiability of solutions with respect to initial conditions. The result is then applied to the problem of determining solution behavior using comparison techniques.

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.

Introduction to Generalized Functions with Applications in Aerodynamics and Aeroacoustics

NASA Technical Reports Server (NTRS)

Farassat, F.

1994-01-01

Generalized functions have many applications in science and engineering. One useful aspect is that discontinuous functions can be handled as easily as continuous or differentiable functions and provide a powerful tool in formulating and solving many problems of aerodynamics and acoustics. Furthermore, generalized function theory elucidates and unifies many ad hoc mathematical approaches used by engineers and scientists. We define generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support, then introduce the concept of generalized differentiation. Generalized differentiation is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some results of classical analysis, are derived with the generalized function theory. Other applications of the generalized function theory in aerodynamics discussed here are the derivations of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of the finite part of divergent integrals, the derivation of the Oswatitsch integral equation of transonic flow, and the analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics include the derivation of the Kirchhoff formula for moving surfaces, the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams-Hawkings equation.

From stochastic processes to numerical methods: A new scheme for solving reaction subdiffusion fractional partial differential equations

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

Angstmann, C.N.; Donnelly, I.C.; Henry, B.I., E-mail: B.Henry@unsw.edu.au

We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also showmore » that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.« less

Exp-function method for solving fractional partial differential equations.

PubMed

Zheng, Bin

2013-01-01

We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.

Network Reconstruction From High-Dimensional Ordinary Differential Equations.

PubMed

Chen, Shizhe; Shojaie, Ali; Witten, Daniela M

2017-01-01

We consider the task of learning a dynamical system from high-dimensional time-course data. For instance, we might wish to estimate a gene regulatory network from gene expression data measured at discrete time points. We model the dynamical system nonparametrically as a system of additive ordinary differential equations. Most existing methods for parameter estimation in ordinary differential equations estimate the derivatives from noisy observations. This is known to be challenging and inefficient. We propose a novel approach that does not involve derivative estimation. We show that the proposed method can consistently recover the true network structure even in high dimensions, and we demonstrate empirical improvement over competing approaches. Supplementary materials for this article are available online.

Analytical approach for the fractional differential equations by using the extended tanh method

NASA Astrophysics Data System (ADS)

Pandir, Yusuf; Yildirim, Ayse

2018-07-01

In this study, we consider analytical solutions of space-time fractional derivative foam drainage equation, the nonlinear Korteweg-de Vries equation with time and space-fractional derivatives and time-fractional reaction-diffusion equation by using the extended tanh method. The fractional derivatives are defined in the modified Riemann-Liouville context. As a result, various exact analytical solutions consisting of trigonometric function solutions, kink-shaped soliton solutions and new exact solitary wave solutions are obtained.

Stable multi-domain spectral penalty methods for fractional partial differential equations

NASA Astrophysics Data System (ADS)

Xu, Qinwu; Hesthaven, Jan S.

2014-01-01

We propose stable multi-domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order. First, a high order discretization is proposed to approximate fractional derivatives of any order on any given grids based on orthogonal polynomials. The approximation order is analyzed and verified through numerical examples. Based on the discrete fractional derivative, we introduce stable multi-domain spectral penalty methods for solving fractional advection and diffusion equations. The equations are discretized in each sub-domain separately and the global schemes are obtained by weakly imposed boundary and interface conditions through a penalty term. Stability of the schemes are analyzed and numerical examples based on both uniform and nonuniform grids are considered to highlight the flexibility and high accuracy of the proposed schemes.

Analytical solutions for coupling fractional partial differential equations with Dirichlet boundary conditions

NASA Astrophysics Data System (ADS)

Ding, Xiao-Li; Nieto, Juan J.

2017-11-01

In this paper, we consider the analytical solutions of coupling fractional partial differential equations (FPDEs) with Dirichlet boundary conditions on a finite domain. Firstly, the method of successive approximations is used to obtain the analytical solutions of coupling multi-term time fractional ordinary differential equations. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the coupling FPDEs to the coupling multi-term time fractional ordinary differential equations. By applying the obtained analytical solutions to the resulting multi-term time fractional ordinary differential equations, the desired analytical solutions of the coupling FPDEs are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.

q-Gaussian distributions and multiplicative stochastic processes for analysis of multiple financial time series

NASA Astrophysics Data System (ADS)

Sato, Aki-Hiro

2010-12-01

This study considers q-Gaussian distributions and stochastic differential equations with both multiplicative and additive noises. In the M-dimensional case a q-Gaussian distribution can be theoretically derived as a stationary probability distribution of the multiplicative stochastic differential equation with both mutually independent multiplicative and additive noises. By using the proposed stochastic differential equation a method to evaluate a default probability under a given risk buffer is proposed.

Modeling biological gradient formation: combining partial differential equations and Petri nets.

PubMed

Bertens, Laura M F; Kleijn, Jetty; Hille, Sander C; Heiner, Monika; Koutny, Maciej; Verbeek, Fons J

2016-01-01

Both Petri nets and differential equations are important modeling tools for biological processes. In this paper we demonstrate how these two modeling techniques can be combined to describe biological gradient formation. Parameters derived from partial differential equation describing the process of gradient formation are incorporated in an abstract Petri net model. The quantitative aspects of the resulting model are validated through a case study of gradient formation in the fruit fly.

Equations for the Filled Inelastic Membrane: A More General Derivation

ERIC Educational Resources Information Center

Deakin, Michael A. B.

2011-01-01

An earlier paper discussed the case of a flexible but inextensible membrane filled to capacity with incompressible fluid. It was found that the resulting shape satisfies a set of three simultaneous partial differential equations. This article gives a more general derivation of these equations and shows their form in an interesting special case.

Analytic solution for the space-time fractional Klein-Gordon and coupled conformable Boussinesq equations

NASA Astrophysics Data System (ADS)

Shallal, Muhannad A.; Jabbar, Hawraz N.; Ali, Khalid K.

2018-03-01

In this paper, we constructed a travelling wave solution for space-time fractional nonlinear partial differential equations by using the modified extended Tanh method with Riccati equation. The method is used to obtain analytic solutions for the space-time fractional Klein-Gordon and coupled conformable space-time fractional Boussinesq equations. The fractional complex transforms and the properties of modified Riemann-Liouville derivative have been used to convert these equations into nonlinear ordinary differential equations.

The mu-derivative and its applications to finding exact solutions of the Cahn-Hilliard, Korteveg-de Vries, and Burgers equations.

PubMed

Mitlin, Vlad

2005-10-15

A new transformation termed the mu-derivative is introduced. Applying it to the Cahn-Hilliard equation yields dynamical exact solutions. It is shown that the mu-transformed Cahn-Hilliard equation can be presented in a separable form. This transformation also yields dynamical exact solutions and separable forms for other nonlinear models such as the modified Korteveg-de Vries and the Burgers equations. The general structure of a nonlinear partial differential equation that becomes separable upon applying the mu-derivative is described.

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.

Efficiently and easily integrating differential equations with JiTCODE, JiTCDDE, and JiTCSDE

NASA Astrophysics Data System (ADS)

Ansmann, Gerrit

2018-04-01

We present a family of Python modules for the numerical integration of ordinary, delay, or stochastic differential equations. The key features are that the user enters the derivative symbolically and it is just-in-time-compiled, allowing the user to efficiently integrate differential equations from a higher-level interpreted language. The presented modules are particularly suited for large systems of differential equations such as those used to describe dynamics on complex networks. Through the selected method of input, the presented modules also allow almost complete automatization of the process of estimating regular as well as transversal Lyapunov exponents for ordinary and delay differential equations. We conceptually discuss the modules' design, analyze their performance, and demonstrate their capabilities by application to timely problems.

Variational differential equations for engineering type trajectories close to a planet with an atmosphere

NASA Technical Reports Server (NTRS)

Dickmanns, E. D.

1972-01-01

The differential equations for the adjoint variables are derived and coded in FORTRAN. The program is written in a form to either take into account or neglect thrust, aerodynamic forces, planet rotation and oblateness, and altitude dependent winds.

Topics in spectral methods

NASA Technical Reports Server (NTRS)

Gottlieb, D.; Turkel, E.

1985-01-01

After detailing the construction of spectral approximations to time-dependent mixed initial boundary value problems, a study is conducted of differential equations of the form 'partial derivative of u/partial derivative of t = Lu + f', where for each t, u(t) belongs to a Hilbert space such that u satisfies homogeneous boundary conditions. For the sake of simplicity, it is assumed that L is an unbounded, time-independent linear operator. Attention is given to Fourier methods of both Galerkin and pseudospectral method types, the Galerkin method, the pseudospectral Chebyshev and Legendre methods, the error equation, hyperbolic partial differentiation equations, and time discretization and iterative methods.

The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations.

PubMed

Khader, M M

2013-10-01

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

Systems-based decomposition schemes for the approximate solution of multi-term fractional differential equations

NASA Astrophysics Data System (ADS)

Ford, Neville J.; Connolly, Joseph A.

2009-07-01

We give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi-term fractional differential equations using the Caputo form of the fractional derivative. The schemes we compare are based on conversion of the original problem into a system of equations. We review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation.

A higher order numerical method for time fractional partial differential equations with nonsmooth data

NASA Astrophysics Data System (ADS)

Xing, Yanyuan; Yan, Yubin

2018-03-01

Gao et al. [11] (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate O (k 3 - α), 0 < α < 1 by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu [20] (2016), where k is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu [20] (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate O (k 3 - α), 0 < α < 1 uniformly with respect to the time variable t. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate O (k 3 - α), 0 < α < 1 uniformly with respect to the time variable t. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate O (k 3 - α), 0 < α < 1 as in Gao et al. [11] (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate O (k 3 - α), 0 < α < 1 for any fixed tn > 0 for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

A modification of \\mathsf {WKB} method for fractional differential operators of Schrödinger's type

NASA Astrophysics Data System (ADS)

Sayevand, K.; Pichaghchi, K.

2017-09-01

In this paper, we were concerned with the description of the singularly perturbed differential equations within the scope of fractional calculus. However, we shall note that one of the main methods used to solve these problems is the so-called WKB method. We should mention that this was not achievable via the existing fractional derivative definitions, because they do not obey the chain rule. In order to accommodate the WKB to the scope of fractional derivative, we proposed a relatively new derivative called the local fractional derivative. By use of properties of local fractional derivative, we extend the WKB method in the scope of the fractional differential equation. By means of this extension, the WKB analysis based on the Borel resummation, for fractional differential operators of WKB type are investigated. The convergence and the Mittag-Leffler stability of the proposed approach is proven. The obtained results are in excellent agreement with the existing ones in open literature and it is shown that the present approach is very effective and accurate. Furthermore, we are mainly interested to construct the solution of fractional Schrödinger equation in the Mittag-Leffler form and how it leads naturally to this semi-classical approximation namely modified WKB.

Solving Nonlinear Fractional Differential Equation by Generalized Mittag-Leffler Function Method

NASA Astrophysics Data System (ADS)

Arafa, A. A. M.; Rida, S. Z.; Mohammadein, A. A.; Ali, H. M.

2013-06-01

In this paper, we use Mittag—Leffler function method for solving some nonlinear fractional differential equations. A new solution is constructed in power series. The fractional derivatives are described by Caputo's sense. To illustrate the reliability of the method, some examples are provided.

Deriving Differential Equations from Process Algebra Models in Reagent-Centric Style

NASA Astrophysics Data System (ADS)

Hillston, Jane; Duguid, Adam

The reagent-centric style of modeling allows stochastic process algebra models of biochemical signaling pathways to be developed in an intuitive way. Furthermore, once constructed, the models are amenable to analysis by a number of different mathematical approaches including both stochastic simulation and coupled ordinary differential equations. In this chapter, we give a tutorial introduction to the reagent-centric style, in PEPA and Bio-PEPA, and the way in which such models can be used to generate systems of ordinary differential equations.

Ordinary differential equation for local accumulation time.

PubMed

Berezhkovskii, Alexander M

2011-08-21

Cell differentiation in a developing tissue is controlled by the concentration fields of signaling molecules called morphogens. Formation of these concentration fields can be described by the reaction-diffusion mechanism in which locally produced molecules diffuse through the patterned tissue and are degraded. The formation kinetics at a given point of the patterned tissue can be characterized by the local accumulation time, defined in terms of the local relaxation function. Here, we show that this time satisfies an ordinary differential equation. Using this equation one can straightforwardly determine the local accumulation time, i.e., without preliminary calculation of the relaxation function by solving the partial differential equation, as was done in previous studies. We derive this ordinary differential equation together with the accompanying boundary conditions and demonstrate that the earlier obtained results for the local accumulation time can be recovered by solving this equation. © 2011 American Institute of Physics

Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations

NASA Astrophysics Data System (ADS)

Gerdt, Vladimir P.; Blinkov, Yuri A.; Mozzhilkin, Vladimir V.

2006-05-01

In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations is derived for the unknown functions and their partial derivatives. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Gröbner basis of the linear difference ideal generated by the polynomials in the discrete system. For these purposes we use the difference form of Janet-like Gröbner bases and their implementation in Maple. As illustration of the described methods and algorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties.

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 Boundary Function Method. Fundamentals

NASA Astrophysics Data System (ADS)

Kot, V. A.

2017-03-01

The boundary function method is proposed for solving applied problems of mathematical physics in the region defined by a partial differential equation of the general form involving constant or variable coefficients with a Dirichlet, Neumann, or Robin boundary condition. In this method, the desired function is defined by a power polynomial, and a boundary function represented in the form of the desired function or its derivative at one of the boundary points is introduced. Different sequences of boundary equations have been set up with the use of differential operators. Systems of linear algebraic equations constructed on the basis of these sequences allow one to determine the coefficients of a power polynomial. Constitutive equations have been derived for initial boundary-value problems of all the main types. With these equations, an initial boundary-value problem is transformed into the Cauchy problem for the boundary function. The determination of the boundary function by its derivative with respect to the time coordinate completes the solution of the problem.

Analyzing Lie symmetry and constructing conservation laws for time-fractional Benny-Lin equation

NASA Astrophysics Data System (ADS)

Rashidi, Saeede; Hejazi, S. Reza

This paper investigates the invariance properties of the time fractional Benny-Lin equation with Riemann-Liouville and Caputo derivatives. This equation can be reduced to the Kawahara equation, fifth-order Kdv equation, the Kuramoto-Sivashinsky equation and Navier-Stokes equation. By using the Lie group analysis method of fractional differential equations (FDEs), we derive Lie symmetries for the Benny-Lin equation. Conservation laws for this equation are obtained with the aid of the concept of nonlinear self-adjointness and the fractional generalization of the Noether’s operators. Furthermore, by means of the invariant subspace method, exact solutions of the equation are also constructed.

The perturbed compound Poisson risk model with constant interest and a threshold dividend strategy

NASA Astrophysics Data System (ADS)

Gao, Shan; Liu, Zaiming

2010-03-01

In this paper, we consider the compound Poisson risk model perturbed by diffusion with constant interest and a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generation function and the nth moment of the present value of all dividends until ruin are derived. We also derive integro-differential equations with boundary conditions for the Gerber-Shiu functions. The special case that the claim size distribution is exponential is considered in some detail.

Partial Fractions via Calculus

ERIC Educational Resources Information Center

Bauldry, William C.

2018-01-01

The standard technique taught in calculus courses for partial fraction expansions uses undetermined coefficients to generate a system of linear equations; we present a derivative-based technique that calculus and differential equations instructors can use to reinforce connections to calculus. Simple algebra shows that we can use the derivative to…

Fractional Stochastic Field Theory

NASA Astrophysics Data System (ADS)

Honkonen, Juha

2018-02-01

Models describing evolution of physical, chemical, biological, social and financial processes are often formulated as differential equations with the understanding that they are large-scale equations for averages of quantities describing intrinsically random processes. Explicit account of randomness may lead to significant changes in the asymptotic behaviour (anomalous scaling) in such models especially in low spatial dimensions, which in many cases may be captured with the use of the renormalization group. Anomalous scaling and memory effects may also be introduced with the use of fractional derivatives and fractional noise. Construction of renormalized stochastic field theory with fractional derivatives and fractional noise in the underlying stochastic differential equations and master equations and the interplay between fluctuation-induced and built-in anomalous scaling behaviour is reviewed and discussed.

Quantum spatial propagation of squeezed light in a degenerate parametric amplifier

NASA Technical Reports Server (NTRS)

Deutsch, Ivan H.; Garrison, John C.

1992-01-01

Differential equations which describe the steady state spatial evolution of nonclassical light are established using standard quantum field theoretic techniques. A Schroedinger equation for the state vector of the optical field is derived using the quantum analog of the slowly varying envelope approximation (SVEA). The steady state solutions are those that satisfy the time independent Schroedinger equation. The resulting eigenvalue problem then leads to the spatial propagation equations. For the degenerate parametric amplifier this method shows that the squeezing parameter obey nonlinear differential equations coupled by the amplifier gain and phase mismatch. The solution to these differential equations is equivalent to one obtained from the classical three wave mixing steady state solution to the parametric amplifier with a nondepleted pump.

Unraveling mirror properties in time-delayed quantum feedback scenarios

NASA Astrophysics Data System (ADS)

Faulstich, Fabian M.; Kraft, Manuel; Carmele, Alexander

2018-06-01

We derive in the Heisenberg picture a widely used phenomenological coupling element to treat feedback effects in quantum optical platforms. Our derivation is based on a microscopic Hamiltonian, which describes the mirror-emitter dynamics based on a dielectric, a mediating fully quantized electromagnetic field and a single two-level system in front of the dielectric. The dielectric is modelled as a system of identical two-state atoms. The Heisenberg equation yields a system of describing differential operator equations, which we solve in the Weisskopf-Wigner limit. Due to a finite round-trip time between emitter and dielectric, we yield delay differential operator equations. Our derivation motivates and justifies the typical phenomenologicalassumed coupling element and allows, furthermore, a generalization to a variety of mirrors, such as dissipative mirrors or mirrors with gain dynamics.

Prolongation structures of nonlinear evolution equations

NASA Technical Reports Server (NTRS)

Wahlquist, H. D.; Estabrook, F. B.

1975-01-01

A technique is developed for systematically deriving a 'prolongation structure' - a set of interrelated potentials and pseudopotentials - for nonlinear partial differential equations in two independent variables. When this is applied to the Korteweg-de Vries equation, a new infinite set of conserved quantities is obtained. Known solution techniques are shown to result from the discovery of such a structure: related partial differential equations for the potential functions, linear 'inverse scattering' equations for auxiliary functions, Backlund transformations. Generalizations of these techniques will result from the use of irreducible matrix representations of the prolongation structure.

Algebraic Riccati equations in zero-sum differential games

NASA Technical Reports Server (NTRS)

Johnson, T. L.; Chao, A.

1974-01-01

The procedure for finding the closed-loop Nash equilibrium solution of two-player zero-sum linear time-invariant differential games with quadratic performance criteria and classical information pattern may be reduced in most cases to the solution of an algebraic Riccati equation. Based on the results obtained by Willems, necessary and sufficient conditions for existence of solutions to these equations are derived, and explicit conditions for a scalar example are given.

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.

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.

Power-law spatial dispersion from fractional Liouville equation

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

Tarasov, Vasily E.

2013-10-15

A microscopic model in the framework of fractional kinetics to describe spatial dispersion of power-law type is suggested. The Liouville equation with the Caputo fractional derivatives is used to obtain the power-law dependence of the absolute permittivity on the wave vector. The fractional differential equations for electrostatic potential in the media with power-law spatial dispersion are derived. The particular solutions of these equations for the electric potential of point charge in this media are considered.

A Simple Derivation of Kepler's Laws without Solving Differential Equations

ERIC Educational Resources Information Center

Provost, J.-P.; Bracco, C.

2009-01-01

Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non-trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple…

THREE-POINT BACKWARD FINITE DIFFERENCE METHOD FOR SOLVING A SYSTEM OF MIXED HYPERBOLIC-PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS. (R825549C019)

EPA Science Inventory

A three-point backward finite-difference method has been derived for a system of mixed hyperbolic_¯¯parabolic (convection_¯¯diffusion) partial differential equations (mixed PDEs). The method resorts to the three-point backward differenci...

The Mathlet Toolkit: Creating Dynamic Applets for Differential Equations and Dynamical Systems

ERIC Educational Resources Information Center

Decker, Robert

2011-01-01

Dynamic/interactive graphing applets can be used to supplement standard computer algebra systems such as Maple, Mathematica, Derive, or TI calculators, in courses such as Calculus, Differential Equations, and Dynamical Systems. The addition of this type of software can lead to discovery learning, with students developing their own conjectures, and…

Application of the principal fractional meta-trigonometric functions for the solution of linear commensurate-order time-invariant fractional differential equations.

PubMed

Lorenzo, C F; Hartley, T T; Malti, R

2013-05-13

A new and simplified method for the solution of linear constant coefficient fractional differential equations of any commensurate order is presented. The solutions are based on the R-function and on specialized Laplace transform pairs derived from the principal fractional meta-trigonometric functions. The new method simplifies the solution of such fractional differential equations and presents the solutions in the form of real functions as opposed to fractional complex exponential functions, and thus is directly applicable to real-world physics.

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.

A study of fractional Schrödinger equation composed of Jumarie fractional derivative

NASA Astrophysics Data System (ADS)

Banerjee, Joydip; Ghosh, Uttam; Sarkar, Susmita; Das, Shantanu

2017-04-01

In this paper we have derived the fractional-order Schrödinger equation composed of Jumarie fractional derivative. The solution of this fractional-order Schrödinger equation is obtained in terms of Mittag-Leffler function with complex arguments, and fractional trigonometric functions. A few important properties of the fractional Schrödinger equation are then described for the case of particles in one-dimensional infinite potential well. One of the motivations for using fractional calculus in physical systems is that the space and time variables, which we often deal with, exhibit coarse-grained phenomena. This means infinitesimal quantities cannot be arbitrarily taken to zero - rather they are non-zero with a minimum spread. This type of non-zero spread arises in the microscopic to mesoscopic levels of system dynamics, which means that, if we denote x as the point in space and t as the point in time, then limit of the differentials d x (and d t) cannot be taken as zero. To take the concept of coarse graining into account, use the infinitesimal quantities as (Δ x) α (and (Δ t) α ) with 0 < α < 1; called as `fractional differentials'. For arbitrarily small Δ x and Δ t (tending towards zero), these `fractional' differentials are greater than Δ x (and Δ t), i.e. (Δ x) α > Δ x and (Δ t) α > Δ t. This way of defining the fractional differentials helps us to use fractional derivatives in the study of dynamic systems.

Exact solutions of fractional mBBM equation and coupled system of fractional Boussinesq-Burgers

NASA Astrophysics Data System (ADS)

Javeed, Shumaila; Saif, Summaya; Waheed, Asif; Baleanu, Dumitru

2018-06-01

The new exact solutions of nonlinear fractional partial differential equations (FPDEs) are established by adopting first integral method (FIM). The Riemann-Liouville (R-L) derivative and the local conformable derivative definitions are used to deal with the fractional order derivatives. The proposed method is applied to get exact solutions for space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation and coupled time-fractional Boussinesq-Burgers equation. The suggested technique is easily applicable and effectual which can be implemented successfully to obtain the solutions for different types of nonlinear FPDEs.

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

Unsteady density-current equations for highly curved terrain

NASA Technical Reports Server (NTRS)

Sivakumaran, N. S.; Dressler, R. F.

1989-01-01

New nonlinear partial differential equations containing terrain curvature and its rate of change are derived that describe the flow of an atmospheric density current. Unlike the classical hydraulic-type equations for density currents, the new equations are valid for two-dimensional, gradually varied flow over highly curved terrain, hence suitable for computing unsteady (or steady) flows over arbitrary mountain/valley profiles. The model assumes the atmosphere above the density current exerts a known arbitrary variable pressure upon the unknown interface. Later this is specialized to the varying hydrostatic pressure of the atmosphere above. The new equations yield the variable velocity distribution, the interface position, and the pressure distribution that contains a centrifugal component, often significantly larger than its hydrostatic component. These partial differential equations are hyperbolic, and the characteristic equations and characteristic directions are derived. Using these to form a characteristic mesh, a hypothetical unsteady curved-flow problem is calculated, not based upon observed data, merely as an example to illustrate the simplicity of their application to unsteady flows over mountains.

Fusion of Imaging and Inertial Sensors for Navigation

DTIC Science & Technology

2006-09-01

combat operations. The Global Positioning System (GPS) was fielded in the 1980’s and first used for precision navigation and targeting in combat...equations [37]. Consider the homogeneous nonlinear differential equation ẋ(t) = f [x(t),u(t), t] ; x(t0) = x0 (2.4) For a given input function , u0(t...differential equation is a time-varying probability density function . The Kalman filter derivation assumes Gaussian distributions for all random

Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of Neolithic transition.

PubMed

Vlad, Marcel Ovidiu; Ross, John

2002-12-01

We introduce a general method for the systematic derivation of nonlinear reaction-diffusion equations with distributed delays. We study the interactions among different types of moving individuals (atoms, molecules, quasiparticles, biological organisms, etc). The motion of each species is described by the continuous time random walk theory, analyzed in the literature for transport problems, whereas the interactions among the species are described by a set of transformation rates, which are nonlinear functions of the local concentrations of the different types of individuals. We use the time interval between two jumps (the transition time) as an additional state variable and obtain a set of evolution equations, which are local in time. In order to make a connection with the transport models used in the literature, we make transformations which eliminate the transition time and derive a set of nonlocal equations which are nonlinear generalizations of the so-called generalized master equations. The method leads under different specified conditions to various types of nonlocal transport equations including a nonlinear generalization of fractional diffusion equations, hyperbolic reaction-diffusion equations, and delay-differential reaction-diffusion equations. Thus in the analysis of a given problem we can fit to the data the type of reaction-diffusion equation and the corresponding physical and kinetic parameters. The method is illustrated, as a test case, by the study of the neolithic transition. We introduce a set of assumptions which makes it possible to describe the transition from hunting and gathering to agriculture economics by a differential delay reaction-diffusion equation for the population density. We derive a delay evolution equation for the rate of advance of agriculture, which illustrates an application of our analysis.

Differential Geometry Based Multiscale Models

PubMed Central

Wei, Guo-Wei

2010-01-01

Large chemical and biological systems such as fuel cells, ion channels, molecular motors, and viruses are of great importance to the scientific community and public health. Typically, these complex systems in conjunction with their aquatic environment pose a fabulous challenge to theoretical description, simulation, and prediction. In this work, we propose a differential geometry based multiscale paradigm to model complex macromolecular systems, and to put macroscopic and microscopic descriptions on an equal footing. In our approach, the differential geometry theory of surfaces and geometric measure theory are employed as a natural means to couple the macroscopic continuum mechanical description of the aquatic environment with the microscopic discrete atom-istic description of the macromolecule. Multiscale free energy functionals, or multiscale action functionals are constructed as a unified framework to derive the governing equations for the dynamics of different scales and different descriptions. Two types of aqueous macromolecular complexes, ones that are near equilibrium and others that are far from equilibrium, are considered in our formulations. We show that generalized Navier–Stokes equations for the fluid dynamics, generalized Poisson equations or generalized Poisson–Boltzmann equations for electrostatic interactions, and Newton's equation for the molecular dynamics can be derived by the least action principle. These equations are coupled through the continuum-discrete interface whose dynamics is governed by potential driven geometric flows. Comparison is given to classical descriptions of the fluid and electrostatic interactions without geometric flow based micro-macro interfaces. The detailed balance of forces is emphasized in the present work. We further extend the proposed multiscale paradigm to micro-macro analysis of electrohydrodynamics, electrophoresis, fuel cells, and ion channels. We derive generalized Poisson–Nernst–Planck equations that are coupled to generalized Navier–Stokes equations for fluid dynamics, Newton's equation for molecular dynamics, and potential and surface driving geometric flows for the micro-macro interface. For excessively large aqueous macromolecular complexes in chemistry and biology, we further develop differential geometry based multiscale fluid-electro-elastic models to replace the expensive molecular dynamics description with an alternative elasticity formulation. PMID:20169418

An introduction to generalized functions with some applications in aerodynamics and aeroacoustics

NASA Technical Reports Server (NTRS)

Farassat, F.

1994-01-01

In this paper, we start with the definition of generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support. The concept of generalization differentiation is introduced next. This is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some of the results of classical analysis, such as Leibniz rule of differentiation under the integral sign and the divergence theorem, are derived using the generalized function theory. It is shown that the divergence theorem remains valid for discontinuous vector fields provided that the derivatives are all viewed as generalized derivatives. This implies that all conservation laws of fluid mechanics are valid as they stand for discontinuous fields with all derivatives treated as generalized deriatives. Once these derivatives are written as ordinary derivatives and jumps in the field parameters across discontinuities, the jump conditions can be easily found. For example, the unsteady shock jump conditions can be derived from mass and momentum conservation laws. By using a generalized function theory, this derivative becomes trivial. Other applications of the generalized function theory in aerodynamics discussed in this paper are derivation of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of finite part of divergent integrals, derivation of Oswatiitsch integral equation of transonic flow, and analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics presented here include the derivation of the Kirchoff formula for moving surfaces,the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams-Hawkings equation.

On mixed derivatives type high dimensional multi-term fractional partial differential equations approximate solutions

NASA Astrophysics Data System (ADS)

Talib, Imran; Belgacem, Fethi Bin Muhammad; Asif, Naseer Ahmad; Khalil, Hammad

2017-01-01

In this research article, we derive and analyze an efficient spectral method based on the operational matrices of three dimensional orthogonal Jacobi polynomials to solve numerically the mixed partial derivatives type multi-terms high dimensions generalized class of fractional order partial differential equations. We transform the considered fractional order problem to an easily solvable algebraic equations with the aid of the operational matrices. Being easily solvable, the associated algebraic system leads to finding the solution of the problem. Some test problems are considered to confirm the accuracy and validity of the proposed numerical method. The convergence of the method is ensured by comparing our Matlab software simulations based obtained results with the exact solutions in the literature, yielding negligible errors. Moreover, comparative results discussed in the literature are extended and improved in this study.

The perturbed Sparre Andersen model with a threshold dividend strategy

NASA Astrophysics Data System (ADS)

Gao, Heli; Yin, Chuancun

2008-10-01

In this paper, we consider a Sparre Andersen model perturbed by diffusion with generalized Erlang(n)-distributed inter-claim times and a threshold dividend strategy. Integro-differential equations with certain boundary conditions for the moment-generation function and the mth moment of the present value of all dividends until ruin are derived. We also derive integro-differential equations with boundary conditions for the Gerber-Shiu functions. The special case where the inter-claim times are Erlang(2) distributed and the claim size distribution is exponential is considered in some details.

The numerical solution of ordinary differential equations by the Taylor series method

NASA Technical Reports Server (NTRS)

Silver, A. H.; Sullivan, E.

1973-01-01

A programming implementation of the Taylor series method is presented for solving ordinary differential equations. The compiler is written in PL/1, and the target language is FORTRAN IV. The reduction of a differential system to rational form is described along with the procedures required for automatic numerical integration. The Taylor method is compared with two other methods for a number of differential equations. Algorithms using the Taylor method to find the zeroes of a given differential equation and to evaluate partial derivatives are presented. An annotated listing of the PL/1 program which performs the reduction and code generation is given. Listings of the FORTRAN routines used by the Taylor series method are included along with a compilation of all the recurrence formulas used to generate the Taylor coefficients for non-rational functions.

Conservation form of the equations of fluid dynamics in general nonsteady coordinates

NASA Astrophysics Data System (ADS)

Zhang, H.; Camarero, R.; Kahawita, R.

1985-11-01

Many of the differential equations arising in fluid dynamics may be stated in conservation-law form. A number of investigations have been conducted with the aim to derive the conservation-law form of the Navier-Stokes equations in general nonsteady coordinate systems. The present note has the objective to illustrate a mathematical methodology with which such forms of the equations may be derived in an easier and more general fashion. For numerical applications, the scalar form of the equations is eventually provided. Attention is given to the conservation form of equations in curvilinear coordinates and numerical considerations.

Blood flow problem in the presence of magnetic particles through a circular cylinder using Caputo-Fabrizio fractional derivative

NASA Astrophysics Data System (ADS)

Uddin, Salah; Mohamad, Mahathir; Khalid, Kamil; Abdulhammed, Mohammed; Saifullah Rusiman, Mohd; Che – Him, Norziha; Roslan, Rozaini

2018-04-01

In this paper, the flow of blood mixed with magnetic particles subjected to uniform transverse magnetic field and pressure gradient in an axisymmetric circular cylinder is studied by using a new trend of fractional derivative without singular kernel. The governing equations are fractional partial differential equations derived based on the Caputo-Fabrizio time-fractional derivatives NFDt. The current result agrees considerably well with that of the previous Caputo fractional derivatives UFDt.

Closed solutions to a differential-difference equation and an associated plate solidification problem.

PubMed

Layeni, Olawanle P; Akinola, Adegbola P; Johnson, Jesse V

2016-01-01

Two distinct and novel formalisms for deriving exact closed solutions of a class of variable-coefficient differential-difference equations arising from a plate solidification problem are introduced. Thereupon, exact closed traveling wave and similarity solutions to the plate solidification problem are obtained for some special cases of time-varying plate surface temperature.

Stability and square integrability of derivatives of solutions of nonlinear fourth order differential equations with delay.

PubMed

Korkmaz, Erdal

2017-01-01

In this paper, we give sufficient conditions for the boundedness, uniform asymptotic stability and square integrability of the solutions to a certain fourth order non-autonomous differential equations with delay by using Lyapunov's second method. The results obtained essentially improve, include and complement the results in the literature.

Control of functional differential equations to target sets in function space

NASA Technical Reports Server (NTRS)

Banks, H. T.; Kent, G. A.

1971-01-01

Optimal control of systems governed by functional differential equations of retarded and neutral type is considered. Problems with function space initial and terminal manifolds are investigated. Existence of optimal controls, regularity, and bang-bang properties are discussed. Necessary and sufficient conditions are derived, and several solved examples which illustrate the theory are presented.

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

Lie symmetry analysis, explicit solutions and conservation laws for the space-time fractional nonlinear evolution equations

NASA Astrophysics Data System (ADS)

Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa; Baleanu, Dumitru

2018-04-01

This paper studies the symmetry analysis, explicit solutions, convergence analysis, and conservation laws (Cls) for two different space-time fractional nonlinear evolution equations with Riemann-Liouville (RL) derivative. The governing equations are reduced to nonlinear ordinary differential equation (ODE) of fractional order using their Lie point symmetries. In the reduced equations, the derivative is in Erdelyi-Kober (EK) sense, power series technique is applied to derive an explicit solutions for the reduced fractional ODEs. The convergence of the obtained power series solutions is also presented. Moreover, the new conservation theorem and the generalization of the Noether operators are developed to construct the nonlocal Cls for the equations . Some interesting figures for the obtained explicit solutions are presented.

A theoretical analysis of fluid flow and energy transport in hydrothermal systems

USGS Publications Warehouse

Faust, Charles R.; Mercer, James W.

1977-01-01

A mathematical derivation for fluid flow and energy transport in hydrothermal systems is presented. Specifically, the mathematical model describes the three-dimensional flow of both single- and two-phase, single-component water and the transport of heat in porous media. The derivation begins with the point balance equations for mass, momentum, and energy. These equations are then averaged over a finite volume to obtain the macroscopic balance equations for a porous medium. The macroscopic equations are combined by appropriate constitutive relationships to form two similified partial differential equations posed in terms of fluid pressure and enthalpy. A two-dimensional formulation of the simplified equations is also derived by partial integration in the vertical dimension. (Woodard-USGS)

Analytical solutions to time-fractional partial differential equations in a two-dimensional multilayer annulus

NASA Astrophysics Data System (ADS)

Chen, Shanzhen; Jiang, Xiaoyun

2012-08-01

In this paper, analytical solutions to time-fractional partial differential equations in a multi-layer annulus are presented. The final solutions are obtained in terms of Mittag-Leffler function by using the finite integral transform technique and Laplace transform technique. In addition, the classical diffusion equation (α=1), the Helmholtz equation (α→0) and the wave equation (α=2) are discussed as special cases. Finally, an illustrative example problem for the three-layer semi-circular annular region is solved and numerical results are presented graphically for various kind of order of fractional derivative.

On exact traveling-wave solutions for local fractional Korteweg-de Vries equation.

PubMed

Yang, Xiao-Jun; Tenreiro Machado, J A; Baleanu, Dumitru; Cattani, Carlo

2016-08-01

This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.

Couple of the Variational Iteration Method and Fractional-Order Legendre Functions Method for Fractional Differential Equations

PubMed Central

Song, Junqiang; Leng, Hongze; Lu, Fengshun

2014-01-01

We present a new numerical method to get the approximate solutions of fractional differential equations. A new operational matrix of integration for fractional-order Legendre functions (FLFs) is first derived. Then a modified variational iteration formula which can avoid “noise terms” is constructed. Finally a numerical method based on variational iteration method (VIM) and FLFs is developed for fractional differential equations (FDEs). Block-pulse functions (BPFs) are used to calculate the FLFs coefficient matrices of the nonlinear terms. Five examples are discussed to demonstrate the validity and applicability of the technique. PMID:24511303

Analytical solutions for the motion of a charged particle in electric and magnetic fields via non-singular fractional derivatives

NASA Astrophysics Data System (ADS)

Morales-Delgado, V. F.; Gómez-Aguilar, J. F.; Taneco-Hernandez, M. A.

2017-12-01

In this work we propose fractional differential equations for the motion of a charged particle in electric, magnetic and electromagnetic fields. Exact solutions are obtained for the fractional differential equations by employing the Laplace transform method. The temporal fractional differential equations are considered in the Caputo-Fabrizio-Caputo and Atangana-Baleanu-Caputo sense. Application examples consider constant, ramp and harmonic fields. In addition, we present numerical results for different values of the fractional order. In all cases, when α = 1, we recover the standard electrodynamics.

From crater functions to partial differential equations: a new approach to ion bombardment induced nonequilibrium pattern formation.

PubMed

Norris, Scott A; Brenner, Michael P; Aziz, Michael J

2009-06-03

We develop a methodology for deriving continuum partial differential equations for the evolution of large-scale surface morphology directly from molecular dynamics simulations of the craters formed from individual ion impacts. Our formalism relies on the separation between the length scale of ion impact and the characteristic scale of pattern formation, and expresses the surface evolution in terms of the moments of the crater function. We demonstrate that the formalism reproduces the classical Bradley-Harper results, as well as ballistic atomic drift, under the appropriate simplifying assumptions. Given an actual set of converged molecular dynamics moments and their derivatives with respect to the incidence angle, our approach can be applied directly to predict the presence and absence of surface morphological instabilities. This analysis represents the first work systematically connecting molecular dynamics simulations of ion bombardment to partial differential equations that govern topographic pattern-forming instabilities.

Unsplit complex frequency shifted perfectly matched layer for second-order wave equation using auxiliary differential equations.

PubMed

Gao, Yingjie; Zhang, Jinhai; Yao, Zhenxing

2015-12-01

The complex frequency shifted perfectly matched layer (CFS-PML) can improve the absorbing performance of PML for nearly grazing incident waves. However, traditional PML and CFS-PML are based on first-order wave equations; thus, they are not suitable for second-order wave equation. In this paper, an implementation of CFS-PML for second-order wave equation is presented using auxiliary differential equations. This method is free of both convolution calculations and third-order temporal derivatives. As an unsplit CFS-PML, it can reduce the nearly grazing incidence. Numerical experiments show that it has better absorption than typical PML implementations based on second-order wave equation.

Modeling electro-magneto-hydrodynamic thermo-fluidic transport of biofluids with new trend of fractional derivative without singular kernel

NASA Astrophysics Data System (ADS)

Abdulhameed, M.; Vieru, D.; Roslan, R.

2017-10-01

This paper investigates the electro-magneto-hydrodynamic flow of the non-Newtonian behavior of biofluids, with heat transfer, through a cylindrical microchannel. The fluid is acted by an arbitrary time-dependent pressure gradient, an external electric field and an external magnetic field. The governing equations are considered as fractional partial differential equations based on the Caputo-Fabrizio time-fractional derivatives without singular kernel. The usefulness of fractional calculus to study fluid flows or heat and mass transfer phenomena was proven. Several experimental measurements led to conclusion that, in such problems, the models described by fractional differential equations are more suitable. The most common time-fractional derivative used in Continuum Mechanics is Caputo derivative. However, two disadvantages appear when this derivative is used. First, the definition kernel is a singular function and, secondly, the analytical expressions of the problem solutions are expressed by generalized functions (Mittag-Leffler, Lorenzo-Hartley, Robotnov, etc.) which, generally, are not adequate to numerical calculations. The new time-fractional derivative Caputo-Fabrizio, without singular kernel, is more suitable to solve various theoretical and practical problems which involve fractional differential equations. Using the Caputo-Fabrizio derivative, calculations are simpler and, the obtained solutions are expressed by elementary functions. Analytical solutions of the biofluid velocity and thermal transport are obtained by means of the Laplace and finite Hankel transforms. The influence of the fractional parameter, Eckert number and Joule heating parameter on the biofluid velocity and thermal transport are numerically analyzed and graphic presented. This fact can be an important in Biochip technology, thus making it possible to use this analysis technique extremely effective to control bioliquid samples of nanovolumes in microfluidic devices used for biological analysis and medical diagnosis.

Differential Equations Models to Study Quorum Sensing.

PubMed

Pérez-Velázquez, Judith; Hense, Burkhard A

2018-01-01

Mathematical models to study quorum sensing (QS) have become an important tool to explore all aspects of this type of bacterial communication. A wide spectrum of mathematical tools and methods such as dynamical systems, stochastics, and spatial models can be employed. In this chapter, we focus on giving an overview of models consisting of differential equations (DE), which can be used to describe changing quantities, for example, the dynamics of one or more signaling molecule in time and space, often in conjunction with bacterial growth dynamics. The chapter is divided into two sections: ordinary differential equations (ODE) and partial differential equations (PDE) models of QS. Rates of change are represented mathematically by derivatives, i.e., in terms of DE. ODE models allow describing changes in one independent variable, for example, time. PDE models can be used to follow changes in more than one independent variable, for example, time and space. Both types of models often consist of systems (i.e., more than one equation) of equations, such as equations for bacterial growth and autoinducer concentration dynamics. Almost from the onset, mathematical modeling of QS using differential equations has been an interdisciplinary endeavor and many of the works we revised here will be placed into their biological context.

Existence and discrete approximation for optimization problems governed by fractional differential equations

NASA Astrophysics Data System (ADS)

Bai, Yunru; Baleanu, Dumitru; Wu, Guo-Cheng

2018-06-01

We investigate a class of generalized differential optimization problems driven by the Caputo derivative. Existence of weak Carathe ´odory solution is proved by using Weierstrass existence theorem, fixed point theorem and Filippov implicit function lemma etc. Then a numerical approximation algorithm is introduced, and a convergence theorem is established. Finally, a nonlinear programming problem constrained by the fractional differential equation is illustrated and the results verify the validity of the algorithm.

On the Well-Definedness of the Order of an Ordinary Differential Equation

ERIC Educational Resources Information Center

Dobbs, David E.

2006-01-01

It is proved that if the differential equations "y[(n)] = f(x,y,y[prime],...,y[(n-1)])" and "y[(m)] = g(x,y,y[prime],...,y[(m-1)])" have the same particular solutions in a suitable region where "f" and "g" are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical…

The Adams formulas for numerical integration of differential equations from 1st to 20th order

NASA Technical Reports Server (NTRS)

Kirkpatrick, J. C.

1976-01-01

The Adams Bashforth predictor coefficients and the Adams Moulton corrector coefficients for the integration of differential equations are presented for methods of 1st to 20th order. The order of the method as presented refers to the highest order difference formula used in Newton's backward difference interpolation formula, on which the Adams method is based. The Adams method is a polynomial approximation method derived from Newton's backward difference interpolation formula. The Newton formula is derived and expanded to 20th order. The Adams predictor and corrector formulas are derived and expressed in terms of differences of the derivatives, as well as in terms of the derivatives themselves. All coefficients are given to 18 significant digits. For the difference formula only, the ratio coefficients are given to 10th order.

SIVA/DIVA- INITIAL VALUE ORDINARY DIFFERENTIAL EQUATION SOLUTION VIA A VARIABLE ORDER ADAMS METHOD

NASA Technical Reports Server (NTRS)

Krogh, F. T.

1994-01-01

The SIVA/DIVA package is a collection of subroutines for the solution of ordinary differential equations. There are versions for single precision and double precision arithmetic. These solutions are applicable to stiff or nonstiff differential equations of first or second order. SIVA/DIVA requires fewer evaluations of derivatives than other variable order Adams predictor-corrector methods. There is an option for the direct integration of second order equations which can make integration of trajectory problems significantly more efficient. Other capabilities of SIVA/DIVA include: monitoring a user supplied function which can be separate from the derivative; dynamically controlling the step size; displaying or not displaying output at initial, final, and step size change points; saving the estimated local error; and reverse communication where subroutines return to the user for output or computation of derivatives instead of automatically performing calculations. The user must supply SIVA/DIVA with: 1) the number of equations; 2) initial values for the dependent and independent variables, integration stepsize, error tolerance, etc.; and 3) the driver program and operational parameters necessary for subroutine execution. SIVA/DIVA contains an extensive diagnostic message library should errors occur during execution. SIVA/DIVA is written in FORTRAN 77 for batch execution and is machine independent. It has a central memory requirement of approximately 120K of 8 bit bytes. This program was developed in 1983 and last updated in 1987.

A new numerical approximation of the fractal ordinary differential equation

NASA Astrophysics Data System (ADS)

Atangana, Abdon; Jain, Sonal

2018-02-01

The concept of fractal medium is present in several real-world problems, for instance, in the geological formation that constitutes the well-known subsurface water called aquifers. However, attention has not been quite devoted to modeling for instance, the flow of a fluid within these media. We deem it important to remind the reader that the concept of fractal derivative is not to represent the fractal sharps but to describe the movement of the fluid within these media. Since this class of ordinary differential equations is highly complex to solve analytically, we present a novel numerical scheme that allows to solve fractal ordinary differential equations. Error analysis of the method is also presented. Application of the method and numerical approximation are presented for fractal order differential equation. The stability and the convergence of the numerical schemes are investigated in detail. Also some exact solutions of fractal order differential equations are presented and finally some numerical simulations are presented.

Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

NASA Astrophysics Data System (ADS)

Baleanu, Dumitru; Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa

2018-06-01

In this work, we investigate the Lie symmetry analysis, exact solutions and conservation laws (Cls) to the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGDK) equation with Riemann-Liouville (RL) derivative. The time fractional CDGDK is reduced to nonlinear ordinary differential equation (ODE) of fractional order. New exact traveling wave solutions for the time fractional CDGDK are obtained by fractional sub-equation method. In the reduced equation, the derivative is in Erdelyi-Kober (EK) sense. Ibragimov's nonlocal conservation method is applied to construct Cls for time fractional CDGDK.

Equations of condition for high order Runge-Kutta-Nystrom formulae

NASA Technical Reports Server (NTRS)

Bettis, D. G.

1974-01-01

Derivation of the equations of condition of order eight for a general system of second-order differential equations approximated by the basic Runge-Kutta-Nystrom algorithm. For this general case, the number of equations of condition is considerably larger than for the special case where the first derivative is not present. Specifically, it is shown that, for orders two through eight, the number of equations for each order is 1, 1, 1, 2, 3, 5, and 9 for the special case and is 1, 1, 2, 5, 13, 34, and 95 for the general case.

One-Dimensional Fokker-Planck Equation with Quadratically Nonlinear Quasilocal Drift

NASA Astrophysics Data System (ADS)

Shapovalov, A. V.

2018-04-01

The Fokker-Planck equation in one-dimensional spacetime with quadratically nonlinear nonlocal drift in the quasilocal approximation is reduced with the help of scaling of the coordinates and time to a partial differential equation with a third derivative in the spatial variable. Determining equations for the symmetries of the reduced equation are derived and the Lie symmetries are found. A group invariant solution having the form of a traveling wave is found. Within the framework of Adomian's iterative method, the first iterations of an approximate solution of the Cauchy problem are obtained. Two illustrative examples of exact solutions are found.

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.

A Potential Function Derivation of a Constitutive Equation for Inelastic Material Response

NASA Technical Reports Server (NTRS)

Stouffer, D. C.; Elfoutouh, N. A.

1983-01-01

Physical and thermodynamic concepts are used to develop a potential function for application to high temperature polycrystalline material response. Inherent in the formulation is a differential relationship between the potential function and constitutive equation in terms of the state variables. Integration of the differential relationship produces a state variable evolution equation that requires specification of the initial value of the state variable and its time derivative. It is shown that the initial loading rate, which is directly related to the initial hardening rate, can significantly influence subsequent material response. This effect is consistent with observed material behavior on the macroscopic and microscopic levels, and may explain the wide scatter in response often found in creep testing.

Simulation of quantum dynamics based on the quantum stochastic differential equation.

PubMed

Li, Ming

2013-01-01

The quantum stochastic differential equation derived from the Lindblad form quantum master equation is investigated. The general formulation in terms of environment operators representing the quantum state diffusion is given. The numerical simulation algorithm of stochastic process of direct photodetection of a driven two-level system for the predictions of the dynamical behavior is proposed. The effectiveness and superiority of the algorithm are verified by the performance analysis of the accuracy and the computational cost in comparison with the classical Runge-Kutta algorithm.

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.

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.

Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics

NASA Astrophysics Data System (ADS)

Motsepa, Tanki; Aziz, Taha; Fatima, Aeeman; Khalique, Chaudry Masood

2018-03-01

The optimal investment-consumption problem under the constant elasticity of variance (CEV) model is investigated from the perspective of Lie group analysis. The Lie symmetry group of the evolution partial differential equation describing the CEV model is derived. The Lie point symmetries are then used to obtain an exact solution of the governing model satisfying a standard terminal condition. Finally, we construct conservation laws of the underlying equation using the general theorem on conservation laws.

Numerical integration of asymptotic solutions of ordinary differential equations

NASA Technical Reports Server (NTRS)

Thurston, Gaylen A.

1989-01-01

Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.

Oxidation Behavior of Carbon Fiber-Reinforced Composites

NASA Technical Reports Server (NTRS)

Sullivan, Roy M.

2008-01-01

OXIMAP is a numerical (FEA-based) solution tool capable of calculating the carbon fiber and fiber coating oxidation patterns within any arbitrarily shaped carbon silicon carbide composite structure as a function of time, temperature, and the environmental oxygen partial pressure. The mathematical formulation is derived from the mechanics of the flow of ideal gases through a chemically reacting, porous solid. The result of the formulation is a set of two coupled, non-linear differential equations written in terms of the oxidant and oxide partial pressures. The differential equations are solved simultaneously to obtain the partial vapor pressures of the oxidant and oxides as a function of the spatial location and time. The local rate of carbon oxidation is determined at each time step using the map of the local oxidant partial vapor pressure along with the Arrhenius rate equation. The non-linear differential equations are cast into matrix equations by applying the Bubnov-Galerkin weighted residual finite element method, allowing for the solution of the differential equations numerically.

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

Differential equations governing slip-induced pore-pressure fluctuations in a water-saturated granular medium

USGS Publications Warehouse

Iverson, R.M.

1993-01-01

Macroscopic frictional slip in water-saturated granular media occurs commonly during landsliding, surface faulting, and intense bedload transport. A mathematical model of dynamic pore-pressure fluctuations that accompany and influence such sliding is derived here by both inductive and deductive methods. The inductive derivation shows how the governing differential equations represent the physics of the steadily sliding array of cylindrical fiberglass rods investigated experimentally by Iverson and LaHusen (1989). The deductive derivation shows how the same equations result from a novel application of Biot's (1956) dynamic mixture theory to macroscopic deformation. The model consists of two linear differential equations and five initial and boundary conditions that govern solid displacements and pore-water pressures. Solid displacements and water pressures are strongly coupled, in part through a boundary condition that ensures mass conservation during irreversible pore deformation that occurs along the bumpy slip surface. Feedback between this deformation and the pore-pressure field may yield complex system responses. The dual derivations of the model help explicate key assumptions. For example, the model requires that the dimensionless parameter B, defined here through normalization of Biot's equations, is much larger than one. This indicates that solid-fluid coupling forces are dominated by viscous rather than inertial effects. A tabulation of physical and kinematic variables for the rod-array experiments of Iverson and LaHusen and for various geologic phenomena shows that the model assumptions commonly are satisfied. A subsequent paper will describe model tests against experimental data. ?? 1993 International Association for Mathematical Geology.

Numerical solution of second order ODE directly by two point block backward differentiation formula

NASA Astrophysics Data System (ADS)

Zainuddin, Nooraini; Ibrahim, Zarina Bibi; Othman, Khairil Iskandar; Suleiman, Mohamed; Jamaludin, Noraini

2015-12-01

Direct Two Point Block Backward Differentiation Formula, (BBDF2) for solving second order ordinary differential equations (ODEs) will be presented throughout this paper. The method is derived by differentiating the interpolating polynomial using three back values. In BBDF2, two approximate solutions are produced simultaneously at each step of integration. The method derived is implemented by using fixed step size and the numerical results that follow demonstrate the advantage of the direct method as compared to the reduction method.

Dimensional analysis yields the general second-order differential equation underlying many natural phenomena: the mathematical properties of a phenomenon's data plot then specify a unique differential equation for it.

PubMed

Kepner, Gordon R

2014-08-27

This study uses dimensional analysis to derive the general second-order differential equation that underlies numerous physical and natural phenomena described by common mathematical functions. It eschews assumptions about empirical constants and mechanisms. It relies only on the data plot's mathematical properties to provide the conditions and constraints needed to specify a second-order differential equation that is free of empirical constants for each phenomenon. A practical example of each function is analyzed using the general form of the underlying differential equation and the observable unique mathematical properties of each data plot, including boundary conditions. This yields a differential equation that describes the relationship among the physical variables governing the phenomenon's behavior. Complex phenomena such as the Standard Normal Distribution, the Logistic Growth Function, and Hill Ligand binding, which are characterized by data plots of distinctly different sigmoidal character, are readily analyzed by this approach. It provides an alternative, simple, unifying basis for analyzing each of these varied phenomena from a common perspective that ties them together and offers new insights into the appropriate empirical constants for describing each phenomenon.

Localization of the eigenvalues of linear integral equations with applications to linear ordinary differential equations.

NASA Technical Reports Server (NTRS)

Sloss, J. M.; Kranzler, S. K.

1972-01-01

The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.

Differential equation models for sharp threshold dynamics.

PubMed

Schramm, Harrison C; Dimitrov, Nedialko B

2014-01-01

We develop an extension to differential equation models of dynamical systems to allow us to analyze probabilistic threshold dynamics that fundamentally and globally change system behavior. We apply our novel modeling approach to two cases of interest: a model of infectious disease modified for malware where a detection event drastically changes dynamics by introducing a new class in competition with the original infection; and the Lanchester model of armed conflict, where the loss of a key capability drastically changes the effectiveness of one of the sides. We derive and demonstrate a step-by-step, repeatable method for applying our novel modeling approach to an arbitrary system, and we compare the resulting differential equations to simulations of the system's random progression. Our work leads to a simple and easily implemented method for analyzing probabilistic threshold dynamics using differential equations. Published by Elsevier Inc.

A perturbative solution to metadynamics ordinary differential equation

NASA Astrophysics Data System (ADS)

Tiwary, Pratyush; Dama, James F.; Parrinello, Michele

2015-12-01

Metadynamics is a popular enhanced sampling scheme wherein by periodic application of a repulsive bias, one can surmount high free energy barriers and explore complex landscapes. Recently, metadynamics was shown to be mathematically well founded, in the sense that the biasing procedure is guaranteed to converge to the true free energy surface in the long time limit irrespective of the precise choice of biasing parameters. A differential equation governing the post-transient convergence behavior of metadynamics was also derived. In this short communication, we revisit this differential equation, expressing it in a convenient and elegant Riccati-like form. A perturbative solution scheme is then developed for solving this differential equation, which is valid for any generic biasing kernel. The solution clearly demonstrates the robustness of metadynamics to choice of biasing parameters and gives further confidence in the widely used method.

A perturbative solution to metadynamics ordinary differential equation.

PubMed

Tiwary, Pratyush; Dama, James F; Parrinello, Michele

2015-12-21

Metadynamics is a popular enhanced sampling scheme wherein by periodic application of a repulsive bias, one can surmount high free energy barriers and explore complex landscapes. Recently, metadynamics was shown to be mathematically well founded, in the sense that the biasing procedure is guaranteed to converge to the true free energy surface in the long time limit irrespective of the precise choice of biasing parameters. A differential equation governing the post-transient convergence behavior of metadynamics was also derived. In this short communication, we revisit this differential equation, expressing it in a convenient and elegant Riccati-like form. A perturbative solution scheme is then developed for solving this differential equation, which is valid for any generic biasing kernel. The solution clearly demonstrates the robustness of metadynamics to choice of biasing parameters and gives further confidence in the widely used method.

Mathematical model of one-man air revitalization system

NASA Technical Reports Server (NTRS)

1976-01-01

A mathematical model was developed for simulating the steady state performance in electrochemical CO2 concentrators which utilize (NMe4)2 CO3 (aq.) electrolyte. This electrolyte, which accommodates a wide range of air relative humidity, is most suitable for one-man air revitalization systems. The model is based on the solution of coupled nonlinear ordinary differential equations derived from mass transport and rate equations for the processes which take place in the cell. The boundary conditions are obtained by solving the mass and energy transport equations. A shooting method is used to solve the differential equations.

A lattice Boltzmann model with an amending function for simulating nonlinear partial differential equations

NASA Astrophysics Data System (ADS)

Chen, Lin-Jie; Ma, Chang-Feng

2010-01-01

This paper proposes a lattice Boltzmann model with an amending function for one-dimensional nonlinear partial differential equations (NPDEs) in the form ut + αuux + βunux + γuxx + δuxxx + ζuxxxx = 0. This model is different from existing models because it lets the time step be equivalent to the square of the space step and derives higher accuracy and nonlinear terms in NPDEs. With the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The numerical results agree well with the analytical solutions.

Statistical theory for the Kardar-Parisi-Zhang equation in (1+1) dimensions.

PubMed

Masoudi, A A; Shahbazi, F; Davoudi, J; Tabar, M Reza Rahimi

2002-02-01

The Kardar-Parisi-Zhang (KPZ) equation in (1+1) dimensions dynamically develops sharply connected valley structures within which the height derivative is not continuous. We develop a statistical theory for the KPZ equation in (1+1) dimensions driven with a random forcing that is white in time and Gaussian-correlated in space. A master equation is derived for the joint probability density function of height difference and height gradient P(h-h*, partial differential(x)h,t) when the forcing correlation length is much smaller than the system size and much larger than the typical sharp valley width. In the time scales before the creation of the sharp valleys, we find the exact generating function of h-h* and partial differential(x)h. The time scale of the sharp valley formation is expressed in terms of the force characteristics. In the stationary state, when the sharp valleys are fully developed, finite-size corrections to the scaling laws of the structure functions left angle bracket(h-h*)(n)(partial differential(x)h)(m)right angle bracket are also obtained.

A boundary value approach for solving three-dimensional elliptic and hyperbolic partial differential equations.

PubMed

Biala, T A; Jator, S N

2015-01-01

In this article, the boundary value method is applied to solve three dimensional elliptic and hyperbolic partial differential equations. The partial derivatives with respect to two of the spatial variables (y, z) are discretized using finite difference approximations to obtain a large system of ordinary differential equations (ODEs) in the third spatial variable (x). Using interpolation and collocation techniques, a continuous scheme is developed and used to obtain discrete methods which are applied via the Block unification approach to obtain approximations to the resulting large system of ODEs. Several test problems are investigated to elucidate the solution process.

Equations of motion for train derailment dynamics

DOT National Transportation Integrated Search

2007-09-11

This paper describes a planar or two-dimensional model to : examine the gross motions of rail cars in a generalized train : derailment. Three coupled, second-order differential equations : are derived from Newton's Laws to calculate rigid-body car : ...

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.

Time-ordered product expansions for computational stochastic system biology.

PubMed

Mjolsness, Eric

2013-06-01

The time-ordered product framework of quantum field theory can also be used to understand salient phenomena in stochastic biochemical networks. It is used here to derive Gillespie's stochastic simulation algorithm (SSA) for chemical reaction networks; consequently, the SSA can be interpreted in terms of Feynman diagrams. It is also used here to derive other, more general simulation and parameter-learning algorithms including simulation algorithms for networks of stochastic reaction-like processes operating on parameterized objects, and also hybrid stochastic reaction/differential equation models in which systems of ordinary differential equations evolve the parameters of objects that can also undergo stochastic reactions. Thus, the time-ordered product expansion can be used systematically to derive simulation and parameter-fitting algorithms for stochastic systems.

Analytical Solutions of the Gravitational Field Equations in de Sitter and Anti-de Sitter Spacetimes

NASA Astrophysics Data System (ADS)

Da Rocha, R.; Capelas Oliveira, E.

2009-01-01

The generalized Laplace partial differential equation, describing gravitational fields, is investigated in de Sitter spacetime from several metric approaches—such as the Riemann, Beltrami, Börner-Dürr, and Prasad metrics—and analytical solutions of the derived Riccati radial differential equations are explicitly obtained. All angular differential equations trivially have solutions given by the spherical harmonics and all radial differential equations can be written as Riccati ordinary differential equations, which analytical solutions involve hypergeometric and Bessel functions. In particular, the radial differential equations predict the behavior of the gravitational field in de Sitter and anti-de Sitter spacetimes, and can shed new light on the investigations of quasinormal modes of perturbations of electromagnetic and gravitational fields in black hole neighborhood. The discussion concerning the geometry of de Sitter and anti-de Sitter spacetimes is not complete without mentioning how the wave equation behaves on such a background. It will prove convenient to begin with a discussion of the Laplace equation on hyperbolic space, partly since this is of interest in itself and also because the wave equation can be investigated by means of an analytic continuation from the hyperbolic space. We also solve the Laplace equation associated to the Prasad metric. After introducing the so called internal and external spaces—corresponding to the symmetry groups SO(3,2) and SO(4,1) respectively—we show that both radial differential equations can be led to Riccati ordinary differential equations, which solutions are given in terms of associated Legendre functions. For the Prasad metric with the radius of the universe independent of the parametrization, the internal and external metrics are shown to be of AdS-Schwarzschild-like type, and also the radial field equations arising are shown to be equivalent to Riccati equations whose solutions can be written in terms of generalized Laguerre polynomials and hypergeometric confluent functions.

Trajectory planning of mobile robots using indirect solution of optimal control method in generalized point-to-point task

NASA Astrophysics Data System (ADS)

Nazemizadeh, M.; Rahimi, H. N.; Amini Khoiy, K.

2012-03-01

This paper presents an optimal control strategy for optimal trajectory planning of mobile robots by considering nonlinear dynamic model and nonholonomic constraints of the system. The nonholonomic constraints of the system are introduced by a nonintegrable set of differential equations which represent kinematic restriction on the motion. The Lagrange's principle is employed to derive the nonlinear equations of the system. Then, the optimal path planning of the mobile robot is formulated as an optimal control problem. To set up the problem, the nonlinear equations of the system are assumed as constraints, and a minimum energy objective function is defined. To solve the problem, an indirect solution of the optimal control method is employed, and conditions of the optimality derived as a set of coupled nonlinear differential equations. The optimality equations are solved numerically, and various simulations are performed for a nonholonomic mobile robot to illustrate effectiveness of the proposed method.

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.

THE FUNDAMENTAL SOLUTIONS FOR MULTI-TERM MODIFIED POWER LAW WAVE EQUATIONS IN A FINITE DOMAIN.

PubMed

Jiang, H; Liu, F; Meerschaert, M M; McGough, R J

2013-01-01

Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n ) ( n > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko's Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi-term time-space fractional models including fractional Laplacian.

Modification of the parallel scattering mean free path of cosmic rays in the presence of adiabatic focusing

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

He, H.-Q.; Schlickeiser, R., E-mail: hqhe@mail.iggcas.ac.cn, E-mail: rsch@tp4.rub.de

The cosmic ray mean free path in a large-scale nonuniform guide magnetic field with superposed magnetostatic turbulence is calculated to clarify some conflicting results in the literature. A new, exact integro-differential equation for the cosmic-ray anisotropy is derived from the Fokker-Planck transport equation. A perturbation analysis of this integro-differential equation leads to an analytical expression for the cosmic ray anisotropy and the focused transport equation for the isotropic part of the cosmic ray distribution function. The derived parallel spatial diffusion coefficient and the associated cosmic ray mean free path include the effect of adiabatic focusing and reduce to the standardmore » forms in the limit of a uniform guide magnetic field. For the illustrative case of isotropic pitch angle scattering, the derived mean free path agrees with the earlier expressions of Beeck and Wibberenz, Bieber and Burger, Kota, and Litvinenko, but disagrees with the result of Shalchi. The disagreement with the expression of Shalchi is particularly strong in the limit of strong adiabatic focusing.« less

Receptor binding kinetics equations: Derivation using the Laplace transform method.

PubMed

Hoare, Sam R J

Measuring unlabeled ligand receptor binding kinetics is valuable in optimizing and understanding drug action. Unfortunately, deriving equations for estimating kinetic parameters is challenging because it involves calculus; integration can be a frustrating barrier to the pharmacologist seeking to measure simple rate parameters. Here, a well-known tool for simplifying the derivation, the Laplace transform, is applied to models of receptor-ligand interaction. The method transforms differential equations to a form in which simple algebra can be applied to solve for the variable of interest, for example the concentration of ligand-bound receptor. The goal is to provide instruction using familiar examples, to enable investigators familiar with handling equilibrium binding equations to derive kinetic equations for receptor-ligand interaction. First, the Laplace transform is used to derive the equations for association and dissociation of labeled ligand binding. Next, its use for unlabeled ligand kinetic equations is exemplified by a full derivation of the kinetics of competitive binding equation. Finally, new unlabeled ligand equations are derived using the Laplace transform. These equations incorporate a pre-incubation step with unlabeled or labeled ligand. Four equations for measuring unlabeled ligand kinetics were compared and the two new equations verified by comparison with numerical solution. Importantly, the equations have not been verified with experimental data because no such experiments are evident in the literature. Equations were formatted for use in the curve-fitting program GraphPad Prism 6.0 and fitted to simulated data. This description of the Laplace transform method will enable pharmacologists to derive kinetic equations for their model or experimental paradigm under study. Application of the transform will expand the set of equations available for the pharmacologist to measure unlabeled ligand binding kinetics, and for other time-dependent pharmacological activities. Copyright © 2017 Elsevier Inc. All rights reserved.

Aerodynamic Design Optimization on Unstructured Meshes Using the Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Nielsen, Eric J.; Anderson, W. Kyle

1998-01-01

A discrete adjoint method is developed and demonstrated for aerodynamic design optimization on unstructured grids. The governing equations are the three-dimensional Reynolds-averaged Navier-Stokes equations coupled with a one-equation turbulence model. A discussion of the numerical implementation of the flow and adjoint equations is presented. Both compressible and incompressible solvers are differentiated and the accuracy of the sensitivity derivatives is verified by comparing with gradients obtained using finite differences. Several simplifying approximations to the complete linearization of the residual are also presented, and the resulting accuracy of the derivatives is examined. Demonstration optimizations for both compressible and incompressible flows are given.

Symmetry investigations on the incompressible stationary axisymmetric Euler equations with swirl

NASA Astrophysics Data System (ADS)

Frewer, M.; Oberlack, M.; Guenther, S.

2007-08-01

We discuss the incompressible stationary axisymmetric Euler equations with swirl, for which we derive via a scalar stream function an equivalent representation, the Bragg-Hawthorne equation [Bragg, S.L., Hawthorne, W.R., 1950. Some exact solutions of the flow through annular cascade actuator discs. J. Aero. Sci. 17, 243]. Despite this obvious equivalence, we will show that under a local Lie point symmetry analysis the Bragg-Hawthorne equation exposes itself as not being fully equivalent to the original Euler equations. This is reflected in the way that it possesses additional symmetries not being admitted by its counterpart. In other words, a symmetry of the Bragg-Hawthorne equation is in general not a symmetry of the Euler equations. Not the differential Euler equations but rather a set of integro-differential equations attains full equivalence to the Bragg-Hawthorne equation. For these intermediate Euler equations, it is interesting to note that local symmetries of the Bragg-Hawthorne equation transform to local as well as to nonlocal symmetries. This behaviour, on the one hand, is in accordance with Zawistowski's result [Zawistowski, Z.J., 2001. Symmetries of integro-differential equations. Rep. Math. Phys. 48, 269; Zawistowski, Z.J., 2004. General criterion of invariance for integro-differential equations. Rep. Math. Phys. 54, 341] that it is possible for integro-differential equations to admit local Lie point symmetries. On the other hand, with this transformation process we collect symmetries which cannot be obtained when carrying out a usual local Lie point symmetry analysis. Finally, the symmetry classification of the Bragg-Hawthorne equation is used to find analytical solutions for the phenomenon of vortex breakdown.

Eigenvalue sensitivity analysis of planar frames with variable joint and support locations

NASA Technical Reports Server (NTRS)

Chuang, Ching H.; Hou, Gene J. W.

1991-01-01

Two sensitivity equations are derived in this study based upon the continuum approach for eigenvalue sensitivity analysis of planar frame structures with variable joint and support locations. A variational form of an eigenvalue equation is first derived in which all of the quantities are expressed in the local coordinate system attached to each member. Material derivative of this variational equation is then sought to account for changes in member's length and orientation resulting form the perturbation of joint and support locations. Finally, eigenvalue sensitivity equations are formulated in either domain quantities (by the domain method) or boundary quantities (by the boundary method). It is concluded that the sensitivity equation derived by the boundary method is more efficient in computation but less accurate than that of the domain method. Nevertheless, both of them in terms of computational efficiency are superior to the conventional direct differentiation method and the finite difference method.

Data-driven discovery of partial differential equations.

PubMed

Rudy, Samuel H; Brunton, Steven L; Proctor, Joshua L; Kutz, J Nathan

2017-04-01

We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework, where the sensors are fixed spatially, or in a Lagrangian framework, where the sensors move with the dynamics. The method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems spanning a number of scientific domains including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially nonunique dynamical terms by using multiple time series taken with different initial data. Thus, for a traveling wave, the method can distinguish between a linear wave equation and the Korteweg-de Vries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parameterized spatiotemporal systems, where first-principles derivations are intractable.

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.

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

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

Gomez, Thomas; Nagayama, Taisuke; Fontes, Chris

Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numericalmore » complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. Here, this technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.« less

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

DOE PAGES

Gomez, Thomas; Nagayama, Taisuke; Fontes, Chris; ...

2018-04-23

Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numericalmore » complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. Here, this technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.« less

The symbolic computation of series solutions to ordinary differential equations using trees (extended abstract)

NASA Technical Reports Server (NTRS)

Grossman, Robert

1991-01-01

Algorithms previously developed by the author give formulas which can be used for the efficient symbolic computation of series expansions to solutions of nonlinear systems of ordinary differential equations. As a by product of this analysis, formulas are derived which relate to trees to the coefficients of the series expansions, similar to the work of Leroux and Viennot, and Lamnabhi, Leroux and Viennot.

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.

Similarity solution of the Boussinesq equation

NASA Astrophysics Data System (ADS)

Lockington, D. A.; Parlange, J.-Y.; Parlange, M. B.; Selker, J.

Similarity transforms of the Boussinesq equation in a semi-infinite medium are available when the boundary conditions are a power of time. The Boussinesq equation is reduced from a partial differential equation to a boundary-value problem. Chen et al. [Trans Porous Media 1995;18:15-36] use a hodograph method to derive an integral equation formulation of the new differential equation which they solve by numerical iteration. In the present paper, the convergence of their scheme is improved such that numerical iteration can be avoided for all practical purposes. However, a simpler analytical approach is also presented which is based on Shampine's transformation of the boundary value problem to an initial value problem. This analytical approximation is remarkably simple and yet more accurate than the analytical hodograph approximations.

Extended method of moments for deterministic analysis of stochastic multistable neurodynamical systems

NASA Astrophysics Data System (ADS)

Deco, Gustavo; Martí, Daniel

2007-03-01

The analysis of transitions in stochastic neurodynamical systems is essential to understand the computational principles that underlie those perceptual and cognitive processes involving multistable phenomena, like decision making and bistable perception. To investigate the role of noise in a multistable neurodynamical system described by coupled differential equations, one usually considers numerical simulations, which are time consuming because of the need for sufficiently many trials to capture the statistics of the influence of the fluctuations on that system. An alternative analytical approach involves the derivation of deterministic differential equations for the moments of the distribution of the activity of the neuronal populations. However, the application of the method of moments is restricted by the assumption that the distribution of the state variables of the system takes on a unimodal Gaussian shape. We extend in this paper the classical moments method to the case of bimodal distribution of the state variables, such that a reduced system of deterministic coupled differential equations can be derived for the desired regime of multistability.

Group invariant solution for a pre-existing fracture driven by a power-law fluid in impermeable rock

NASA Astrophysics Data System (ADS)

Fareo, A. G.; Mason, D. P.

2013-12-01

The effect of power-law rheology on hydraulic fracturing is investigated. The evolution of a two-dimensional fracture with non-zero initial length and driven by a power-law fluid is analyzed. Only fluid injection into the fracture is considered. The surrounding rock mass is impermeable. With the aid of lubrication theory and the PKN approximation a partial differential equation for the fracture half-width is derived. Using a linear combination of the Lie-point symmetry generators of the partial differential equation, the group invariant solution is obtained and the problem is reduced to a boundary value problem for an ordinary differential equation. Exact analytical solutions are derived for hydraulic fractures with constant volume and with constant propagation speed. The asymptotic solution near the fracture tip is found. The numerical solution for general working conditions is obtained by transforming the boundary value problem to a pair of initial value problems. Throughout the paper, hydraulic fracturing with shear thinning, Newtonian and shear thickening fluids are compared.

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.

A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel

NASA Astrophysics Data System (ADS)

Coronel-Escamilla, A.; Gómez-Aguilar, J. F.; Torres, L.; Escobar-Jiménez, R. F.

2018-02-01

A reaction-diffusion system can be represented by the Gray-Scott model. The reaction-diffusion dynamic is described by a pair of time and space dependent Partial Differential Equations (PDEs). In this paper, a generalization of the Gray-Scott model by using variable-order fractional differential equations is proposed. The variable-orders were set as smooth functions bounded in (0 , 1 ] and, specifically, the Liouville-Caputo and the Atangana-Baleanu-Caputo fractional derivatives were used to express the time differentiation. In order to find a numerical solution of the proposed model, the finite difference method together with the Adams method were applied. The simulations results showed the chaotic behavior of the proposed model when different variable-orders are applied.

Propagation of mechanical waves through a stochastic medium with spherical symmetry

NASA Astrophysics Data System (ADS)

Avendaño, Carlos G.; Reyes, J. Adrián

2018-01-01

We theoretically analyze the propagation of outgoing mechanical waves through an infinite isotropic elastic medium possessing spherical symmetry whose Lamé coefficients and density are spatial random functions characterized by well-defined statistical parameters. We derive the differential equation that governs the average displacement for a system whose properties depend on the radial coordinate. We show that such an equation is an extended version of the well-known Bessel differential equation whose perturbative additional terms contain coefficients that depend directly on the squared noise intensities and the autocorrelation lengths in an exponential decay fashion. We numerically solve the second order differential equation for several values of noise intensities and autocorrelation lengths and compare the corresponding displacement profiles with that of the exact analytic solution for the case of absent inhomogeneities.

Numerical methods for stochastic differential equations

NASA Astrophysics Data System (ADS)

Kloeden, Peter; Platen, Eckhard

1991-06-01

The numerical analysis of stochastic differential equations differs significantly from that of ordinary differential equations due to the peculiarities of stochastic calculus. This book provides an introduction to stochastic calculus and stochastic differential equations, both theory and applications. The main emphasise is placed on the numerical methods needed to solve such equations. It assumes an undergraduate background in mathematical methods typical of engineers and physicists, through many chapters begin with a descriptive summary which may be accessible to others who only require numerical recipes. To help the reader develop an intuitive understanding of the underlying mathematicals and hand-on numerical skills exercises and over 100 PC Exercises (PC-personal computer) are included. The stochastic Taylor expansion provides the key tool for the systematic derivation and investigation of discrete time numerical methods for stochastic differential equations. The book presents many new results on higher order methods for strong sample path approximations and for weak functional approximations, including implicit, predictor-corrector, extrapolation and variance-reduction methods. Besides serving as a basic text on such methods. the book offers the reader ready access to a large number of potential research problems in a field that is just beginning to expand rapidly and is widely applicable.

A low dimensional dynamical system for the wall layer

NASA Technical Reports Server (NTRS)

Aubry, N.; Keefe, L. R.

1987-01-01

Low dimensional dynamical systems which model a fully developed turbulent wall layer were derived.The model is based on the optimally fast convergent proper orthogonal decomposition, or Karhunen-Loeve expansion. This decomposition provides a set of eigenfunctions which are derived from the autocorrelation tensor at zero time lag. Via Galerkin projection, low dimensional sets of ordinary differential equations in time, for the coefficients of the expansion, were derived from the Navier-Stokes equations. The energy loss to the unresolved modes was modeled by an eddy viscosity representation, analogous to Heisenberg's spectral model. A set of eigenfunctions and eigenvalues were obtained from direct numerical simulation of a plane channel at a Reynolds number of 6600, based on the mean centerline velocity and the channel width flow and compared with previous work done by Herzog. Using the new eigenvalues and eigenfunctions, a new ten dimensional set of ordinary differential equations were derived using five non-zero cross-stream Fourier modes with a periodic length of 377 wall units. The dynamical system was integrated for a range of the eddy viscosity prameter alpha. This work is encouraging.

A Model for the Oxidation of Carbon Silicon Carbide Composite Structures

NASA Technical Reports Server (NTRS)

Sullivan, Roy M.

2004-01-01

A mathematical theory and an accompanying numerical scheme have been developed for predicting the oxidation behavior of carbon silicon carbide (C/SiC) composite structures. The theory is derived from the mechanics of the flow of ideal gases through a porous solid. The result of the theoretical formulation is a set of two coupled nonlinear differential equations written in terms of the oxidant and oxide partial pressures. The differential equations are solved simultaneously to obtain the partial vapor pressures of the oxidant and oxides as a function of the spatial location and time. The local rate of carbon oxidation is determined using the map of the local oxidant partial vapor pressure along with the Arrhenius rate equation. The nonlinear differential equations are cast into matrix equations by applying the Bubnov-Galerkin weighted residual method, allowing for the solution of the differential equations numerically. The numerical method is demonstrated by utilizing the method to model the carbon oxidation and weight loss behavior of C/SiC specimens during thermogravimetric experiments. The numerical method is used to study the physics of carbon oxidation in carbon silicon carbide composites.

Local and global Hopf bifurcation analysis in a neutral-type neuron system with two delays

NASA Astrophysics Data System (ADS)

Lv, Qiuyu; Liao, Xiaofeng

2018-03-01

In recent years, neutral-type differential-difference equations have been applied extensively in the field of engineering, and their dynamical behaviors are more complex than that of the delay differential-difference equations. In this paper, the equations used to describe a neutral-type neural network system of differential difference equation with two delays are studied (i.e. neutral-type differential equations). Firstly, by selecting τ1, τ2 respectively as a parameter, we provide an analysis about the local stability of the zero equilibrium point of the equations, and sufficient conditions of asymptotic stability for the system are derived. Secondly, by using the theory of normal form and applying the theorem of center manifold introduced by Hassard et al., the Hopf bifurcation is found and some formulas for deciding the stability of periodic solutions and the direction of Hopf bifurcation are given. Moreover, by applying the theorem of global Hopf bifurcation, the existence of global periodic solution of the system is studied. Finally, an example is given, and some computer numerical simulations are taken to demonstrate and certify the correctness of the presented results.

Fractional analysis for nonlinear electrical transmission line and nonlinear Schroedinger equations with incomplete sub-equation

NASA Astrophysics Data System (ADS)

Fendzi-Donfack, Emmanuel; Nguenang, Jean Pierre; Nana, Laurent

2018-02-01

We use the fractional complex transform with the modified Riemann-Liouville derivative operator to establish the exact and generalized solutions of two fractional partial differential equations. We determine the solutions of fractional nonlinear electrical transmission lines (NETL) and the perturbed nonlinear Schroedinger (NLS) equation with the Kerr law nonlinearity term. The solutions are obtained for the parameters in the range (0<α≤1) of the derivative operator and we found the traditional solutions for the limiting case of α =1. We show that according to the modified Riemann-Liouville derivative, the solutions found can describe physical systems with memory effect, transient effects in electrical systems and nonlinear transmission lines, and other systems such as optical fiber.

Fractional calculus in hydrologic modeling: A numerical perspective

PubMed Central

Benson, David A.; Meerschaert, Mark M.; Revielle, Jordan

2013-01-01

Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus. PMID:23524449

A remark on fractional differential equation involving I-function

NASA Astrophysics Data System (ADS)

Mishra, Jyoti

2018-02-01

The present paper deals with the solution of the fractional differential equation using the Laplace transform operator and its corresponding properties in the fractional calculus; we derive an exact solution of a complex fractional differential equation involving a special function known as I-function. The analysis of the some fractional integral with two parameters is presented using the suggested Theorem 1. In addition, some very useful corollaries are established and their proofs presented in detail. Some obtained exact solutions are depicted to see the effect of each fractional order. Owing to the wider applicability of the I-function, we can conclude that, the obtained results in our work generalize numerous well-known results obtained by specializing the parameters.

Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM.

PubMed

Singh, Brajesh K; Srivastava, Vineet K

2015-04-01

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.

Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM

PubMed Central

Singh, Brajesh K.; Srivastava, Vineet K.

2015-01-01

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations. PMID:26064639

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

Momentum Maps and Stochastic Clebsch Action Principles

NASA Astrophysics Data System (ADS)

Cruzeiro, Ana Bela; Holm, Darryl D.; Ratiu, Tudor S.

2018-01-01

We derive stochastic differential equations whose solutions follow the flow of a stochastic nonlinear Lie algebra operation on a configuration manifold. For this purpose, we develop a stochastic Clebsch action principle, in which the noise couples to the phase space variables through a momentum map. This special coupling simplifies the structure of the resulting stochastic Hamilton equations for the momentum map. In particular, these stochastic Hamilton equations collectivize for Hamiltonians that depend only on the momentum map variable. The Stratonovich equations are derived from the Clebsch variational principle and then converted into Itô form. In comparing the Stratonovich and Itô forms of the stochastic dynamical equations governing the components of the momentum map, we find that the Itô contraction term turns out to be a double Poisson bracket. Finally, we present the stochastic Hamiltonian formulation of the collectivized momentum map dynamics and derive the corresponding Kolmogorov forward and backward equations.

Mechanical modeling for magnetorheological elastomer isolators based on constitutive equations and electromagnetic analysis

NASA Astrophysics Data System (ADS)

Wang, Qi; Dong, Xufeng; Li, Luyu; Ou, Jinping

2018-06-01

As constitutive models are too complicated and existing mechanical models lack universality, these models are beyond satisfaction for magnetorheological elastomer (MRE) devices. In this article, a novel universal method is proposed to build concise mechanical models. Constitutive model and electromagnetic analysis were applied in this method to ensure universality, while a series of derivations and simplifications were carried out to obtain a concise formulation. To illustrate the proposed modeling method, a conical MRE isolator was introduced. Its basic mechanical equations were built based on equilibrium, deformation compatibility, constitutive equations and electromagnetic analysis. An iteration model and a highly efficient differential equation editor based model were then derived to solve the basic mechanical equations. The final simplified mechanical equations were obtained by re-fitting the simulations with a novel optimal algorithm. In the end, verification test of the isolator has proved the accuracy of the derived mechanical model and the modeling method.

An implicit semianalytic numerical method for the solution of nonequilibrium chemistry problems

NASA Technical Reports Server (NTRS)

Graves, R. A., Jr.; Gnoffo, P. A.; Boughner, R. E.

1974-01-01

The first order differential equation form systems of equations. They are solved by a simple and relatively accurate implicit semianalytic technique which is derived from a quadrature solution of the governing equation. This method is mathematically simpler than most implicit methods and has the exponential nature of the problem embedded in the solution.

Exact renormalization group equation for the Lifshitz critical point

NASA Astrophysics Data System (ADS)

Bervillier, C.

2004-10-01

An exact renormalization equation (ERGE) accounting for an anisotropic scaling is derived. The critical and tricritical Lifshitz points are then studied at leading order of the derivative expansion which is shown to involve two differential equations. The resulting estimates of the Lifshitz critical exponents compare well with the O(ε) calculations. In the case of the Lifshitz tricritical point, it is shown that a marginally relevant coupling defies the perturbative approach since it actually makes the fixed point referred to in the previous perturbative calculations O(ε) finally unstable.

Computer program for the load and trajectory analysis of two DOF bodies connected by an elastic tether: Users manual

NASA Technical Reports Server (NTRS)

Doyle, G. R., Jr.; Burbick, J. W.

1973-01-01

The derivation of the differential equations of motion of a 3 Degrees of Freedom body joined to a 3 Degrees of Freedom body by an elastic tether. The tether is represented by a spring and dashpot in parallel. A computer program which integrates the equations of motion is also described. Although the derivation of the equations of motions are for a general system, the computer program is written for defining loads in large boosters recovered by parachutes.

Genetic network inference as a series of discrimination tasks.

PubMed

Kimura, Shuhei; Nakayama, Satoshi; Hatakeyama, Mariko

2009-04-01

Genetic network inference methods based on sets of differential equations generally require a great deal of time, as the equations must be solved many times. To reduce the computational cost, researchers have proposed other methods for inferring genetic networks by solving sets of differential equations only a few times, or even without solving them at all. When we try to obtain reasonable network models using these methods, however, we must estimate the time derivatives of the gene expression levels with great precision. In this study, we propose a new method to overcome the drawbacks of inference methods based on sets of differential equations. Our method infers genetic networks by obtaining classifiers capable of predicting the signs of the derivatives of the gene expression levels. For this purpose, we defined a genetic network inference problem as a series of discrimination tasks, then solved the defined series of discrimination tasks with a linear programming machine. Our experimental results demonstrated that the proposed method is capable of correctly inferring genetic networks, and doing so more than 500 times faster than the other inference methods based on sets of differential equations. Next, we applied our method to actual expression data of the bacterial SOS DNA repair system. And finally, we demonstrated that our approach relates to the inference method based on the S-system model. Though our method provides no estimation of the kinetic parameters, it should be useful for researchers interested only in the network structure of a target system. Supplementary data are available at Bioinformatics online.

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.

Methodology for sensitivity analysis, approximate analysis, and design optimization in CFD for multidisciplinary applications

NASA Technical Reports Server (NTRS)

Taylor, Arthur C., III; Hou, Gene W.

1994-01-01

The straightforward automatic-differentiation and the hand-differentiated incremental iterative methods are interwoven to produce a hybrid scheme that captures some of the strengths of each strategy. With this compromise, discrete aerodynamic sensitivity derivatives are calculated with the efficient incremental iterative solution algorithm of the original flow code. Moreover, the principal advantage of automatic differentiation is retained (i.e., all complicated source code for the derivative calculations is constructed quickly with accuracy). The basic equations for second-order sensitivity derivatives are presented; four methods are compared. Each scheme requires that large systems are solved first for the first-order derivatives and, in all but one method, for the first-order adjoint variables. Of these latter three schemes, two require no solutions of large systems thereafter. For the other two for which additional systems are solved, the equations and solution procedures are analogous to those for the first order derivatives. From a practical viewpoint, implementation of the second-order methods is feasible only with software tools such as automatic differentiation, because of the extreme complexity and large number of terms. First- and second-order sensitivities are calculated accurately for two airfoil problems, including a turbulent flow example; both geometric-shape and flow-condition design variables are considered. Several methods are tested; results are compared on the basis of accuracy, computational time, and computer memory. For first-order derivatives, the hybrid incremental iterative scheme obtained with automatic differentiation is competitive with the best hand-differentiated method; for six independent variables, it is at least two to four times faster than central finite differences and requires only 60 percent more memory than the original code; the performance is expected to improve further in the future.

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.

Learning partial differential equations via data discovery and sparse optimization

NASA Astrophysics Data System (ADS)

Schaeffer, Hayden

2017-01-01

We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.

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.

Learning partial differential equations via data discovery and sparse optimization.

PubMed

Schaeffer, Hayden

2017-01-01

We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.

Learning partial differential equations via data discovery and sparse optimization

PubMed Central

2017-01-01

We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection. PMID:28265183

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I

NASA Astrophysics Data System (ADS)

Chelnokov, Yu. N.

2017-11-01

We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo-Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper. A survey of papers related to the regularization of the differential equations of the two- and threebody problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues-Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given. Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo-Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventhorder differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass. The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classicalmethods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses.

Theory of biaxial graded-index optical fiber. M.S. Thesis

NASA Technical Reports Server (NTRS)

Kawalko, Stephen F.

1990-01-01

A biaxial graded-index fiber with a homogeneous cladding is studied. Two methods, wave equation and matrix differential equation, of formulating the problem and their respective solutions are discussed. For the wave equation formulation of the problem it is shown that for the case of a diagonal permittivity tensor the longitudinal electric and magnetic fields satisfy a pair of coupled second-order differential equations. Also, a generalized dispersion relation is derived in terms of the solutions for the longitudinal electric and magnetic fields. For the case of a step-index fiber, either isotropic or uniaxial, these differential equations can be solved exactly in terms of Bessel functions. For the cases of an istropic graded-index and a uniaxial graded-index fiber, a solution using the Wentzel, Krammers and Brillouin (WKB) approximation technique is shown. Results for some particular permittivity profiles are presented. Also the WKB solutions is compared with the vector solution found by Kurtz and Streifer. For the matrix formulation it is shown that the tangential components of the electric and magnetic fields satisfy a system of four first-order differential equations which can be conveniently written in matrix form. For the special case of meridional modes, the system of equations splits into two systems of two equations. A general iterative technique, asymptotic partitioning of systems of equations, for solving systems of differential equations is presented. As a simple example, Bessel's differential equation is written in matrix form and is solved using this asymptotic technique. Low order solutions for particular examples of a biaxial and uniaxial graded-index fiber are presented. Finally numerical results obtained using the asymptotic technique are presented for particular examples of isotropic and uniaxial step-index fibers and isotropic, uniaxial and biaxial graded-index fibers.

Efficient numerical method for solving Cauchy problem for the Gamma equation

NASA Astrophysics Data System (ADS)

Koleva, Miglena N.

2011-12-01

In this work we consider Cauchy problem for the so called Gamma equation, derived by transforming the fully nonlinear Black-Scholes equation for option price into a quasilinear parabolic equation for the second derivative (Greek) Γ = VSS of the option price V. We develop an efficient numerical method for solving the model problem concerning different volatility terms. Using suitable change of variables the problem is transformed on finite interval, keeping original behavior of the solution at the infinity. Then we construct Picard-Newton algorithm with adaptive mesh step in time, which can be applied also in the case of non-differentiable functions. Results of numerical simulations are given.

Exact Solutions for Stokes' Flow of a Non-Newtonian Nanofluid Model: A Lie Similarity Approach

NASA Astrophysics Data System (ADS)

Aziz, Taha; Aziz, A.; Khalique, C. M.

2016-07-01

The fully developed time-dependent flow of an incompressible, thermodynamically compatible non-Newtonian third-grade nanofluid is investigated. The classical Stokes model is considered in which the flow is generated due to the motion of the plate in its own plane with an impulsive velocity. The Lie symmetry approach is utilised to convert the governing nonlinear partial differential equation into different linear and nonlinear ordinary differential equations. The reduced ordinary differential equations are then solved by using the compatibility and generalised group method. Exact solutions for the model equation are deduced in the form of closed-form exponential functions which are not available in the literature before. In addition, we also derived the conservation laws associated with the governing model. Finally, the physical features of the pertinent parameters are discussed in detail through several graphs.

Modified Taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on finite intervals.

PubMed

Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel Antonio; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Marin-Hernandez, Antonio; Herrera-May, Agustin Leobardo; Diaz-Sanchez, Alejandro; Huerta-Chua, Jesus

2014-01-01

In this article, we propose the application of a modified Taylor series method (MTSM) for the approximation of nonlinear problems described on finite intervals. The issue of Taylor series method with mixed boundary conditions is circumvented using shooting constants and extra derivatives of the problem. In order to show the benefits of this proposal, three different kinds of problems are solved: three-point boundary valued problem (BVP) of third-order with a hyperbolic sine nonlinearity, two-point BVP for a second-order nonlinear differential equation with an exponential nonlinearity, and a two-point BVP for a third-order nonlinear differential equation with a radical nonlinearity. The result shows that the MTSM method is capable to generate easily computable and highly accurate approximations for nonlinear equations. 34L30.

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.

Comparison of numerical techniques for integration of stiff ordinary differential equations arising in combustion chemistry

NASA Technical Reports Server (NTRS)

Radhakrishnan, K.

1984-01-01

The efficiency and accuracy of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations are compared. The methods examined include two general-purpose codes, EPISODE and LSODE, and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an interactive solution of the algebraic energy conservation equation to compute the temperature does not result in significant errors. In addition, this method is more efficient than evaluating the temperature by integrating its time derivative. Significant reductions in computational work are realized by updating the rate constants (k = at(supra N) N exp(-E/RT) only when the temperature change exceeds an amount delta T that is problem dependent. An approximate expression for the automatic evaluation of delta T is derived and is shown to result in increased efficiency.

Data-driven discovery of partial differential equations

PubMed Central

Rudy, Samuel H.; Brunton, Steven L.; Proctor, Joshua L.; Kutz, J. Nathan

2017-01-01

We propose a sparse regression method capable of discovering the governing partial differential equation(s) of a given system by time series measurements in the spatial domain. The regression framework relies on sparsity-promoting techniques to select the nonlinear and partial derivative terms of the governing equations that most accurately represent the data, bypassing a combinatorially large search through all possible candidate models. The method balances model complexity and regression accuracy by selecting a parsimonious model via Pareto analysis. Time series measurements can be made in an Eulerian framework, where the sensors are fixed spatially, or in a Lagrangian framework, where the sensors move with the dynamics. The method is computationally efficient, robust, and demonstrated to work on a variety of canonical problems spanning a number of scientific domains including Navier-Stokes, the quantum harmonic oscillator, and the diffusion equation. Moreover, the method is capable of disambiguating between potentially nonunique dynamical terms by using multiple time series taken with different initial data. Thus, for a traveling wave, the method can distinguish between a linear wave equation and the Korteweg–de Vries equation, for instance. The method provides a promising new technique for discovering governing equations and physical laws in parameterized spatiotemporal systems, where first-principles derivations are intractable. PMID:28508044

Conformational statistics of stiff macromolecules as solutions to partial differential equations on the rotation and motion groups

PubMed

Chirikjian; Wang

2000-07-01

Partial differential equations (PDE's) for the probability density function (PDF) of the position and orientation of the distal end of a stiff macromolecule relative to its proximal end are derived and solved. The Kratky-Porod wormlike chain, the Yamakawa helical wormlike chain, and the original and revised Marko-Siggia models are examples of stiffness models to which the present formulation is applied. The solution technique uses harmonic analysis on the rotation and motion groups to convert PDE's governing the PDF's of interest into linear algebraic equations which have mathematically elegant solutions.

Fluid limit of nonintegrable continuous-time random walks in terms of fractional differential equations.

PubMed

Sánchez, R; Carreras, B A; van Milligen, B Ph

2005-01-01

The fluid limit of a recently introduced family of nonintegrable (nonlinear) continuous-time random walks is derived in terms of fractional differential equations. In this limit, it is shown that the formalism allows for the modeling of the interaction between multiple transport mechanisms with not only disparate spatial scales but also different temporal scales. For this reason, the resulting fluid equations may find application in the study of a large number of nonlinear multiscale transport problems, ranging from the study of self-organized criticality to the modeling of turbulent transport in fluids and plasmas.

Analytical Derivation of Power Laws in Firm Size Variables from Gibrat's Law and Quasi-inversion Symmetry: A Geomorphological Approach

NASA Astrophysics Data System (ADS)

Ishikawa, Atushi; Fujimoto, Shouji; Mizuno, Takayuki; Watanabe, Tsutomu

2014-03-01

We start from Gibrat's law and quasi-inversion symmetry for three firm size variables (i.e., tangible fixed assets K, number of employees L, and sales Y) and derive a partial differential equation to be satisfied by the joint probability density function of K and L. We then transform K and L, which are correlated, into two independent variables by applying surface openness used in geomorphology and provide an analytical solution to the partial differential equation. Using worldwide data on the firm size variables for companies, we confirm that the estimates on the power-law exponents of K, L, and Y satisfy a relationship implied by the theory.

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.

Analysis and testing of numerical formulas for the initial value problem

NASA Technical Reports Server (NTRS)

Brown, R. L.; Kovach, K. R.; Popyack, J. L.

1980-01-01

Three computer programs for evaluating and testing numerical integration formulas used with fixed stepsize programs to solve initial value systems of ordinary differential equations are described. A program written in PASCAL SERIES, takes as input the differential equations and produces a FORTRAN subroutine for the derivatives of the system and for computing the actual solution through recursive power series techniques. Both of these are used by STAN, a FORTRAN program that interactively displays a discrete analog of the Liapunov stability region of any two dimensional subspace of the system. The derivatives may be used by CLMP, a FORTRAN program, to test the fixed stepsize formula against a good numerical result and interactively display the solutions.

General Tricomi-Rassias problem and oblique derivative problem for generalized Chaplygin equations

NASA Astrophysics Data System (ADS)

Wen, Guochun; Chen, Dechang; Cheng, Xiuzhen

2007-09-01

Many authors have discussed the Tricomi problem for some second order equations of mixed type, which has important applications in gas dynamics. In particular, Bers proposed the Tricomi problem for Chaplygin equations in multiply connected domains [L. Bers, Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York, 1958]. And Rassias proposed the exterior Tricomi problem for mixed equations in a doubly connected domain and proved the uniqueness of solutions for the problem [J.M. Rassias, Lecture Notes on Mixed Type Partial Differential Equations, World Scientific, Singapore, 1990]. In the present paper, we discuss the general Tricomi-Rassias problem for generalized Chaplygin equations. This is one general oblique derivative problem that includes the exterior Tricomi problem as a special case. We first give the representation of solutions of the general Tricomi-Rassias problem, and then prove the uniqueness and existence of solutions for the problem by a new method. In this paper, we shall also discuss another general oblique derivative problem for generalized Chaplygin equations.

Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type

NASA Astrophysics Data System (ADS)

Doha, E. H.; Abd-Elhameed, W. M.; Youssri, Y. H.

2013-10-01

In this paper, we present a new second kind Chebyshev (S2KC) operational matrix of derivatives. With the aid of S2KC, an algorithm is described to obtain numerical solutions of a class of linear and nonlinear Lane-Emden type singular initial value problems (IVPs). The idea of obtaining such solutions is essentially based on reducing the differential equation with its initial conditions to a system of algebraic equations. Two illustrative examples concern relevant physical problems (the Lane-Emden equations of the first and second kind) are discussed to demonstrate the validity and applicability of the suggested algorithm. Numerical results obtained are comparing favorably with the analytical known solutions.

A class of traveling wave solutions for space-time fractional biological population model in mathematical physics

NASA Astrophysics Data System (ADS)

Akram, Ghazala; Batool, Fiza

2017-10-01

The (G'/G)-expansion method is utilized for a reliable treatment of space-time fractional biological population model. The method has been applied in the sense of the Jumarie's modified Riemann-Liouville derivative. Three classes of exact traveling wave solutions, hyperbolic, trigonometric and rational solutions of the associated equation are characterized with some free parameters. A generalized fractional complex transform is applied to convert the fractional equations to ordinary differential equations which subsequently resulted in number of exact solutions. It should be mentioned that the (G'/G)-expansion method is very effective and convenient for solving nonlinear partial differential equations of fractional order whose balancing number is a negative integer.

Differential renormalization-group generators for static and dynamic critical phenomena

NASA Astrophysics Data System (ADS)

Chang, T. S.; Vvedensky, D. D.; Nicoll, J. F.

1992-09-01

The derivation of differential renormalization-group (DRG) equations for applications to static and dynamic critical phenomena is reviewed. The DRG approach provides a self-contained closed-form representation of the Wilson renormalization group (RG) and should be viewed as complementary to the Callan-Symanzik equations used in field-theoretic approaches to the RG. The various forms of DRG equations are derived to illustrate the general mathematical structure of each approach and to point out the advantages and disadvantages for performing practical calculations. Otherwise, the review focuses upon the one-particle-irreducible DRG equations derived by Nicoll and Chang and by Chang, Nicoll, and Young; no attempt is made to provide a general treatise of critical phenomena. A few specific examples are included to illustrate the utility of the DRG approach: the large- n limit of the classical n-vector model (the spherical model), multi- or higher-order critical phenomena, and crit ical dynamics far from equilibrium. The large- n limit of the n-vector model is used to introduce the application of DRG equations to a well-known example, with exact solution obtained for the nonlinear trajectories, generating functions for nonlinear scaling fields, and the equation of state. Trajectory integrals and nonlinear scaling fields within the framework of ɛ-expansions are then discussed for tricritical crossover, and briefly for certain aspects of multi- or higher-order critical points, including the derivation of the Helmholtz free energy and the equation of state. The discussion then turns to critical dynamics with a development of the path integral formulation for general dynamic processes. This is followed by an application to a model far-from-equilibrium system that undergoes a phase transformation analogous to a second-order critical point, the Schlögl model for a chemical instability.

Chandrasekhar equations and computational algorithms for distributed parameter systems

NASA Technical Reports Server (NTRS)

Burns, J. A.; Ito, K.; Powers, R. K.

1984-01-01

The Chandrasekhar equations arising in optimal control problems for linear distributed parameter systems are considered. The equations are derived via approximation theory. This approach is used to obtain existence, uniqueness, and strong differentiability of the solutions and provides the basis for a convergent computation scheme for approximating feedback gain operators. A numerical example is presented to illustrate these ideas.

Non-classical and potential symmetry analysis of Richard's equation for moisture flow in soil

NASA Astrophysics Data System (ADS)

Wiltshire, Ron; El-Kafri, Manal

2004-01-01

This paper focuses upon the derivation of the non-classical symmetries of Bluman and Cole as they apply to Richard's equation for water flow in an unsaturated uniform soil. It is shown that the determining equations for the non-classical case lead to four highly non-linear equations which have been solved in five particular cases. In each case the corresponding similarity ansatz has been derived and Richard's equation is reduced to an ordinary differential equation. Explicit solutions are produced when possible. Richard's equation is also expressed as a potential system and in reviewing the classical Lie solutions a new symmetry is derived together with its similarity ansatz. Determining equations are then produced for the potential system using the non-classical algorithm. This results in an under-determined set of equations and an example symmetry that reveals a missing classical case is presented. An example of a classical and a non-classical symmetry reduction applied to the infiltration of moisture in soil is presented. The condition for surface invariance is used to demonstrate the equivalence of a classical Lie and a potential symmetry.

Errors in finite-difference computations on curvilinear coordinate systems

NASA Technical Reports Server (NTRS)

Mastin, C. W.; Thompson, J. F.

1980-01-01

Curvilinear coordinate systems were used extensively to solve partial differential equations on arbitrary regions. An analysis of truncation error in the computation of derivatives revealed why numerical results may be erroneous. A more accurate method of computing derivatives is presented.

Feynman-Kac formula for stochastic hybrid systems.

PubMed

Bressloff, Paul C

2017-01-01

We derive a Feynman-Kac formula for functionals of a stochastic hybrid system evolving according to a piecewise deterministic Markov process. We first derive a stochastic Liouville equation for the moment generator of the stochastic functional, given a particular realization of the underlying discrete Markov process; the latter generates transitions between different dynamical equations for the continuous process. We then analyze the stochastic Liouville equation using methods recently developed for diffusion processes in randomly switching environments. In particular, we obtain dynamical equations for the moment generating function, averaged with respect to realizations of the discrete Markov process. The resulting Feynman-Kac formula takes the form of a differential Chapman-Kolmogorov equation. We illustrate the theory by calculating the occupation time for a one-dimensional velocity jump process on the infinite or semi-infinite real line. Finally, we present an alternative derivation of the Feynman-Kac formula based on a recent path-integral formulation of stochastic hybrid systems.

Analytic theory of orbit contraction

NASA Technical Reports Server (NTRS)

Vinh, N. X.; Longuski, J. M.; Busemann, A.; Culp, R. D.

1977-01-01

The motion of a satellite in orbit, subject to atmospheric force and the motion of a reentry vehicle are governed by gravitational and aerodynamic forces. This suggests the derivation of a uniform set of equations applicable to both cases. For the case of satellite motion, by a proper transformation and by the method of averaging, a technique appropriate for long duration flight, the classical nonlinear differential equation describing the contraction of the major axis is derived. A rigorous analytic solution is used to integrate this equation with a high degree of accuracy, using Poincare's method of small parameters and Lagrange's expansion to explicitly express the major axis as a function of the eccentricity. The solution is uniformly valid for moderate and small eccentricities. For highly eccentric orbits, the asymptotic equation is derived directly from the general equation. Numerical solutions were generated to display the accuracy of the analytic theory.

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.

Anomalous diffusion associated with nonlinear fractional derivative fokker-planck-like equation: exact time-dependent solutions

PubMed

Bologna; Tsallis; Grigolini

2000-08-01

We consider the d=1 nonlinear Fokker-Planck-like equation with fractional derivatives ( partial differential/ partial differentialt)P(x,t)=D( partial differential(gamma)/ partial differentialx(gamma))[P(x,t)](nu). Exact time-dependent solutions are found for nu=(2-gamma)/(1+gamma)(-infinity

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 the expected discounted penalty functions for two classes of risk processes under a threshold dividend strategy

NASA Astrophysics Data System (ADS)

Lu, Zhaoyang; Xu, Wei; Sun, Decai; Han, Weiguo

2009-10-01

In this paper, the discounted penalty (Gerber-Shiu) functions for a risk model involving two independent classes of insurance risks under a threshold dividend strategy are developed. We also assume that the two claim number processes are independent Poisson and generalized Erlang (2) processes, respectively. When the surplus is above this threshold level, dividends are paid at a constant rate that does not exceed the premium rate. Two systems of integro-differential equations for discounted penalty functions are derived, based on whether the surplus is above this threshold level. Laplace transformations of the discounted penalty functions when the surplus is below the threshold level are obtained. And we also derive a system of renewal equations satisfied by the discounted penalty function with initial surplus above the threshold strategy via the Dickson-Hipp operator. Finally, analytical solutions of the two systems of integro-differential equations are presented.

Flap-lag-torsional dynamics of helicopter rotor blades in forward flight

NASA Technical Reports Server (NTRS)

Crespodasilva, M. R. M.

1986-01-01

A perturbation/numerical methodology to analyze the flap-lead/lag motion of a centrally hinged spring restrained rotor blade that is valid for both hover and for forward flight was developed. The derivation of the nonlinear differential equations of motion and the analysis of the stability of the steady state response of the blade were conducted entirely in a Symbolics 3670 Machine using MACSYMA to perform all the lengthy symbolic manipulations. It also includes generation of the fortran codes and plots of the results. The Floquet theory was also applied to the differential equations of motion in order to compare results with those obtained from the perturbation analysis. The results obtained from the perturbation methodology and from Floquet theory were found to be very close to each other, which demonstrates the usefullness of the perturbation methodology. Another problem under study consisted in the analysis of the influence of higher order terms in the response and stability of a flexible rotor blade in forward flight using Computerized Symbolic Manipulation and a perturbation technique to bypass the Floquet theory. The derivation of the partial differential equations of motion is presented.

Fem Formulation of Heat Transfer in Cylindrical Porous Medium

NASA Astrophysics Data System (ADS)

Azeem; Khaleed, H. M. T.; Soudagar, Manzoor Elahi M.

2017-08-01

Heat transfer in porous medium can be derived from the fundamental laws of flow in porous region ass given by Henry Darcy. The fluid flow and energy transport inside the porous medium can be described with the help of momentum and energy equations. The heat transfer in cylindrical porous medium differs from its counterpart in radial and axial coordinates. The present work is focused to discuss the finite element formulation of heat transfer in cylindrical porous medium. The basic partial differential equations are derived using Darcy law which is the converted into a set of algebraic equations with the help of finite element method. The resulting equations are solved by matrix method for two solution variables involved in the coupled equations.

Thermal diffusion of Boussinesq solitons.

PubMed

Arévalo, Edward; Mertens, Franz G

2007-10-01

We consider the problem of the soliton dynamics in the presence of an external noisy force for the Boussinesq type equations. A set of ordinary differential equations (ODEs) of the relevant coordinates of the system is derived. We show that for the improved Boussinesq (IBq) equation the set of ODEs has limiting cases leading to a set of ODEs which can be directly derived either from the ill-posed Boussinesq equation or from the Korteweg-de Vries (KdV) equation. The case of a soliton propagating in the presence of damping and thermal noise is considered for the IBq equation. A good agreement between theory and simulations is observed showing the strong robustness of these excitations. The results obtained here generalize previous results obtained in the frame of the KdV equation for lattice solitons in the monatomic chain of atoms.

Baecklund transformation for the Ernst equation of general relativity

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

Harrison, B.K.

A Baecklund transformation for the Ernst equation arising in general relativity in connection with several physical problems is derived, using the pseudopotential method of Wahlquist and Estabrook. A prolongation structure is also constructed, using a method of writing the equations in terms of differential forms, and an equation in the spirit of Lax is constructed, somewhat different from that given by Maison. Possible uses of the Baecklund transformation to generate new solutions are mentioned.

Coulomb Thrusting Application Study

DTIC Science & Technology

2006-01-20

Acceleration Magnitudes To study the relative motion of spacecraft in nearly circular orbits, the Clohessy - Wiltshire - Hill equations are commonly...Modeling The Clohessy - Wiltshire -Hill’s equations12–14 for one of the spacecraft in the 2-craft Coulomb tether formation is given by ẍ1 − 2nẏ1...equa- tion was obtained using the Clohessy - Wiltshire - Hill equations, while the linearized differential equations of ψ and θ were derived from the full

The Fokker-Planck equation for coupled Brown-Néel-rotation.

PubMed

Weizenecker, Jürgen

2018-01-22

Calculating the dynamic properties of magnetization of single-domain particles is of great importance for the tomographic imaging modality known as magnetic particle imaging (MPI). Although the assumption of instantaneous thermodynamic equilibrium (Langevin function) after application of time-dependent magnetic fields is sufficient for understanding the fundamental behavior, it is essential to consider the finite response times of magnetic particles for optimizing or analyzing various aspects, e.g. interpreting spectra, optimizing MPI sequences, developing new contrasts, and evaluating simplified models. The change in magnetization following the application of the fields is caused by two different movements: the geometric rotation of the particle and the rotation of magnetization with respect to the fixed particle axes. These individual rotations can be well described using the Langevin equations or the Fokker-Planck equation. However, because the two rotations generally exhibit interdependence, it is necessary to consider coupling between the two equations. This article shows how a coupled Fokker-Planck equation can be derived on the basis of coupled Langevin equations. Two physically equivalent Fokker-Planck equations are derived and transformed by means of an appropriate series expansion into a system of ordinary differential equations, which can be solved numerically. Finally, this system is also used to specify a system of differential equations for various limiting cases (Néel, Brown, uniaxial symmetry). Generally, the system exhibits a sparsely populated matrix and can therefore be handled well numerically.

The Fokker-Planck equation for coupled Brown-Néel-rotation

NASA Astrophysics Data System (ADS)

Weizenecker, Jürgen

2018-02-01

Calculating the dynamic properties of magnetization of single-domain particles is of great importance for the tomographic imaging modality known as magnetic particle imaging (MPI). Although the assumption of instantaneous thermodynamic equilibrium (Langevin function) after application of time-dependent magnetic fields is sufficient for understanding the fundamental behavior, it is essential to consider the finite response times of magnetic particles for optimizing or analyzing various aspects, e.g. interpreting spectra, optimizing MPI sequences, developing new contrasts, and evaluating simplified models. The change in magnetization following the application of the fields is caused by two different movements: the geometric rotation of the particle and the rotation of magnetization with respect to the fixed particle axes. These individual rotations can be well described using the Langevin equations or the Fokker-Planck equation. However, because the two rotations generally exhibit interdependence, it is necessary to consider coupling between the two equations. This article shows how a coupled Fokker-Planck equation can be derived on the basis of coupled Langevin equations. Two physically equivalent Fokker-Planck equations are derived and transformed by means of an appropriate series expansion into a system of ordinary differential equations, which can be solved numerically. Finally, this system is also used to specify a system of differential equations for various limiting cases (Néel, Brown, uniaxial symmetry). Generally, the system exhibits a sparsely populated matrix and can therefore be handled well numerically.

Bending of Euler-Bernoulli nanobeams based on the strain-driven and stress-driven nonlocal integral models: a numerical approach

NASA Astrophysics Data System (ADS)

Oskouie, M. Faraji; Ansari, R.; Rouhi, H.

2018-04-01

Eringen's nonlocal elasticity theory is extensively employed for the analysis of nanostructures because it is able to capture nanoscale effects. Previous studies have revealed that using the differential form of the strain-driven version of this theory leads to paradoxical results in some cases, such as bending analysis of cantilevers, and recourse must be made to the integral version. In this article, a novel numerical approach is developed for the bending analysis of Euler-Bernoulli nanobeams in the context of strain- and stress-driven integral nonlocal models. This numerical approach is proposed for the direct solution to bypass the difficulties related to converting the integral governing equation into a differential equation. First, the governing equation is derived based on both strain-driven and stress-driven nonlocal models by means of the minimum total potential energy. Also, in each case, the governing equation is obtained in both strong and weak forms. To solve numerically the derived equations, matrix differential and integral operators are constructed based upon the finite difference technique and trapezoidal integration rule. It is shown that the proposed numerical approach can be efficiently applied to the strain-driven nonlocal model with the aim of resolving the mentioned paradoxes. Also, it is able to solve the problem based on the strain-driven model without inconsistencies of the application of this model that are reported in the literature.

Estimating Dynamical Systems: Derivative Estimation Hints from Sir Ronald A. Fisher

ERIC Educational Resources Information Center

Deboeck, Pascal R.

2010-01-01

The fitting of dynamical systems to psychological data offers the promise of addressing new and innovative questions about how people change over time. One method of fitting dynamical systems is to estimate the derivatives of a time series and then examine the relationships between derivatives using a differential equation model. One common…

Computing anticipatory systems with incursion and hyperincursion

NASA Astrophysics Data System (ADS)

Dubois, Daniel M.

1998-07-01

An anticipatory system is a system which contains a model of itself and/or of its environment in view of computing its present state as a function of the prediction of the model. With the concepts of incursion and hyperincursion, anticipatory discrete systems can be modelled, simulated and controlled. By definition an incursion, an inclusive or implicit recursion, can be written as: x(t+1)=F[…,x(t-1),x(t),x(t+1),…] where the value of a variable x(t+1) at time t+1 is a function of this variable at past, present and future times. This is an extension of recursion. Hyperincursion is an incursion with multiple solutions. For example, chaos in the Pearl-Verhulst map model: x(t+1)=a.x(t).[1-x(t)] is controlled by the following anticipatory incursive model: x(t+1)=a.x(t).[1-x(t+1)] which corresponds to the differential anticipatory equation: dx(t)/dt=a.x(t).[1-x(t+1)]-x(t). The main part of this paper deals with the discretisation of differential equation systems of linear and non-linear oscillators. The non-linear oscillator is based on the Lotka-Volterra equations model. The discretisation is made by incursion. The incursive discrete equation system gives the same stability condition than the original differential equations without numerical instabilities. The linearisation of the incursive discrete non-linear Lotka-Volterra equation system gives rise to the classical harmonic oscillator. The incursive discretisation of the linear oscillator is similar to define backward and forward discrete derivatives. A generalized complex derivative is then considered and applied to the harmonic oscillator. Non-locality seems to be a property of anticipatory systems. With some mathematical assumption, the Schrödinger quantum equation is derived for a particle in a uniform potential. Finally an hyperincursive system is given in the case of a neural stack memory.

Positive solutions to logistic type equations with harvesting

NASA Astrophysics Data System (ADS)

Girão, Pedro; Tehrani, Hossein

We use comparison principles, variational arguments and a truncation method to obtain positive solutions to logistic type equations with harvesting both in R and in a bounded domain Ω⊂R, with N⩾3, when the carrying capacity of the environment is not constant. By relaxing the growth assumption on the coefficients of the differential equation we derive a new equation which is easily solved. The solution of this new equation is then used to produce a positive solution of our original problem.

Derivation of exact master equation with stochastic description: dissipative harmonic oscillator.

PubMed

Li, Haifeng; Shao, Jiushu; Wang, Shikuan

2011-11-01

A systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description. For the Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and elementary derivation of the bath-induced stochastic field is presented. The dynamics of the system is thereby fully described by a stochastic differential equation, and the desired master equation would be acquired with statistical averaging. It is shown that the existence of a closed-form master equation depends on the specificity of the system as well as the feature of the dissipation characterized by the spectral density function. For a dissipative harmonic oscillator it is observed that the correlation between the stochastic field due to the bath and the system can be decoupled, and the master equation naturally results. Such an equation possesses the Lindblad form in which time-dependent coefficients are determined by a set of integral equations. It is proved that the obtained master equation is equivalent to the well-known Hu-Paz-Zhang equation based on the path-integral technique. The procedure is also used to obtain the master equation of a dissipative harmonic oscillator in time-dependent fields.

Synergies from using higher order symplectic decompositions both for ordinary differential equations and quantum Monte Carlo methods

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

Matuttis, Hans-Georg; Wang, Xiaoxing

Decomposition methods of the Suzuki-Trotter type of various orders have been derived in different fields. Applying them both to classical ordinary differential equations (ODEs) and quantum systems allows to judge their effectiveness and gives new insights for many body quantum mechanics where reference data are scarce. Further, based on data for 6 × 6 system we conclude that sampling with sign (minus-sign problem) is probably detrimental to the accuracy of fermionic simulations with determinant algorithms.

Solution of transonic flows by an integro-differential equation method

NASA Technical Reports Server (NTRS)

Ogana, W.

1978-01-01

Solutions of steady transonic flow past a two-dimensional airfoil are obtained from a singular integro-differential equation which involves a tangential derivative of the perturbation velocity potential. Subcritical flows are solved by taking central differences everywhere. For supercritical flows with shocks, central differences are taken in subsonic flow regions and backward differences in supersonic flow regions. The method is applied to a nonlifting parabolic-arc airfoil and to a lifting NACA 0012 airfoil. Results compare favorably with those of finite-difference schemes.

On Chaotic Behavior of Temperature Distribution in a Heat Exchanger

NASA Astrophysics Data System (ADS)

Bagyalakshmi, Morachan; Gangadharan, Saisundarakrishnan; Ganesh, Madhu

The objective of this paper is to introduce the notion of fractional derivatives in the energy equations and to study the chaotic nature of the temperature distribution in a heat exchanger with variation of temperature dependent transport properties. The governing fractional partial differential equations are transformed to a set of recurrence relations using fractional differential transform method and solved using inverse transform. The approximate analytical solution obtained by the proposed method has good agreement with the existing results.

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.

Extension of Gibbs-Duhem equation including influences of external fields

NASA Astrophysics Data System (ADS)

Guangze, Han; Jianjia, Meng

2018-03-01

Gibbs-Duhem equation is one of the fundamental equations in thermodynamics, which describes the relation among changes in temperature, pressure and chemical potential. Thermodynamic system can be affected by external field, and this effect should be revealed by thermodynamic equations. Based on energy postulate and the first law of thermodynamics, the differential equation of internal energy is extended to include the properties of external fields. Then, with homogeneous function theorem and a redefinition of Gibbs energy, a generalized Gibbs-Duhem equation with influences of external fields is derived. As a demonstration of the application of this generalized equation, the influences of temperature and external electric field on surface tension, surface adsorption controlled by external electric field, and the derivation of a generalized chemical potential expression are discussed, which show that the extended Gibbs-Duhem equation developed in this paper is capable to capture the influences of external fields on a thermodynamic system.

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.

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

Thompson, S.

This report describes the use of several subroutines from the CORLIB core mathematical subroutine library for the solution of a model fluid flow problem. The model consists of the Euler partial differential equations. The equations are spatially discretized using the method of pseudo-characteristics. The resulting system of ordinary differential equations is then integrated using the method of lines. The stiff ordinary differential equation solver LSODE (2) from CORLIB is used to perform the time integration. The non-stiff solver ODE (4) is used to perform a related integration. The linear equation solver subroutines DECOMP and SOLVE are used to solve linearmore » systems whose solutions are required in the calculation of the time derivatives. The monotone cubic spline interpolation subroutines PCHIM and PCHFE are used to approximate water properties. The report describes the use of each of these subroutines in detail. It illustrates the manner in which modules from a standard mathematical software library such as CORLIB can be used as building blocks in the solution of complex problems of practical interest. 9 refs., 2 figs., 4 tabs.« less

Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem

NASA Astrophysics Data System (ADS)

Li, Lei; Liu, Jian-Guo; Lu, Jianfeng

2017-10-01

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.

Algebraic features of some generalizations of the Lotka-Volterra system

NASA Astrophysics Data System (ADS)

Bibik, Yu. V.; Sarancha, D. A.

2010-10-01

For generalizations of the Lotka-Volterra system, an integration method is proposed based on the nontrivial algebraic structure of these generalizations. The method makes use of an auxiliary first-order differential equation derived from the phase curve equation with the help of this algebraic structure. Based on this equation, a Hamiltonian approach can be developed and canonical variables (moreover, action-angle variables) can be constructed.

Modeling extracellular electrical stimulation: I. Derivation and interpretation of neurite equations.

PubMed

Meffin, Hamish; Tahayori, Bahman; Grayden, David B; Burkitt, Anthony N

2012-12-01

Neuroprosthetic devices, such as cochlear and retinal implants, work by directly stimulating neurons with extracellular electrodes. This is commonly modeled using the cable equation with an applied extracellular voltage. In this paper a framework for modeling extracellular electrical stimulation is presented. To this end, a cylindrical neurite with confined extracellular space in the subthreshold regime is modeled in three-dimensional space. Through cylindrical harmonic expansion of Laplace's equation, we derive the spatio-temporal equations governing different modes of stimulation, referred to as longitudinal and transverse modes, under types of boundary conditions. The longitudinal mode is described by the well-known cable equation, however, the transverse modes are described by a novel ordinary differential equation. For the longitudinal mode, we find that different electrotonic length constants apply under the two different boundary conditions. Equations connecting current density to voltage boundary conditions are derived that are used to calculate the trans-impedance of the neurite-plus-thin-extracellular-sheath. A detailed explanation on depolarization mechanisms and the dominant current pathway under different modes of stimulation is provided. The analytic results derived here enable the estimation of a neurite's membrane potential under extracellular stimulation, hence bypassing the heavy computational cost of using numerical methods.

On the origins of generalized fractional calculus

NASA Astrophysics Data System (ADS)

Kiryakova, Virginia

2015-11-01

In Fractional Calculus (FC), as in the (classical) Calculus, the notions of derivatives and integrals (of first, second, etc. or arbitrary, incl. non-integer order) are basic and co-related. One of the most frequent approach in FC is to define first the Riemann-Liouville (R-L) integral of fractional order, and then by means of suitable integer-order differentiation operation applied over it (or under its sign) a fractional derivative is defined - in the R-L sense (or in Caputo sense). The first mentioned (R-L type) is closer to the theoretical studies in analysis, but has some shortages - from the point of view of interpretation of the initial conditions for Cauchy problems for fractional differential equations (stated also by means of fractional order derivatives/ integrals), and also for the analysts' confusion that such a derivative of a constant is not zero in general. The Caputo (C-) derivative, arising first in geophysical studies, helps to overcome these problems and to describe models of applied problems with physically consistent initial conditions. The operators of the Generalized Fractional Calculus - GFC (integrals and derivatives) are based on commuting m-tuple (m = 1, 2, 3, …) compositions of operators of the classical FC with power weights (the so-called Erdélyi-Kober operators), but represented in compact and explicit form by means of integral, integro-differential (R-L type) or differential-integral (C-type) operators, where the kernels are special functions of most general hypergeometric kind. The foundations of this theory are given in Kiryakova 18. In this survey we present the genesis of the definitions of the GFC - the generalized fractional integrals and derivatives (of fractional multi-order) of R-L type and Caputo type, analyze their properties and applications. Their special cases are all the known operators of classical FC, their generalizations introduced by other authors, the hyper-Bessel differential operators of higher integer order m as a multi-order (1, 1,…, 1), the Gelfond-Leontiev generalized differentiation operators, many other integral and differential operators in Calculus that have been used in various topics, some of them not related to FC at all, others involved in differential and integral equations for treating fractional order models.

Partner symmetries and non-invariant solutions of four-dimensional heavenly equations

NASA Astrophysics Data System (ADS)

Malykh, A. A.; Nutku, Y.; Sheftel, M. B.

2004-07-01

We extend our method of partner symmetries to the hyperbolic complex Monge-Ampère equation and the second heavenly equation of Plebañski. We show the existence of partner symmetries and derive the relations between them. For certain simple choices of partner symmetries the resulting differential constraints together with the original heavenly equations are transformed to systems of linear equations by an appropriate Legendre transformation. The solutions of these linear equations are generically non-invariant. As a consequence we obtain explicitly new classes of heavenly metrics without Killing vectors.

Symmetry analysis of the high-order equations for the description of the Fermi - Pasta - Ulam problem

NASA Astrophysics Data System (ADS)

Kudryashov, N. A.; Volkov, A. K.

2017-01-01

Recently some new nonlinear equations for the description of the Fermi - Pasta - Ulam problem have been derived. The main aim of this work is to use the symmetry test to investigate these equations. We consider equations for the description of the α and α + β Fermi - Pasta - Ulam model. We find the infinitesimal operators and Lie groups, admitted by the equations. Using the groups we find the self-similar variables as well as the reductions to the ordinary differential equations. Some exact solutions are also constructed.

Algorithm for Stabilizing a POD-Based Dynamical System

NASA Technical Reports Server (NTRS)

Kalb, Virginia L.

2010-01-01

This algorithm provides a new way to improve the accuracy and asymptotic behavior of a low-dimensional system based on the proper orthogonal decomposition (POD). Given a data set representing the evolution of a system of partial differential equations (PDEs), such as the Navier-Stokes equations for incompressible flow, one may obtain a low-dimensional model in the form of ordinary differential equations (ODEs) that should model the dynamics of the flow. Temporal sampling of the direct numerical simulation of the PDEs produces a spatial time series. The POD extracts the temporal and spatial eigenfunctions of this data set. Truncated to retain only the most energetic modes followed by Galerkin projection of these modes onto the PDEs obtains a dynamical system of ordinary differential equations for the time-dependent behavior of the flow. In practice, the steps leading to this system of ODEs entail numerically computing first-order derivatives of the mean data field and the eigenfunctions, and the computation of many inner products. This is far from a perfect process, and often results in the lack of long-term stability of the system and incorrect asymptotic behavior of the model. This algorithm describes a new stabilization method that utilizes the temporal eigenfunctions to derive correction terms for the coefficients of the dynamical system to significantly reduce these errors.

ADAPTIVE METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS VIA NATURAL EMBEDDINGS AND REJECTION SAMPLING WITH MEMORY.

PubMed

Rackauckas, Christopher; Nie, Qing

2017-01-01

Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.

ADAPTIVE METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS VIA NATURAL EMBEDDINGS AND REJECTION SAMPLING WITH MEMORY

PubMed Central

Rackauckas, Christopher

2017-01-01

Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs. PMID:29527134

Sensitivity analysis of dynamic biological systems with time-delays.

PubMed

Wu, Wu Hsiung; Wang, Feng Sheng; Chang, Maw Shang

2010-10-15

Mathematical modeling has been applied to the study and analysis of complex biological systems for a long time. Some processes in biological systems, such as the gene expression and feedback control in signal transduction networks, involve a time delay. These systems are represented as delay differential equation (DDE) models. Numerical sensitivity analysis of a DDE model by the direct method requires the solutions of model and sensitivity equations with time-delays. The major effort is the computation of Jacobian matrix when computing the solution of sensitivity equations. The computation of partial derivatives of complex equations either by the analytic method or by symbolic manipulation is time consuming, inconvenient, and prone to introduce human errors. To address this problem, an automatic approach to obtain the derivatives of complex functions efficiently and accurately is necessary. We have proposed an efficient algorithm with an adaptive step size control to compute the solution and dynamic sensitivities of biological systems described by ordinal differential equations (ODEs). The adaptive direct-decoupled algorithm is extended to solve the solution and dynamic sensitivities of time-delay systems describing by DDEs. To save the human effort and avoid the human errors in the computation of partial derivatives, an automatic differentiation technique is embedded in the extended algorithm to evaluate the Jacobian matrix. The extended algorithm is implemented and applied to two realistic models with time-delays: the cardiovascular control system and the TNF-α signal transduction network. The results show that the extended algorithm is a good tool for dynamic sensitivity analysis on DDE models with less user intervention. By comparing with direct-coupled methods in theory, the extended algorithm is efficient, accurate, and easy to use for end users without programming background to do dynamic sensitivity analysis on complex biological systems with time-delays.

Numerical analysis for the fractional diffusion and fractional Buckmaster equation by the two-step Laplace Adam-Bashforth method

NASA Astrophysics Data System (ADS)

Jain, Sonal

2018-01-01

In this paper, we aim to use the alternative numerical scheme given by Gnitchogna and Atangana for solving partial differential equations with integer and non-integer differential operators. We applied this method to fractional diffusion model and fractional Buckmaster models with non-local fading memory. The method yields a powerful numerical algorithm for fractional order derivative to implement. Also we present in detail the stability analysis of the numerical method for solving the diffusion equation. This proof shows that this method is very stable and also converges very quickly to exact solution and finally some numerical simulation is presented.

Stability and bifurcation analysis of a generalized scalar delay differential equation.

PubMed

Bhalekar, Sachin

2016-08-01

This paper deals with the stability and bifurcation analysis of a general form of equation D(α)x(t)=g(x(t),x(t-τ)) involving the derivative of order α ∈ (0, 1] and a constant delay τ ≥ 0. The stability of equilibrium points is presented in terms of the stability regions and critical surfaces. We provide a necessary condition to exist chaos in the system also. A wide range of delay differential equations involving a constant delay can be analyzed using the results proposed in this paper. The illustrative examples are provided to explain the theory.

High-order FDTD methods for transverse electromagnetic systems in dispersive inhomogeneous media.

PubMed

Zhao, Shan

2011-08-15

This Letter introduces a novel finite-difference time-domain (FDTD) formulation for solving transverse electromagnetic systems in dispersive media. Based on the auxiliary differential equation approach, the Debye dispersion model is coupled with Maxwell's equations to derive a supplementary ordinary differential equation for describing the regularity changes in electromagnetic fields at the dispersive interface. The resulting time-dependent jump conditions are rigorously enforced in the FDTD discretization by means of the matched interface and boundary scheme. High-order convergences are numerically achieved for the first time in the literature in the FDTD simulations of dispersive inhomogeneous media. © 2011 Optical Society of America

The fifth-order partial differential equation for the description of the α + β Fermi-Pasta-Ulam model

NASA Astrophysics Data System (ADS)

Kudryashov, Nikolay A.; Volkov, Alexandr K.

2017-01-01

We study a new nonlinear partial differential equation of the fifth order for the description of perturbations in the Fermi-Pasta-Ulam mass chain. This fifth-order equation is an expansion of the Gardner equation for the description of the Fermi-Pasta-Ulam model. We use the potential of interaction between neighbouring masses with both quadratic and cubic terms. The equation is derived using the continuous limit. Unlike the previous works, we take into account higher order terms in the Taylor series expansions. We investigate the equation using the Painlevé approach. We show that the equation does not pass the Painlevé test and can not be integrated by the inverse scattering transform. We use the logistic function method and the Laurent expansion method to find travelling wave solutions of the fifth-order equation. We use the pseudospectral method for the numerical simulation of wave processes, described by the equation.

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.

An ansatz for solving nonlinear partial differential equations in mathematical physics.

PubMed

Akbar, M Ali; Ali, Norhashidah Hj Mohd

2016-01-01

In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.

Theoretical Investigation of Thermo-Mechanical Behavior of Carbon Nanotube-Based Composites Using the Integral Transform Method

NASA Technical Reports Server (NTRS)

Pawloski, Janice S.

2001-01-01

This project uses the integral transform technique to model the problem of nanotube behavior as an axially symmetric system of shells. Assuming that the nanotube behavior can be described by the equations of elasticity, we seek a stress function x which satisfies the biharmonic equation: del(exp 4) chi = [partial deriv(r(exp 2)) + partial deriv(r) + partial deriv(z(exp 2))] chi = 0. The method of integral transformations is used to transform the differential equation. The symmetry with respect to the z-axis indicates that we only need to consider the sine transform of the stress function: X(bar)(r,zeta) = integral(from 0 to infinity) chi(r,z)sin(zeta,z) dz.

Self-gravitating static non-critical black holes in 4 D Einstein-Klein-Gordon system with nonminimal derivative coupling

NASA Astrophysics Data System (ADS)

Gunara, Bobby Eka; Yaqin, Ainol

2018-06-01

We study static non-critical hairy black holes of four dimensional gravitational model with nonminimal derivative coupling and a scalar potential turned on. By taking an ansatz, namely, the first derivative of the scalar field is proportional to square root of a metric function, we reduce the Einstein field equation and the scalar field equation of motions into a single highly nonlinear differential equation. This setup implies that the hair is secondary-like since the scalar charge-like depends on the non-constant mass-like quantity in the asymptotic limit. Then, we show that near boundaries the solution is not the critical point of the scalar potential and the effective geometries become spaces of constant scalar curvature.

Global collocation methods for approximation and the solution of partial differential equations

NASA Technical Reports Server (NTRS)

Solomonoff, A.; Turkel, E.

1986-01-01

Polynomial interpolation methods are applied both to the approximation of functions and to the numerical solutions of hyperbolic and elliptic partial differential equations. The derivative matrix for a general sequence of the collocation points is constructed. The approximate derivative is then found by a matrix times vector multiply. The effects of several factors on the performance of these methods including the effect of different collocation points are then explored. The resolution of the schemes for both smooth functions and functions with steep gradients or discontinuities in some derivative are also studied. The accuracy when the gradients occur both near the center of the region and in the vicinity of the boundary is investigated. The importance of the aliasing limit on the resolution of the approximation is investigated in detail. Also examined is the effect of boundary treatment on the stability and accuracy of the scheme.

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.

Einstein-Cartan Theory of Gravitation: Kinematical Parameters and Maxwell Equations

NASA Astrophysics Data System (ADS)

Katkar, L. N.

2015-03-01

In the space-time manifold of Einstein-Cartan Theory (ECT) of gravitation, the expressions for the time-like kinematical parameters are derived and the propagation equation for expansion is obtained.It has been observed that when the spin tensor is u-orthogonal the spin of the gravitating matter has no influence on the propagation equation of expansion while it has influence when it is not u-orthogonal. The usual formula for the curl of gradient of a scalar function is not zero in ECT. So is the case with the divergence of the curl of a vector.Their expressions on the space-time manifold of ECT are derived. A new derivative operator d ∗ is introduced to develop the calculus on space-time manifold of ECT. It is obtained by taking the covariant derivative of an associated tensor of a form with respect to an asymmetric connections. We have used this differential operator to obtain the form of the Maxwell's equations in the ECT of gravitation. Cartan's equations of structure are also derived through the new derivative operator. It has been shown that unlike the consequences of exterior derivative in Einstein space-time, the repetition of d ∗ on a form of any degree is not zero.

Canonical coordinates for partial differential equations

NASA Technical Reports Server (NTRS)

Hunt, L. R.; Villarreal, Ramiro

1988-01-01

Necessary and sufficient conditions are found under which operators of the form Sigma (m, j=1) x (2) sub j + X sub O can be made constant coefficient. In addition, necessary and sufficient conditions are derived which classify those linear partial differential operators that can be moved to the Kolmogorov type.

Canonical coordinates for partial differential equations

NASA Technical Reports Server (NTRS)

Hunt, L. R.; Villarreal, Ramiro

1987-01-01

Necessary and sufficient conditions are found under which operators of the form Sigma(m, j=1) X(2)sub j + X sub 0 can be made constant coefficient. In addition, necessary and sufficient conditions are derived which classify those linear partial differential operators that can be moved to the Kolmogorov type.

Exact analytical solutions of continuity equation for electron beams precipitating in Coulomb collisions

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

Dobranskis, R. R.; Zharkova, V. V., E-mail: valentina.zharkova@northumbria.ac.uk

2014-06-10

The original continuity equation (CE) used for the interpretation of the power law energy spectra of beam electrons in flares was written and solved for an electron beam flux while ignoring an additional free term with an electron density. In order to remedy this omission, the original CE for electron flux, considering beam's energy losses in Coulomb collisions, was first differentiated by the two independent variables: depth and energy leading to partial differential equation for an electron beam density instead of flux with the additional free term. The analytical solution of this partial differential continuity equation (PDCE) is obtained bymore » using the method of characteristics. This solution is further used to derive analytical expressions for mean electron spectra for Coulomb collisions and to carry out numeric calculations of hard X-ray (HXR) photon spectra for beams with different parameters. The solutions revealed a significant departure of electron densities at lower energies from the original results derived from the CE for the flux obtained for Coulomb collisions. This departure is caused by the additional exponential term that appeared in the updated solutions for electron differential density leading to its faster decrease at lower energies (below 100 keV) with every precipitation depth similar to the results obtained with numerical Fokker-Planck solutions. The effects of these updated solutions for electron densities on mean electron spectra and HXR photon spectra are also discussed.« less

Universal and integrable nonlinear evolution systems of equations in 2+1 dimensions

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

Maccari, A.

1997-08-01

Integrable systems of nonlinear partial differential equations (PDEs) are obtained from integrable equations in 2+1 dimensions, by means of a reduction method of broad applicability based on Fourier expansion and spatio{endash}temporal rescalings, which is asymptotically exact in the limit of weak nonlinearity. The integrability by the spectral transform is explicitly demonstrated, because the corresponding Lax pairs have been derived, applying the same reduction method to the Lax pair of the initial equation. These systems of nonlinear PDEs are likely to be of applicative relevance and have a {open_quotes}universal{close_quotes} character, inasmuch as they may be derived from a very large classmore » of nonlinear evolution equations with a linear dispersive part. {copyright} {ital 1997 American Institute of Physics.}« less

Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks

NASA Astrophysics Data System (ADS)

Shirvany, Yazdan; Hayati, Mohsen; Moradian, Rostam

2008-12-01

We present a method to solve boundary value problems using artificial neural networks (ANN). A trial solution of the differential equation is written as a feed-forward neural network containing adjustable parameters (the weights and biases). From the differential equation and its boundary conditions we prepare the energy function which is used in the back-propagation method with momentum term to update the network parameters. We improved energy function of ANN which is derived from Schrodinger equation and the boundary conditions. With this improvement of energy function we can use unsupervised training method in the ANN for solving the equation. Unsupervised training aims to minimize a non-negative energy function. We used the ANN method to solve Schrodinger equation for few quantum systems. Eigenfunctions and energy eigenvalues are calculated. Our numerical results are in agreement with their corresponding analytical solution and show the efficiency of ANN method for solving eigenvalue problems.

Blending and nudging in fluid dynamics: some simple observations

NASA Astrophysics Data System (ADS)

Germano, M.

2017-10-01

Blending and nudging methods have been recently applied in fluid dynamics, particularly regarding the assimilation of experimental data into the computations. In the paper we formally derive the differential equation associated to blending and compare it to the standard nudging equation. Some simple considerations related to these techniques and their mutual relations are exposed.

A Multiscale Model for Virus Capsid Dynamics

PubMed Central

Chen, Changjun; Saxena, Rishu; Wei, Guo-Wei

2010-01-01

Viruses are infectious agents that can cause epidemics and pandemics. The understanding of virus formation, evolution, stability, and interaction with host cells is of great importance to the scientific community and public health. Typically, a virus complex in association with its aquatic environment poses a fabulous challenge to theoretical description and prediction. In this work, we propose a differential geometry-based multiscale paradigm to model complex biomolecule systems. In our approach, the differential geometry theory of surfaces and geometric measure theory are employed as a natural means to couple the macroscopic continuum domain of the fluid mechanical description of the aquatic environment from the microscopic discrete domain of the atomistic description of the biomolecule. A multiscale action functional is constructed as a unified framework to derive the governing equations for the dynamics of different scales. We show that the classical Navier-Stokes equation for the fluid dynamics and Newton's equation for the molecular dynamics can be derived from the least action principle. These equations are coupled through the continuum-discrete interface whose dynamics is governed by potential driven geometric flows. PMID:20224756

Numerical Asymptotic Solutions Of Differential Equations

NASA Technical Reports Server (NTRS)

Thurston, Gaylen A.

1992-01-01

Numerical algorithms derived and compared with classical analytical methods. In method, expansions replaced with integrals evaluated numerically. Resulting numerical solutions retain linear independence, main advantage of asymptotic solutions.

Modelling the radiotherapy effect in the reaction-diffusion equation.

PubMed

Borasi, Giovanni; Nahum, Alan

2016-09-01

In recent years, the reaction-diffusion (Fisher-Kolmogorov) equation has received much attention from the oncology research community due to its ability to describe the infiltrating nature of glioblastoma multiforme and its extraordinary resistance to any type of therapy. However, in a number of previous papers in the literature on applications of this equation, the term (R) expressing the 'External Radiotherapy effect' was incorrectly derived. In this note we derive an analytical expression for this term in the correct form to be included in the reaction-diffusion equation. The R term has been derived starting from the Linear-Quadratic theory of cell killing by ionizing radiation. The correct definition of R was adopted and the basic principles of differential calculus applied. The compatibility of the R term derived here with the reaction-diffusion equation was demonstrated. Referring to a typical glioblastoma tumour, we have compared the results obtained using our expression for the R term with the 'incorrect' expression proposed by other authors. Copyright © 2016 Associazione Italiana di Fisica Medica. Published by Elsevier Ltd. All rights reserved.

Dynamic crack propagation in a 2D elastic body: The out-of-plane case

NASA Astrophysics Data System (ADS)

Nicaise, Serge; Sandig, Anna-Margarete

2007-05-01

Already in 1920 Griffith has formulated an energy balance criterion for quasistatic crack propagation in brittle elastic materials. Nowadays, a generalized energy balance law is used in mechanics [F. Erdogan, Crack propagation theories, in: H. Liebowitz (Ed.), Fracture, vol. 2, Academic Press, New York, 1968, pp. 498-586; L.B. Freund, Dynamic Fracture Mechanics, Cambridge Univ. Press, Cambridge, 1990; D. Gross, Bruchmechanik, Springer-Verlag, Berlin, 1996] in order to predict how a running crack will grow. We discuss this situation in a rigorous mathematical way for the out-of-plane state. This model is described by two coupled equations in the reference configuration: a two-dimensional scalar wave equation for the displacement fields in a cracked bounded domain and an ordinary differential equation for the crack position derived from the energy balance law. We handle both equations separately, assuming at first that the crack position is known. Then the weak and strong solvability of the wave equation will be studied and the crack tip singularities will be derived under the assumption that the crack is straight and moves tangentially. Using the energy balance law and the crack tip behavior of the displacement fields we finally arrive at an ordinary differential equation for the motion of the crack tip.

Ambipolarity in a tokamak with magnetic field ripple

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

Hazeltine, R. D.

In view of the recognized importance of electrostatic fields regarding turbulent transport, the radial electric field in a tokamak with magnetic field ripple is reconsidered. Terms in the ambipolarity condition involving the radial derivative of the field are derived from an extended drift-kinetic equation, including effects of second order in the gyroradius. Such terms are of interest in part because of their known importance in rotational relaxation equations for the axisymmetric case. The electric field is found to satisfy a nonlinear differential equation that is universal in a certain sense, and that implies spatial relaxation of the potential to itsmore » conventionally predicted value.« less

Discrete sensitivity derivatives of the Navier-Stokes equations with a parallel Krylov solver

NASA Technical Reports Server (NTRS)

Ajmani, Kumud; Taylor, Arthur C., III

1994-01-01

This paper solves an 'incremental' form of the sensitivity equations derived by differentiating the discretized thin-layer Navier Stokes equations with respect to certain design variables of interest. The equations are solved with a parallel, preconditioned Generalized Minimal RESidual (GMRES) solver on a distributed-memory architecture. The 'serial' sensitivity analysis code is parallelized by using the Single Program Multiple Data (SPMD) programming model, domain decomposition techniques, and message-passing tools. Sensitivity derivatives are computed for low and high Reynolds number flows over a NACA 1406 airfoil on a 32-processor Intel Hypercube, and found to be identical to those computed on a single-processor Cray Y-MP. It is estimated that the parallel sensitivity analysis code has to be run on 40-50 processors of the Intel Hypercube in order to match the single-processor processing time of a Cray Y-MP.

Group iterative methods for the solution of two-dimensional time-fractional diffusion equation

NASA Astrophysics Data System (ADS)

Balasim, Alla Tareq; Ali, Norhashidah Hj. Mohd.

2016-06-01

Variety of problems in science and engineering may be described by fractional partial differential equations (FPDE) in relation to space and/or time fractional derivatives. The difference between time fractional diffusion equations and standard diffusion equations lies primarily in the time derivative. Over the last few years, iterative schemes derived from the rotated finite difference approximation have been proven to work well in solving standard diffusion equations. However, its application on time fractional diffusion counterpart is still yet to be investigated. In this paper, we will present a preliminary study on the formulation and analysis of new explicit group iterative methods in solving a two-dimensional time fractional diffusion equation. These methods were derived from the standard and rotated Crank-Nicolson difference approximation formula. Several numerical experiments were conducted to show the efficiency of the developed schemes in terms of CPU time and iteration number. At the request of all authors of the paper an updated version of this article was published on 7 July 2016. The original version supplied to AIP Publishing contained an error in Table 1 and References 15 and 16 were incomplete. These errors have been corrected in the updated and republished article.

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)

Arbitrarily Curved and Twisted Space Beams. Ph.D. Thesis - Va. Polytech. Inst. and State Univ.; [Elastic Deformation, Stress Analysis

NASA Technical Reports Server (NTRS)

Hunter, W. F.

1974-01-01

A derivation of the equations which govern the deformation of an arbitrarily curved and twisted space beam is presented. These equations differ from those of the classical theory in that (1) extensional effects are included; (2) the strain-displacement relations are derived; and (3) the expressions for the stress resultants are developed from the strain displacement relations. It is shown that the torsional stress resultant obtained by the classical approach is basically incorrect except when the cross-section is circular. The governing equations are given in the form of first-order differential equations. A numerical algorithm is given for obtaining the natural vibration characteristics and example problems are presented.

Modeling of Inverted Annular Film Boiling using an integral method

NASA Astrophysics Data System (ADS)

Sridharan, Arunkumar

In modeling Inverted Annular Film Boiling (IAFB), several important phenomena such as interaction between the liquid and the vapor phases and irregular nature of the interface, which greatly influence the momentum and heat transfer at the interface, need to be accounted for. However, due to the complexity of these phenomena, they were not modeled in previous studies. Since two-phase heat transfer equations and relationships rely heavily on experimental data, many closure relationships that were used in previous studies to solve the problem are empirical in nature. Also, in deriving the relationships, the experimental data were often extrapolated beyond the intended range of conditions, causing errors in predictions. In some cases, empirical correlations that were derived from situations other than IAFB, and whose applicability to IAFB was questionable, were used. Moreover, arbitrary constants were introduced in the model developed in previous studies to provide good fit to the experimental data. These constants have no physical basis, thereby leading to questionable accuracy in the model predictions. In the present work, modeling of Inverted Annular Film Boiling (IAFB) is done using Integral Method. Two-dimensional formulation of IAFB is presented. Separate equations for the conservation of mass, momentum and energy are derived from first principles, for the vapor film and the liquid core. Turbulence is incorporated in the formulation. The system of second-order partial differential equations is integrated over the radial direction to obtain a system of integral differential equations. In order to solve the system of equations, second order polynomial profiles are used to describe the nondimensional velocity and temperatures. The unknown coefficients in the profiles are functions of the axial direction alone. Using the boundary conditions that govern the physical problem, equations for the unknown coefficients are derived in terms of the primary dependent variables: wall shear stress, interfacial shear stress, film thickness, pressure, wall temperature and the mass transfer rate due to evaporation. A system of non-linear first order coupled ordinary differential equations is obtained. Due to the inherent mathematical complexity of the system of equations, simplifying assumptions are made to obtain a numerical solution. The system of equations is solved numerically to obtain values of the unknown quantities at each subsequent axial location. Derived quantities like void fraction and heat transfer coefficient are calculated at each axial location. The calculation is terminated when the void fraction reaches a value of 0.6, the upper limit of IAFB. The results obtained agree with the experimental trends observed. Void fraction increases along the heated length, while the heat transfer coefficient drops due to the increased resistance of the vapor film as expected.

Analytical properties of a three-compartmental dynamical demographic model

NASA Astrophysics Data System (ADS)

Postnikov, E. B.

2015-07-01

The three-compartmental demographic model by Korotaeyv-Malkov-Khaltourina, connecting population size, economic surplus, and education level, is considered from the point of view of dynamical systems theory. It is shown that there exist two integrals of motion, which enables the system to be reduced to one nonlinear ordinary differential equation. The study of its structure provides analytical criteria for the dominance ranges of the dynamics of Malthus and Kremer. Additionally, the particular ranges of parameters enable the derived general ordinary differential equations to be reduced to the models of Gompertz and Thoularis-Wallace.

Analytical Energy Gradients for Excited-State Coupled-Cluster Methods

NASA Astrophysics Data System (ADS)

Wladyslawski, Mark; Nooijen, Marcel

The equation-of-motion coupled-cluster (EOM-CC) and similarity transformed equation-of-motion coupled-cluster (STEOM-CC) methods have been firmly established as accurate and routinely applicable extensions of single-reference coupled-cluster theory to describe electronically excited states. An overview of these methods is provided, with emphasis on the many-body similarity transform concept that is the key to a rationalization of their accuracy. The main topic of the paper is the derivation of analytical energy gradients for such non-variational electronic structure approaches, with an ultimate focus on obtaining their detailed algebraic working equations. A general theoretical framework using Lagrange's method of undetermined multipliers is presented, and the method is applied to formulate the EOM-CC and STEOM-CC gradients in abstract operator terms, following the previous work in [P.G. Szalay, Int. J. Quantum Chem. 55 (1995) 151] and [S.R. Gwaltney, R.J. Bartlett, M. Nooijen, J. Chem. Phys. 111 (1999) 58]. Moreover, the systematics of the Lagrange multiplier approach is suitable for automation by computer, enabling the derivation of the detailed derivative equations through a standardized and direct procedure. To this end, we have developed the SMART (Symbolic Manipulation and Regrouping of Tensors) package of automated symbolic algebra routines, written in the Mathematica programming language. The SMART toolkit provides the means to expand, differentiate, and simplify equations by manipulation of the detailed algebraic tensor expressions directly. The Lagrangian multiplier formulation establishes a uniform strategy to perform the automated derivation in a standardized manner: A Lagrange multiplier functional is constructed from the explicit algebraic equations that define the energy in the electronic method; the energy functional is then made fully variational with respect to all of its parameters, and the symbolic differentiations directly yield the explicit equations for the wavefunction amplitudes, the Lagrange multipliers, and the analytical gradient via the perturbation-independent generalized Hellmann-Feynman effective density matrix. This systematic automated derivation procedure is applied to obtain the detailed gradient equations for the excitation energy (EE-), double ionization potential (DIP-), and double electron affinity (DEA-) similarity transformed equation-of-motion coupled-cluster singles-and-doubles (STEOM-CCSD) methods. In addition, the derivatives of the closed-shell-reference excitation energy (EE-), ionization potential (IP-), and electron affinity (EA-) equation-of-motion coupled-cluster singles-and-doubles (EOM-CCSD) methods are derived. Furthermore, the perturbative EOM-PT and STEOM-PT gradients are obtained. The algebraic derivative expressions for these dozen methods are all derived here uniformly through the automated Lagrange multiplier process and are expressed compactly in a chain-rule/intermediate-density formulation, which facilitates a unified modular implementation of analytic energy gradients for CCSD/PT-based electronic methods. The working equations for these analytical gradients are presented in full detail, and their factorization and implementation into an efficient computer code are discussed.

Whitham modulation theory for the two-dimensional Benjamin-Ono equation.

PubMed

Ablowitz, Mark; Biondini, Gino; Wang, Qiao

2017-09-01

Whitham modulation theory for the two-dimensional Benjamin-Ono (2DBO) equation is presented. A system of five quasilinear first-order partial differential equations is derived. The system describes modulations of the traveling wave solutions of the 2DBO equation. These equations are transformed to a singularity-free hydrodynamic-like system referred to here as the 2DBO-Whitham system. Exact reductions of this system are discussed, the formulation of initial value problems is considered, and the system is used to study the transverse stability of traveling wave solutions of the 2DBO equation.

A deterministic particle method for one-dimensional reaction-diffusion equations

NASA Technical Reports Server (NTRS)

Mascagni, Michael

1995-01-01

We derive a deterministic particle method for the solution of nonlinear reaction-diffusion equations in one spatial dimension. This deterministic method is an analog of a Monte Carlo method for the solution of these problems that has been previously investigated by the author. The deterministic method leads to the consideration of a system of ordinary differential equations for the positions of suitably defined particles. We then consider the time explicit and implicit methods for this system of ordinary differential equations and we study a Picard and Newton iteration for the solution of the implicit system. Next we solve numerically this system and study the discretization error both analytically and numerically. Numerical computation shows that this deterministic method is automatically adaptive to large gradients in the solution.

Scaling and scale invariance of conservation laws in Reynolds transport theorem framework

NASA Astrophysics Data System (ADS)

Haltas, Ismail; Ulusoy, Suleyman

2015-07-01

Scale invariance is the case where the solution of a physical process at a specified time-space scale can be linearly related to the solution of the processes at another time-space scale. Recent studies investigated the scale invariance conditions of hydrodynamic processes by applying the one-parameter Lie scaling transformations to the governing equations of the processes. Scale invariance of a physical process is usually achieved under certain conditions on the scaling ratios of the variables and parameters involved in the process. The foundational axioms of hydrodynamics are the conservation laws, namely, conservation of mass, conservation of linear momentum, and conservation of energy from continuum mechanics. They are formulated using the Reynolds transport theorem. Conventionally, Reynolds transport theorem formulates the conservation equations in integral form. Yet, differential form of the conservation equations can also be derived for an infinitesimal control volume. In the formulation of the governing equation of a process, one or more than one of the conservation laws and, some times, a constitutive relation are combined together. Differential forms of the conservation equations are used in the governing partial differential equation of the processes. Therefore, differential conservation equations constitute the fundamentals of the governing equations of the hydrodynamic processes. Applying the one-parameter Lie scaling transformation to the conservation laws in the Reynolds transport theorem framework instead of applying to the governing partial differential equations may lead to more fundamental conclusions on the scaling and scale invariance of the hydrodynamic processes. This study will investigate the scaling behavior and scale invariance conditions of the hydrodynamic processes by applying the one-parameter Lie scaling transformation to the conservation laws in the Reynolds transport theorem framework.

A Tutorial Review on Fractal Spacetime and Fractional Calculus

NASA Astrophysics Data System (ADS)

He, Ji-Huan

2014-11-01

This tutorial review of fractal-Cantorian spacetime and fractional calculus begins with Leibniz's notation for derivative without limits which can be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. Fractal spacetime is used to elucidate some basic properties of fractal which is the foundation of fractional calculus, and El Naschie's mass-energy equation for the dark energy. The variational iteration method is used to introduce the definition of fractional derivatives. Fractal derivative is explained geometrically and q-derivative is motivated by quantum mechanics. Some effective analytical approaches to fractional differential equations, e.g., the variational iteration method, the homotopy perturbation method, the exp-function method, the fractional complex transform, and Yang-Laplace transform, are outlined and the main solution processes are given.

Fractal ladder models and power law wave equations

PubMed Central

Kelly, James F.; McGough, Robert J.

2009-01-01

The ultrasonic attenuation coefficient in mammalian tissue is approximated by a frequency-dependent power law for frequencies less than 100 MHz. To describe this power law behavior in soft tissue, a hierarchical fractal network model is proposed. The viscoelastic and self-similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite-ladder topology involving alternating springs and dashpots. In the low-frequency limit, this ladder network yields a stress-strain constitutive equation with a time-fractional derivative. By combining this constitutive equation with linearized conservation principles and an adiabatic equation of state, a fractional partial differential equation that describes power law attenuation is derived. The resulting attenuation coefficient is a power law with exponent ranging between 1 and 2, while the phase velocity is in agreement with the Kramers–Kronig relations. The fractal ladder model is compared to published attenuation coefficient data, thus providing equivalent lumped parameters. PMID:19813816

Bifurcation Analysis and Nonlinear Decay of a Tumor in the Presence of an Immune Response

NASA Astrophysics Data System (ADS)

López, Álvaro G.; Seoane, Jesús M.; Sanjuán, Miguel A. F.

2017-12-01

The decay of a planar compact surface that is reduced through its boundary is considered. The interest of this problem lies in the fact that it can represent the destruction of a solid tumor by a population of immune cells. The theory of curves is utilized to derive the rate at which the area of the set decreases. Firstly, the process is represented as a discrete dynamical system. A recurrence equation describing the shrinkage of the area at any step is deduced. Then, a continuum limit is attained to derive an ordinary differential equation that governs the decay of the set. The solutions to the differential equation and its implications are discussed, and numerical simulations are carried out to test the accuracy of the decay law. Finally, the dynamics of a tumor-immune aggregate is inspected using this law and the theory of bifurcations. As the ratio of immune destruction to tumor growth increases, a saddle-node bifurcation stabilizes the tumor-free fixed point.

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.

The large discretization step method for time-dependent partial differential equations

NASA Technical Reports Server (NTRS)

Haras, Zigo; Taasan, Shlomo

1995-01-01

A new method for the acceleration of linear and nonlinear time dependent calculations is presented. It is based on the Large Discretization Step (LDS) approximation, defined in this work, which employs an extended system of low accuracy schemes to approximate a high accuracy discrete approximation to a time dependent differential operator. Error bounds on such approximations are derived. These approximations are efficiently implemented in the LDS methods for linear and nonlinear hyperbolic equations, presented here. In these algorithms the high and low accuracy schemes are interpreted as the same discretization of a time dependent operator on fine and coarse grids, respectively. Thus, a system of correction terms and corresponding equations are derived and solved on the coarse grid to yield the fine grid accuracy. These terms are initialized by visiting the fine grid once in many coarse grid time steps. The resulting methods are very general, simple to implement and may be used to accelerate many existing time marching schemes.

Quadratically Convergent Method for Simultaneously Approaching the Roots of Polynomial Solutions of a Class of Differential Equations

NASA Astrophysics Data System (ADS)

Recchioni, Maria Cristina

2001-12-01

This paper investigates the application of the method introduced by L. Pasquini (1989) for simultaneously approaching the zeros of polynomial solutions to a class of second-order linear homogeneous ordinary differential equations with polynomial coefficients to a particular case in which these polynomial solutions have zeros symmetrically arranged with respect to the origin. The method is based on a family of nonlinear equations which is associated with a given class of differential equations. The roots of the nonlinear equations are related to the roots of the polynomial solutions of differential equations considered. Newton's method is applied to find the roots of these nonlinear equations. In (Pasquini, 1994) the nonsingularity of the roots of these nonlinear equations is studied. In this paper, following the lines in (Pasquini, 1994), the nonsingularity of the roots of these nonlinear equations is studied. More favourable results than the ones in (Pasquini, 1994) are proven in the particular case of polynomial solutions with symmetrical zeros. The method is applied to approximate the roots of Hermite-Sobolev type polynomials and Freud polynomials. A lower bound for the smallest positive root of Hermite-Sobolev type polynomials is given via the nonlinear equation. The quadratic convergence of the method is proven. A comparison with a classical method that uses the Jacobi matrices is carried out. We show that the algorithm derived by the proposed method is sometimes preferable to the classical QR type algorithms for computing the eigenvalues of the Jacobi matrices even if these matrices are real and symmetric.

New analytical exact solutions of time fractional KdV-KZK equation by Kudryashov methods

NASA Astrophysics Data System (ADS)

S Saha, Ray

2016-04-01

In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov (KdV-KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.

Influence of tides in viscoelastic bodies of planet and satellite on the satellite's orbital motion

NASA Astrophysics Data System (ADS)

Emelyanov, N. V.

2018-06-01

The problem of influence of tidal friction in both planetary and satellite bodies upon satellite's orbital motion is considered. Using the differential equations in satellite's rectangular planetocentric coordinates, the differential equations describing the changes in semimajor axis and eccentricity are derived. The equations in rectangular coordinates were taken from earlier works on the problem. The calcultations carried out for a number of test examples prove that the averaged solutions of equations in coordinates and precise solutions of averaged equations in the Keplerian elements are identical. For the problem of tides raised on planet's body, it was found that, if satellite's mean motion n is equal to 11/18 Ω, where Ω is the planet's angular rotation rate, the orbital eccentricity does not change. This conclusion is in agreement with the results of other authors. It was also found that there is essential discrepancy between the equations in the elements obtained in this paper and analogous equations published by earlier researchers.

Compacton solutions in a class of generalized fifth-order Korteweg-de Vries equations.

PubMed

Cooper, F; Hyman, J M; Khare, A

2001-08-01

Solitons play a fundamental role in the evolution of general initial data for quasilinear dispersive partial differential equations, such as the Korteweg-de Vries (KdV), nonlinear Schrödinger, and the Kadomtsev-Petviashvili equations. These integrable equations have linear dispersion and the solitons have infinite support. We have derived and investigate a new KdV-like Hamiltonian partial differential equation from a four-parameter Lagrangian where the nonlinear dispersion gives rise to solitons with compact support (compactons). The new equation does not seem to be integrable and only mass, momentum, and energy seem to be conserved; yet, the solitons display almost the same modal decompositions and structural stability observed in integrable partial differential equations. The compactons formed from arbitrary initial data, are nonlinearly self-stabilizing, and maintain their coherence after multiple collisions. The robustness of these compactons and the inapplicability of the inverse scattering tools, that worked so well for the KdV equation, make it clear that there is a fundamental mechanism underlying the processes beyond integrability. We have found explicit formulas for multiple classes of compact traveling wave solutions. When there are more than one compacton solution for a particular set of parameters, the wider compacton is the minimum of a reduced Hamiltonian and is the only one that is stable.

When is quasi-linear theory exact. [particle acceleration

NASA Technical Reports Server (NTRS)

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

1975-01-01

We use the cumulant expansion technique of Kubo (1962, 1963) to derive an integrodifferential equation for 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 equation for this function degenerates exactly to a differential equation of Fokker-Planck type. Quasi-linear theory, including the adiabatic assumption, is an exact theory only for this limited class of fluctuations.

An Anharmonic Solution to the Equation of Motion for the Simple Pendulum

ERIC Educational Resources Information Center

Johannessen, Kim

2011-01-01

An anharmonic solution to the differential equation describing the oscillations of a simple pendulum at large angles is discussed. The solution is expressed in terms of functions not involving the Jacobi elliptic functions. In the derivation, a sinusoidal expression, including a linear and a Fourier sine series in the argument, has been applied.…

Coupled out of plane vibrations of spiral beams for micro-scale applications

NASA Astrophysics Data System (ADS)

Amin Karami, M.; Yardimoglu, Bulent; Inman, Daniel J.

2010-12-01

An analytical method is proposed to calculate the natural frequencies and the corresponding mode shape functions of an Archimedean spiral beam. The deflection of the beam is due to both bending and torsion, which makes the problem coupled in nature. The governing partial differential equations and the boundary conditions are derived using Hamilton's principle. Two factors make the vibrations of spirals different from oscillations of constant radius arcs. The first is the presence of terms with derivatives of the radius in the governing equations of spirals and the second is the fact that variations of radius of the beam causes the coefficients of the differential equations to be variable. It is demonstrated, using perturbation techniques that the derivative of the radius terms have negligible effect on structure's dynamics. The spiral is then approximated with many merging constant-radius curved sections joined together to approximate the slow change of radius along the spiral. The equations of motion are formulated in non-dimensional form and the effect of all the key parameters on natural frequencies is presented. Non-dimensional curves are used to summarize the results for clarity. We also solve the governing equations using Rayleigh's approximate method. The fundamental frequency results of the exact and Rayleigh's method are in close agreement. This to some extent verifies the exact solutions. The results show that the vibration of spirals is mostly torsional which complicates using the spiral beam as a host for a sensor or energy harvesting device.

Advanced control concepts. [trim solution for space shuttle

NASA Technical Reports Server (NTRS)

Hutton, M. F.; Friedland, B.

1973-01-01

The selection of a trim solution that provides the space shuttle with the highest level of performance and dynamic control in the presense of wind disturbances and bias torques due to misalignment of rocket engines is described. It was determined that engine gimballing is insufficient to provide control to trim the vehicle for headwind and sidewind disturbances, and that it is necessary to use aerodynamic surfaces in conjunction with engine gimballing to achieve trim. The algebraic equations for computing the trim solution were derived from the differential equations describing the motion of the vehicle by substituting the desired trim conditions. The general problem of showing how the trim equations are derived from the equations of motion and the mathematical forms of the performance criterion is discussed in detail, along with the general equations for studying the dynamic response of the trim solution.

Analytic study of solutions for a (3 + 1) -dimensional generalized KP equation

NASA Astrophysics Data System (ADS)

Gao, Hui; Cheng, Wenguang; Xu, Tianzhou; Wang, Gangwei

2018-03-01

The (3 + 1) -dimensional generalized KP (gKP) equation is an important nonlinear partial differential equation in theoretical and mathematical physics which can be used to describe nonlinear wave motion. Through the Hirota bilinear method, one-solition, two-solition and N-solition solutions are derived via symbolic computation. Two classes of lump solutions, rationally localized in all directions in space, to the dimensionally reduced cases in (2 + 1)-dimensions, are constructed by using a direct method based on the Hirota bilinear form of the equation. It implies that we can derive the lump solutions of the reduced gKP equation from positive quadratic function solutions to the aforementioned bilinear equation. Meanwhile, we get interaction solutions between a lump and a kink of the gKP equation. The lump appears from a kink and is swallowed by it with the change of time. This work offers a possibility which can enrich the variety of the dynamical features of solutions for higher-dimensional nonlinear evolution equations.

First- and Second-Order Sensitivity Analysis of a P-Version Finite Element Equation Via Automatic Differentiation

NASA Technical Reports Server (NTRS)

Hou, Gene

1998-01-01

Sensitivity analysis is a technique for determining derivatives of system responses with respect to design parameters. Among many methods available for sensitivity analysis, automatic differentiation has been proven through many applications in fluid dynamics and structural mechanics to be an accurate and easy method for obtaining derivatives. Nevertheless, the method can be computational expensive and can require a high memory space. This project will apply an automatic differentiation tool, ADIFOR, to a p-version finite element code to obtain first- and second- order then-nal derivatives, respectively. The focus of the study is on the implementation process and the performance of the ADIFOR-enhanced codes for sensitivity analysis in terms of memory requirement, computational efficiency, and accuracy.

Elimination of secular terms from the differential equations for the elements of perturbed two-body motion

NASA Technical Reports Server (NTRS)

Bond, Victor R.; Fraietta, Michael F.

1991-01-01

In 1961, Sperling linearized and regularized the differential equations of motion of the two-body problem by changing the independent variable from time to fictitious time by Sundman's transformation (r = dt/ds) and by embedding the two-body energy integral and the Laplace vector. In 1968, Burdet developed a perturbation theory which was uniformly valid for all types of orbits using a variation of parameters approach on the elements which appeared in Sperling's equations for the two-body solution. In 1973, Bond and Hanssen improved Burdet's set of differential equations by embedding the total energy (which is a constant when the potential function is explicitly dependent upon time.) The Jacobian constant was used as an element to replace the total energy in a reformulation of the differential equations of motion. In the process, another element which is proportional to a component of the angular momentum was introduced. Recently trajectories computed during numerical studies of atmospheric entry from circular orbits and low thrust beginning in near-circular orbits exhibited numerical instability when solved by the method of Bond and Gottlieb (1989) for long time intervals. It was found that this instability was due to secular terms which appear on the righthand sides of the differential equations of some of the elements. In this paper, this instability is removed by the introduction of another vector integral called the delta integral (which replaces the Laplace Vector) and another scalar integral which removes the secular terms. The introduction of these integrals requires a new derivation of the differential equations for most of the elements. For this rederivation, the Lagrange method of variation of parameters is used, making the development more concise. Numerical examples of this improvement are presented.

Stable boundary conditions and difference schemes for Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Dutt, P.

1985-01-01

The Navier-Stokes equations can be viewed as an incompletely elliptic perturbation of the Euler equations. By using the entropy function for the Euler equations as a measure of energy for the Navier-Stokes equations, it was possible to obtain nonlinear energy estimates for the mixed initial boundary value problem. These estimates are used to derive boundary conditions which guarantee L2 boundedness even when the Reynolds number tends to infinity. Finally, a new difference scheme for modelling the Navier-Stokes equations in multidimensions for which it is possible to obtain discrete energy estimates exactly analogous to those we obtained for the differential equation was proposed.

A semi-analytical solution for elastic analysis of rotating thick cylindrical shells with variable thickness using disk form multilayers.

PubMed

Zamani Nejad, Mohammad; Jabbari, Mehdi; Ghannad, Mehdi

2014-01-01

Using disk form multilayers, a semi-analytical solution has been derived for determination of displacements and stresses in a rotating cylindrical shell with variable thickness under uniform pressure. The thick cylinder is divided into disk form layers form with their thickness corresponding to the thickness of the cylinder. Due to the existence of shear stress in the thick cylindrical shell with variable thickness, the equations governing disk layers are obtained based on first-order shear deformation theory (FSDT). These equations are in the form of a set of general differential equations. Given that the cylinder is divided into n disks, n sets of differential equations are obtained. The solution of this set of equations, applying the boundary conditions and continuity conditions between the layers, yields displacements and stresses. A numerical solution using finite element method (FEM) is also presented and good agreement was found.

A Semi-Analytical Solution for Elastic Analysis of Rotating Thick Cylindrical Shells with Variable Thickness Using Disk Form Multilayers

PubMed Central

Zamani Nejad, Mohammad; Jabbari, Mehdi; Ghannad, Mehdi

2014-01-01

Using disk form multilayers, a semi-analytical solution has been derived for determination of displacements and stresses in a rotating cylindrical shell with variable thickness under uniform pressure. The thick cylinder is divided into disk form layers form with their thickness corresponding to the thickness of the cylinder. Due to the existence of shear stress in the thick cylindrical shell with variable thickness, the equations governing disk layers are obtained based on first-order shear deformation theory (FSDT). These equations are in the form of a set of general differential equations. Given that the cylinder is divided into n disks, n sets of differential equations are obtained. The solution of this set of equations, applying the boundary conditions and continuity conditions between the layers, yields displacements and stresses. A numerical solution using finite element method (FEM) is also presented and good agreement was found. PMID:24719582

A generalized simplest equation method and its application to the Boussinesq-Burgers equation.

PubMed

Sudao, Bilige; Wang, Xiaomin

2015-01-01

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.

A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

PubMed Central

Sudao, Bilige; Wang, Xiaomin

2015-01-01

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method. PMID:25973605

Wavefronts for a global reaction-diffusion population model with infinite distributed delay

NASA Astrophysics Data System (ADS)

Weng, Peixuan; Xu, Zhiting

2008-09-01

We consider a global reaction-diffusion population model with infinite distributed delay which includes models of Nicholson's blowflies and hematopoiesis derived by Gurney, Mackey and Glass, respectively. The existence of monotone wavefronts is derived by using the abstract settings of functional differential equations and Schauder fixed point theory.

A unifying fractional wave equation for compressional and shear waves.

PubMed

Holm, Sverre; Sinkus, Ralph

2010-01-01

This study has been motivated by the observed difference in the range of the power-law attenuation exponent for compressional and shear waves. Usually compressional attenuation increases with frequency to a power between 1 and 2, while shear wave attenuation often is described with powers less than 1. Another motivation is the apparent lack of partial differential equations with desirable properties such as causality that describe such wave propagation. Starting with a constitutive equation which is a generalized Hooke's law with a loss term containing a fractional derivative, one can derive a causal fractional wave equation previously given by Caputo [Geophys J. R. Astron. Soc. 13, 529-539 (1967)] and Wismer [J. Acoust. Soc. Am. 120, 3493-3502 (2006)]. In the low omegatau (low-frequency) case, this equation has an attenuation with a power-law in the range from 1 to 2. This is consistent with, e.g., attenuation in tissue. In the often neglected high omegatau (high-frequency) case, it describes attenuation with a power-law between 0 and 1, consistent with what is observed in, e.g., dynamic elastography. Thus a unifying wave equation derived properly from constitutive equations can describe both cases.

Dynamic system classifier.

PubMed

Pumpe, Daniel; Greiner, Maksim; Müller, Ewald; Enßlin, Torsten A

2016-07-01

Stochastic differential equations describe well many physical, biological, and sociological systems, despite the simplification often made in their derivation. Here the usage of simple stochastic differential equations to characterize and classify complex dynamical systems is proposed within a Bayesian framework. To this end, we develop a dynamic system classifier (DSC). The DSC first abstracts training data of a system in terms of time-dependent coefficients of the descriptive stochastic differential equation. Thereby the DSC identifies unique correlation structures within the training data. For definiteness we restrict the presentation of the DSC to oscillation processes with a time-dependent frequency ω(t) and damping factor γ(t). Although real systems might be more complex, this simple oscillator captures many characteristic features. The ω and γ time lines represent the abstract system characterization and permit the construction of efficient signal classifiers. Numerical experiments show that such classifiers perform well even in the low signal-to-noise regime.

Sensitivity of Satellite Altimetry Data Assimilation on a Naval Anti-Submarine Warfare Weapon System

DTIC Science & Technology

2004-09-01

representing the actual ocean structure than static climatology databases (Fox et. al., 2002; Chu et. al., 2004). It is expected that this 2...pressure amplitude function and ( , )P P r z= is the phase function, or eikonal . Doing this and collecting real and imaginary terms yields an equation... eikonal equation, [ ]2 2P k∇ = , (13) from which differential equations for rays can be derived (Etter, 1991). The rays are the normals to surfaces

A Model for the Oxidation of C/SiC Composite Structures

NASA Technical Reports Server (NTRS)

Sullivan, Roy M.

2003-01-01

A mathematical theory and an accompanying numerical scheme have been developed for predicting the oxidation behavior of C/SiC composite structures. The theory is derived from the mechanics of the flow of ideal gases through a porous solid. Within the mathematical formulation, two diffusion mechanisms are possible: (1) the relative diffusion of one species with respect to the mixture, which is concentration gradient driven and (2) the diffusion associated with the average velocity of the gas mixture, which is total gas pressure gradient driven. The result of the theoretical formulation is a set of two coupled nonlinear differential equations written in terms of the oxidant and oxide partial pressures. The differential equations must be solved simultaneously to obtain the partial vapor pressures of the oxidant and oxides as a function of space and time. The local rate of carbon oxidation is determined as a function of space and time using the map of the local oxidant partial vapor pressure along with the Arrhenius rate equation. The nonlinear differential equations are cast into matrix equations by applying the Bubnov-Galerkin weighted residual method, allowing for the solution of the differential equations numerically. The end result is a numerical scheme capable of determining the variation of the local carbon oxidation rates as a function of space and time for any arbitrary C/SiC composite structures.

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.

A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives

NASA Astrophysics Data System (ADS)

Abro, Kashif Ali; Memon, Anwar Ahmed; Uqaili, Muhammad Aslam

2018-03-01

This research article is analyzed for the comparative study of RL and RC electrical circuits by employing newly presented Atangana-Baleanu and Caputo-Fabrizio fractional derivatives. The governing ordinary differential equations of RL and RC electrical circuits have been fractionalized in terms of fractional operators in the range of 0 ≤ ξ ≤ 1 and 0 ≤ η ≤ 1. The analytic solutions of fractional differential equations for RL and RC electrical circuits have been solved by using the Laplace transform with its inversions. General solutions have been investigated for periodic and exponential sources by implementing the Atangana-Baleanu and Caputo-Fabrizio fractional operators separately. The investigated solutions have been expressed in terms of simple elementary functions with convolution product. On the basis of newly fractional derivatives with and without singular kernel, the voltage and current have interesting behavior with several similarities and differences for the periodic and exponential sources.

On the validity of the modified equation approach to the stability analysis of finite-difference methods

NASA Technical Reports Server (NTRS)

Chang, Sin-Chung

1987-01-01

The validity of the modified equation stability analysis introduced by Warming and Hyett was investigated. It is shown that the procedure used in the derivation of the modified equation is flawed and generally leads to invalid results. Moreover, the interpretation of the modified equation as the exact partial differential equation solved by a finite-difference method generally cannot be justified even if spatial periodicity is assumed. For a two-level scheme, due to a series of mathematical quirks, the connection between the modified equation approach and the von Neuman method established by Warming and Hyett turns out to be correct despite its questionable original derivation. However, this connection is only partially valid for a scheme involving more than two time levels. In the von Neumann analysis, the complex error multiplication factor associated with a wave number generally has (L-1) roots for an L-level scheme. It is shown that the modified equation provides information about only one of these roots.

Boundary-layer equations in generalized curvilinear coordinates

NASA Technical Reports Server (NTRS)

Panaras, Argyris G.

1987-01-01

A set of higher-order boundary-layer equations is derived valid for three-dimensional compressible flows. The equations are written in a generalized curvilinear coordinate system, in which the surface coordinates are nonorthogonal; the third axis is restricted to be normal to the surface. Also, higher-order viscous terms which are retained depend on the surface curvature of the body. Thus, the equations are suitable for the calculation of the boundary layer about arbitrary vehicles. As a starting point, the Navier-Stokes equations are derived in a tensorian notation. Then by means of an order-of-magnitude analysis, the boundary-layer equations are developed. To provide an interface between the analytical partial differentiation notation and the compact tensor notation, a brief review of the most essential theorems of the tensor analysis related to the equations of the fluid dynamics is given. Many useful quantities, such as the contravariant and the covariant metrics and the physical velocity components, are written in both notations.

An efficient mode-splitting method for a curvilinear nearshore circulation model

USGS Publications Warehouse

Shi, Fengyan; Kirby, James T.; Hanes, Daniel M.

2007-01-01

A mode-splitting method is applied to the quasi-3D nearshore circulation equations in generalized curvilinear coordinates. The gravity wave mode and the vorticity wave mode of the equations are derived using the two-step projection method. Using an implicit algorithm for the gravity mode and an explicit algorithm for the vorticity mode, we combine the two modes to derive a mixed difference–differential equation with respect to surface elevation. McKee et al.'s [McKee, S., Wall, D.P., and Wilson, S.K., 1996. An alternating direction implicit scheme for parabolic equations with mixed derivative and convective terms. J. Comput. Phys., 126, 64–76.] ADI scheme is then used to solve the parabolic-type equation in dealing with the mixed derivative and convective terms from the curvilinear coordinate transformation. Good convergence rates are found in two typical cases which represent respectively the motions dominated by the gravity mode and the vorticity mode. Time step limitations imposed by the vorticity convective Courant number in vorticity-mode-dominant cases are discussed. Model efficiency and accuracy are verified in model application to tidal current simulations in San Francisco Bight.

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.

A solution procedure for behavior of thick plates on a nonlinear foundation and postbuckling behavior of long plates

NASA Technical Reports Server (NTRS)

Stein, M.; Stein, P. A.

1978-01-01

Approximate solutions for three nonlinear orthotropic plate problems are presented: (1) a thick plate attached to a pad having nonlinear material properties which, in turn, is attached to a substructure which is then deformed; (2) a long plate loaded in inplane longitudinal compression beyond its buckling load; and (3) a long plate loaded in inplane shear beyond its buckling load. For all three problems, the two dimensional plate equations are reduced to one dimensional equations in the y-direction by using a one dimensional trigonometric approximation in the x-direction. Each problem uses different trigonometric terms. Solutions are obtained using an existing algorithm for simultaneous, first order, nonlinear, ordinary differential equations subject to two point boundary conditions. Ordinary differential equations are derived to determine the variable coefficients of the trigonometric terms.

Geometry of the submanifolds of SEXn. II. The generalized fundamental equations for the hypersubmanifold of SEXn

NASA Astrophysics Data System (ADS)

Chung, Kyung Tae; Lee, Jong Woo

1989-08-01

A connection which is both Einstein and semisymmetric is called an SE connection, and a generalized n-dimensional Riemannian manifold on which the differential geometric structure is imposed by g λμ through an SE connection is called an n-dimensional SE manifold and denoted by SEXn. This paper is a direct continuation of earlier work. In this paper, we derive the generalized fundamental equations for the hypersubmanifold of SEXn, including generalized Gauss formulas, generalized Weingarten equations, and generalized Gauss-Codazzi equations.

Geometric approach to nuclear pasta phases

NASA Astrophysics Data System (ADS)

Kubis, Sebastian; Wójcik, Włodzimierz

2016-12-01

By use of the variational methods and differential geometry in the framework of the liquid drop model we formulate appropriate equilibrium equations for pasta phases with imposed periodicity. The extension of the Young-Laplace equation in the case of charged fluid is obtained. The β equilibrium and virial theorem are also generalized. All equations are shown in gauge invariant form. For the first time, the pasta shape stability analysis is carried out. The proper stability condition in the form of the generalized Jacobi equation is derived. The presented formalism is tested on some particular cases.

Mass conservation: 1-D open channel flow equations

USGS Publications Warehouse

DeLong, Lewis L.

1989-01-01

Unsteady flow simulation in natural rivers is often complicated by meandering channels of compound section. Hydraulic properties and the length of the wetted channel may vary significantly as a meandering river inundates its adjacent floodplain. The one-dimensional, unsteady, open-channel flow equations can be extended to simulate floods in channels of compound section. It will be shown that equations derived from the addition of differential equations individually describing flow in main and overbank channels do not in general conserve mass when overbank and main channels are of different lengths.

Two ways to solve, using Lie group analysis, the fundamental valuation equation in the double-square-root model of the term structure

NASA Astrophysics Data System (ADS)

Sinkala, W.

2011-01-01

Two approaches based on Lie group analysis are employed to obtain the closed-form solution of a partial differential equation derived by Francis A. Longstaff [J Financial Econom 1989;23:195-224] for the price of a discount bond in the double-square-root model of the term structure.

Dynamics of curved fronts in systems with power-law memory

NASA Astrophysics Data System (ADS)

Abu Hamed, M.; Nepomnyashchy, A. A.

2016-08-01

The dynamics of a curved front in a plane between two stable phases with equal potentials is modeled via two-dimensional fractional in time partial differential equation. A closed equation governing a slow motion of a small-curvature front is derived and applied for two typical examples of the potential function. Approximate axisymmetric and non-axisymmetric solutions are obtained.

Statistical Mechanics of Node-perturbation Learning with Noisy Baseline

NASA Astrophysics Data System (ADS)

Hara, Kazuyuki; Katahira, Kentaro; Okada, Masato

2017-02-01

Node-perturbation learning is a type of statistical gradient descent algorithm that can be applied to problems where the objective function is not explicitly formulated, including reinforcement learning. It estimates the gradient of an objective function by using the change in the object function in response to the perturbation. The value of the objective function for an unperturbed output is called a baseline. Cho et al. proposed node-perturbation learning with a noisy baseline. In this paper, we report on building the statistical mechanics of Cho's model and on deriving coupled differential equations of order parameters that depict learning dynamics. We also show how to derive the generalization error by solving the differential equations of order parameters. On the basis of the results, we show that Cho's results are also apply in general cases and show some general performances of Cho's model.

Analysis of the cable equation with non-local and non-singular kernel fractional derivative

NASA Astrophysics Data System (ADS)

Karaagac, Berat

2018-02-01

Recently a new concept of differentiation was introduced in the literature where the kernel was converted from non-local singular to non-local and non-singular. One of the great advantages of this new kernel is its ability to portray fading memory and also well defined memory of the system under investigation. In this paper the cable equation which is used to develop mathematical models of signal decay in submarine or underwater telegraphic cables will be analysed using the Atangana-Baleanu fractional derivative due to the ability of the new fractional derivative to describe non-local fading memory. The existence and uniqueness of the more generalized model is presented in detail via the fixed point theorem. A new numerical scheme is used to solve the new equation. In addition, stability, convergence and numerical simulations are presented.

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

Symmetry Reductions and Group-Invariant Radial Solutions to the n-Dimensional Wave Equation

NASA Astrophysics Data System (ADS)

Feng, Wei; Zhao, Songlin

2018-01-01

In this paper, we derive explicit group-invariant radial solutions to a class of wave equation via symmetry group method. The optimal systems of one-dimensional subalgebras for the corresponding radial wave equation are presented in terms of the known point symmetries. The reductions of the radial wave equation into second-order ordinary differential equations (ODEs) with respect to each symmetry in the optimal systems are shown. Then we solve the corresponding reduced ODEs explicitly in order to write out the group-invariant radial solutions for the wave equation. Finally, several analytical behaviours and smoothness of the resulting solutions are discussed.

Sensitivity Equation Derivation for Transient Heat Transfer Problems

NASA Technical Reports Server (NTRS)

Hou, Gene; Chien, Ta-Cheng; Sheen, Jeenson

2004-01-01

The focus of the paper is on the derivation of sensitivity equations for transient heat transfer problems modeled by different discretization processes. Two examples will be used in this study to facilitate the discussion. The first example is a coupled, transient heat transfer problem that simulates the press molding process in fabrication of composite laminates. These state equations are discretized into standard h-version finite elements and solved by a multiple step, predictor-corrector scheme. The sensitivity analysis results based upon the direct and adjoint variable approaches will be presented. The second example is a nonlinear transient heat transfer problem solved by a p-version time-discontinuous Galerkin's Method. The resulting matrix equation of the state equation is simply in the form of Ax = b, representing a single step, time marching scheme. A direct differentiation approach will be used to compute the thermal sensitivities of a sample 2D problem.

Finite difference methods for transient signal propagation in stratified dispersive media

NASA Technical Reports Server (NTRS)

Lam, D. H.

1975-01-01

Explicit difference equations are presented for the solution of a signal of arbitrary waveform propagating in an ohmic dielectric, a cold plasma, a Debye model dielectric, and a Lorentz model dielectric. These difference equations are derived from the governing time-dependent integro-differential equations for the electric fields by a finite difference method. A special difference equation is derived for the grid point at the boundary of two different media. Employing this difference equation, transient signal propagation in an inhomogeneous media can be solved provided that the medium is approximated in a step-wise fashion. The solutions are generated simply by marching on in time. It is concluded that while the classical transform methods will remain useful in certain cases, with the development of the finite difference methods described, an extensive class of problems of transient signal propagating in stratified dispersive media can be effectively solved by numerical methods.

A framework for qualitative reasoning about solid objects

NASA Technical Reports Server (NTRS)

Davis, E.

1987-01-01

Predicting the behavior of a qualitatively described system of solid objects requires a combination of geometrical, temporal, and physical reasoning. Methods based upon formulating and solving differential equations are not adequate for robust prediction, since the behavior of a system over extended time may be much simpler than its behavior over local time. A first-order logic, in which one can state simple physical problems and derive their solution deductively, without recourse to solving the differential equations, is discussed. This logic is substantially more expressive and powerful than any previous AI representational system in this domain.

2–stage stochastic Runge–Kutta for stochastic delay differential equations

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

Rosli, Norhayati; Jusoh Awang, Rahimah; Bahar, Arifah

2015-05-15

This paper proposes a newly developed one-step derivative-free method, that is 2-stage stochastic Runge-Kutta (SRK2) to approximate the solution of stochastic delay differential equations (SDDEs) with a constant time lag, r > 0. General formulation of stochastic Runge-Kutta for SDDEs is introduced and Stratonovich Taylor series expansion for numerical solution of SRK2 is presented. Local truncation error of SRK2 is measured by comparing the Stratonovich Taylor expansion of the exact solution with the computed solution. Numerical experiment is performed to assure the validity of the method in simulating the strong solution of SDDEs.

Dynamic stiffness of chemically and physically ageing rubber vibration isolators in the audible frequency range. Part 1: constitutive equations

NASA Astrophysics Data System (ADS)

Kari, Leif

2017-09-01

The constitutive equations of chemically and physically ageing rubber in the audible frequency range are modelled as a function of ageing temperature, ageing time, actual temperature, time and frequency. The constitutive equations are derived by assuming nearly incompressible material with elastic spherical response and viscoelastic deviatoric response, using Mittag-Leffler relaxation function of fractional derivative type, the main advantage being the minimum material parameters needed to successfully fit experimental data over a broad frequency range. The material is furthermore assumed essentially entropic and thermo-mechanically simple while using a modified William-Landel-Ferry shift function to take into account temperature dependence and physical ageing, with fractional free volume evolution modelled by a nonlinear, fractional differential equation with relaxation time identical to that of the stress response and related to the fractional free volume by Doolittle equation. Physical ageing is a reversible ageing process, including trapping and freeing of polymer chain ends, polymer chain reorganizations and free volume changes. In contrast, chemical ageing is an irreversible process, mainly attributed to oxygen reaction with polymer network either damaging the network by scission or reformation of new polymer links. The chemical ageing is modelled by inner variables that are determined by inner fractional evolution equations. Finally, the model parameters are fitted to measurements results of natural rubber over a broad audible frequency range, and various parameter studies are performed including comparison with results obtained by ordinary, non-fractional ageing evolution differential equations.

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

Ghafarian, M.; Ariaei, A., E-mail: ariaei@eng.ui.ac.ir

The free vibration analysis of a multiple rotating nanobeams' system applying the nonlocal Eringen elasticity theory is presented. Multiple nanobeams' systems are of great importance in nano-optomechanical applications. At nanoscale, the nonlocal effects become non-negligible. According to the nonlocal Euler-Bernoulli beam theory, the governing partial differential equations are derived by incorporating the nonlocal scale effects. Assuming a structure of n parallel nanobeams, the vibration of the system is described by a coupled set of n partial differential equations. The method involves a change of variables to uncouple the equations and the differential transform method as an efficient mathematical technique tomore » solve the nonlocal governing differential equations. Then a number of parametric studies are conducted to assess the effect of the nonlocal scaling parameter, rotational speed, boundary conditions, hub radius, and the stiffness coefficients of the elastic interlayer media on the vibration behavior of the coupled rotating multiple-carbon-nanotube-beam system. It is revealed that the bending vibration of the system is significantly influenced by the rotational speed, elastic mediums, and the nonlocal scaling parameters. This model is validated by comparing the results with those available in the literature. The natural frequencies are in a reasonably good agreement with the reported results.« less

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

Célérier, Marie-Noëlle; Nottale, Laurent, E-mail: marie-noelle.celerier@obspm.fr, E-mail: laurent.nottale@obspm.fr

Owing to the non-differentiable nature of the theory of Scale Relativity, the emergence of complex wave functions, then of spinors and bi-spinors occurs naturally in its framework. The wave function is here a manifestation of the velocity field of geodesics of a continuous and non-differentiable (therefore fractal) space-time. In a first paper (Paper I), we have presented the general argument which leads to this result using an elaborate and more detailed derivation than previously displayed. We have therefore been able to show how the complex wave function emerges naturally from the doubling of the velocity field and to revisit themore » derivation of the non-relativistic Schrödinger equation of motion. In the present paper (Paper II), we deal with relativistic motion and detail the natural emergence of the bi-spinors from such first principles of the theory. Moreover, while Lorentz invariance has been up to now inferred from mathematical results obtained in stochastic mechanics, we display here a new and detailed derivation of the way one can obtain a Lorentz invariant expression for the expectation value of the product of two independent fractal fluctuation fields in the sole framework of the theory of Scale Relativity. These new results allow us to enhance the robustness of our derivation of the two main equations of motion of relativistic quantum mechanics (the Klein-Gordon and Dirac equations) which we revisit here at length.« less

Molar mass, radius of gyration and second virial coefficient from new static light scattering equations for dilute solutions: application to 21 (macro)molecules.

PubMed

Illien, Bertrand; Ying, Ruifeng

2009-05-11

New static light scattering (SLS) equations for dilute binary solutions are derived. Contrarily to the usual SLS equations [Carr-Zimm (CZ)], the new equations have no need for the experimental absolute Rayleigh ratio of a reference liquid and solely rely on the ratio of scattered intensities of solutions and solvent. The new equations, which are based on polarizability equations, take into account the usual refractive index increment partial differential n/partial differential rho(2) complemented by the solvent specific polarizability and a term proportional to the slope of the solution density rho versus the solute mass concentration rho(2) (density increment). Then all the equations are applied to 21 (macro)molecules with a wide range of molar mass (0.2500 kg mol(-1)), for which the scattered intensity is no longer independent of the scattering angle, the new equations give the same value of the radius of gyration as the CZ equation and consistent values of the second virial coefficient.

Mechanical Balance Laws for Boussinesq Models of Surface Water Waves

NASA Astrophysics Data System (ADS)

Ali, Alfatih; Kalisch, Henrik

2012-06-01

Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and it appears that the expressions for many of these quantities are not known. This work presents a systematic derivation of mass, momentum and energy densities and fluxes associated with a general family of Boussinesq systems. The derivation is based on a reconstruction of the velocity field and the pressure in the fluid column below the free surface, and the derivation of differential balance equations which are of the same asymptotic validity as the evolution equations. It is shown that all these mechanical quantities can be expressed in terms of the principal dependent variables of the Boussinesq system: the surface excursion η and the horizontal velocity w at a given level in the fluid.

Adjoint-Based Methodology for Time-Dependent Optimization

NASA Technical Reports Server (NTRS)

Yamaleev, N. K.; Diskin, B.; Nielsen, E. J.

2008-01-01

This paper presents a discrete adjoint method for a broad class of time-dependent optimization problems. The time-dependent adjoint equations are derived in terms of the discrete residual of an arbitrary finite volume scheme which approximates unsteady conservation law equations. Although only the 2-D unsteady Euler equations are considered in the present analysis, this time-dependent adjoint method is applicable to the 3-D unsteady Reynolds-averaged Navier-Stokes equations with minor modifications. The discrete adjoint operators involving the derivatives of the discrete residual and the cost functional with respect to the flow variables are computed using a complex-variable approach, which provides discrete consistency and drastically reduces the implementation and debugging cycle. The implementation of the time-dependent adjoint method is validated by comparing the sensitivity derivative with that obtained by forward mode differentiation. Our numerical results show that O(10) optimization iterations of the steepest descent method are needed to reduce the objective functional by 3-6 orders of magnitude for test problems considered.

Solution of the Bagley Torvik equation by fractional DTM

NASA Astrophysics Data System (ADS)

Arora, Geeta; Pratiksha

2017-07-01

In this paper, fractional differential transform method(DTM) is implemented on the Bagley Torvik equation. This equation models the viscoelastic behavior of geological strata, metals, glasses etc. It explains the motion of a rigid plate immersed in a Newtonian fluid. DTM is a simple, reliable and efficient method that gives a series solution. Caputo fractional derivative is considered throughout this work. Two examples are given to demonstrate the validity and applicability of the method and comparison is made with the existing results.

Properties of finite difference models of non-linear conservative oscillators

NASA Technical Reports Server (NTRS)

Mickens, R. E.

1988-01-01

Finite-difference (FD) approaches to the numerical solution of the differential equations describing the motion of a nonlinear conservative oscillator are investigated analytically. A generalized formulation of the Duffing and modified Duffing equations is derived and analyzed using several FD techniques, and it is concluded that, although it is always possible to contstruct FD models of conservative oscillators which are themselves conservative, caution is required to avoid numerical solutions which do not accurately reflect the properties of the original equation.

Second- and Higher-Order Virial Coefficients Derived from Equations of State for Real Gases

ERIC Educational Resources Information Center

Parkinson, William A.

2009-01-01

Derivation of the second- and higher-order virial coefficients for models of the gaseous state is demonstrated by employing a direct differential method and subsequent term-by-term comparison to power series expansions. This communication demonstrates the application of this technique to van der Waals representations of virial coefficients.…

Differential geometry-based solvation and electrolyte transport models for biomolecular modeling: a review

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

Wei, Guowei; Baker, Nathan A.

2016-11-11

This chapter reviews the differential geometry-based solvation and electrolyte transport for biomolecular solvation that have been developed over the past decade. A key component of these methods is the differential geometry of surfaces theory, as applied to the solvent-solute boundary. In these approaches, the solvent-solute boundary is determined by a variational principle that determines the major physical observables of interest, for example, biomolecular surface area, enclosed volume, electrostatic potential, ion density, electron density, etc. Recently, differential geometry theory has been used to define the surfaces that separate the microscopic (solute) domains for biomolecules from the macroscopic (solvent) domains. In thesemore » approaches, the microscopic domains are modeled with atomistic or quantum mechanical descriptions, while continuum mechanics models (including fluid mechanics, elastic mechanics, and continuum electrostatics) are applied to the macroscopic domains. This multiphysics description is integrated through an energy functional formalism and the resulting Euler-Lagrange equation is employed to derive a variety of governing partial differential equations for different solvation and transport processes; e.g., the Laplace-Beltrami equation for the solvent-solute interface, Poisson or Poisson-Boltzmann equations for electrostatic potentials, the Nernst-Planck equation for ion densities, and the Kohn-Sham equation for solute electron density. Extensive validation of these models has been carried out over hundreds of molecules, including proteins and ion channels, and the experimental data have been compared in terms of solvation energies, voltage-current curves, and density distributions. We also propose a new quantum model for electrolyte transport.« less

The Modelling of Axially Translating Flexible Beams

NASA Astrophysics Data System (ADS)

Theodore, R. J.; Arakeri, J. H.; Ghosal, A.

1996-04-01

The axially translating flexible beam with a prismatic joint can be modelled by using the Euler-Bernoulli beam equation together with the convective terms. In general, the method of separation of variables cannot be applied to solve this partial differential equation. In this paper, a non-dimensional form of the Euler Bernoulli beam equation is presented, obtained by using the concept of group velocity, and also the conditions under which separation of variables and assumed modes method can be used. The use of clamped-mass boundary conditions leads to a time-dependent frequency equation for the translating flexible beam. A novel method is presented for solving this time dependent frequency equation by using a differential form of the frequency equation. The assume mode/Lagrangian formulation of dynamics is employed to derive closed form equations of motion. It is shown by using Lyapunov's first method that the dynamic responses of flexural modal variables become unstable during retraction of the flexible beam, which the dynamic response during extension of the beam is stable. Numerical simulation results are presented for the uniform axial motion induced transverse vibration for a typical flexible beam.

High-accuracy power series solutions with arbitrarily large radius of convergence for the fractional nonlinear Schrödinger-type equations

NASA Astrophysics Data System (ADS)

Khawaja, U. Al; Al-Refai, M.; Shchedrin, Gavriil; Carr, Lincoln D.

2018-06-01

Fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories: nonlinearity and fractional derivatives. These effective descriptions thus appear commonly in physical and mathematical modeling. We present a new series method providing systematic controlled accuracy for solutions of fractional nonlinear differential equations, including the fractional nonlinear Schrödinger equation and the fractional nonlinear diffusion equation. The method relies on spatially iterative use of power series expansions. Our approach permits an arbitrarily large radius of convergence and thus solves the typical divergence problem endemic to power series approaches. In the specific case of the fractional nonlinear Schrödinger equation we find fractional generalizations of cnoidal waves of Jacobi elliptic functions as well as a fractional bright soliton. For the fractional nonlinear diffusion equation we find the combination of fractional and nonlinear effects results in a more strongly localized solution which nevertheless still exhibits power law tails, albeit at a much lower density.

Initial value formulation of dynamical Chern-Simons gravity

NASA Astrophysics Data System (ADS)

Delsate, Térence; Hilditch, David; Witek, Helvi

2015-01-01

We derive an initial value formulation for dynamical Chern-Simons gravity, a modification of general relativity involving parity-violating higher derivative terms. We investigate the structure of the resulting system of partial differential equations thinking about linearization around arbitrary backgrounds. This type of consideration is necessary if we are to establish well-posedness of the Cauchy problem. Treating the field equations as an effective field theory we find that weak necessary conditions for hyperbolicity are satisfied. For the full field equations we find that there are states from which subsequent evolution is not determined. Generically the evolution system closes, but is not hyperbolic in any sense that requires a first order pseudodifferential reduction. In a cursory mode analysis we find that the equations of motion contain terms that may cause ill-posedness of the initial value problem.

Analytical solution of the time-dependent Bloch NMR flow equations: a translational mechanical analysis

NASA Astrophysics Data System (ADS)

Awojoyogbe, O. B.

2004-08-01

Various biological and physiological properties of living tissue can be studied by means of nuclear magnetic resonance techniques. Unfortunately, the basic physics of extracting the relevant information from the solution of Bloch nuclear magnetic resource (NMR) equations to accurately monitor the clinical state of biological systems is still not yet fully understood. Presently, there are no simple closed solutions known to the Bloch equations for a general RF excitation. Therefore the translational mechanical analysis of the Bloch NMR equations presented in this study, which can be taken as definitions of new functions to be studied in detail may reveal very important information from which various NMR flow parameters can be derived. Fortunately, many of the most important but hidden applications of blood flow parameters can be revealed without too much difficulty if appropriate mathematical techniques are used to solve the equations. In this study we are concerned with a mathematical study of the laws of NMR physics from the point of view of translational mechanical theory. The important contribution of this study is that solutions to the Bloch NMR flow equations do always exist and can be found as accurately as desired. We shall restrict our attention to cases where the radio frequency field can be treated by simple analytical methods. First we shall derive a time dependant second-order non-homogeneous linear differential equation from the Bloch NMR equation in term of the equilibrium magnetization M0, RF B1( t) field, T1 and T2 relaxation times. Then, we would develop a general method of solving the differential equation for the cases when RF B1( t)=0, and when RF B1( t)≠0. This allows us to obtain the intrinsic or natural behavior of the NMR system as well as the response of the system under investigation to a specific influence of external force to the system. Specifically, we consider the case where the RF B1 varies harmonically with time. Here the complete motion of the system consists of two parts. The first part describes the motion of the transverse magnetization My in the absence of RF B( t) field. The second part of the motion described by the particular integral of the derived differential equation does not decay with time but continues its periodic behavior indefinitely. The complete motion of the NMR flow system is then quantitatively and qualitatively described.

A Modeling Insight into Adipose-Derived Stem Cell Myogenesis.

PubMed

Deshpande, Rajiv S; Grayson, Warren L; Spector, Alexander A

2015-01-01

Adipose-derived stem cells (ASCs) are clinically important in regenerative medicine as they are relatively easy to obtain, are characterized by low morbidity, and can differentiate into myogenic progenitor cells. Although studies have elucidated the principal markers, PAX7, Desmin, MyoD, and MHC, the underlying mechanisms are not completely understood. This motivates the application of computational methods to facilitate greater understanding of such processes. In the following, we present a multi-stage kinetic model comprising a system of ordinary differential equations (ODEs). We sought to model ASC differentiation using data from a static culture, where no strain is applied, and a dynamic culture, where 10% strain is applied. The coefficients of the equations have been modulated by those experimental data points. To correctly represent the trajectories, various switches and a feedback factor based on total cell number have been introduced to better represent the biology of ASC differentiation. Furthermore, the model has then been applied to predict ASC fate for strains different from those used in the experimental conditions and for times longer than the duration of the experiment. Analysis of the results reveals unique characteristics of ASC myogenesis under dynamic conditions of the applied strain.

Fokker-Planck equation of the reduced Wigner function associated to an Ohmic quantum Langevin dynamics

NASA Astrophysics Data System (ADS)

Colmenares, Pedro J.

2018-05-01

This article has to do with the derivation and solution of the Fokker-Planck equation associated to the momentum-integrated Wigner function of a particle subjected to a harmonic external field in contact with an ohmic thermal bath of quantum harmonic oscillators. The strategy employed is a simplified version of the phenomenological approach of Schramm, Jung, and Grabert of interpreting the operators as c numbers to derive the quantum master equation arising from a twofold transformation of the Wigner function of the entire phase space. The statistical properties of the random noise comes from the integral functional theory of Grabert, Schramm, and Ingold. By means of a single Wigner transformation, a simpler equation than that mentioned before is found. The Wigner function reproduces the known results of the classical limit. This allowed us to rewrite the underdamped classical Langevin equation as a first-order stochastic differential equation with time-dependent drift and diffusion terms.

Pure quasi-P-wave calculation in transversely isotropic media using a hybrid method

NASA Astrophysics Data System (ADS)

Wu, Zedong; Liu, Hongwei; Alkhalifah, Tariq

2018-07-01

The acoustic approximation for anisotropic media is widely used in current industry imaging and inversion algorithms mainly because Pwaves constitute the majority of the energy recorded in seismic exploration. The resulting acoustic formulae tend to be simpler, resulting in more efficient implementations, and depend on fewer medium parameters. However, conventional solutions of the acoustic wave equation with higher-order derivatives suffer from shear wave artefacts. Thus, we derive a new acoustic wave equation for wave propagation in transversely isotropic (TI) media, which is based on a partially separable approximation of the dispersion relation for TI media and free of shear wave artefacts. Even though our resulting equation is not a partial differential equation, it is still a linear equation. Thus, we propose to implement this equation efficiently by combining the finite difference approximation with spectral evaluation of the space-independent parts. The resulting algorithm provides solutions without the constraint ɛ ≥ δ. Numerical tests demonstrate the effectiveness of the approach.

An Approximate Solution to the Equation of Motion for Large-Angle Oscillations of the Simple Pendulum with Initial Velocity

ERIC Educational Resources Information Center

Johannessen, Kim

2010-01-01

An analytic approximation of the solution to the differential equation describing the oscillations of a simple pendulum at large angles and with initial velocity is discussed. In the derivation, a sinusoidal approximation has been applied, and an analytic formula for the large-angle period of the simple pendulum is obtained, which also includes…

Flux Jacobian Matrices For Equilibrium Real Gases

NASA Technical Reports Server (NTRS)

Vinokur, Marcel

1990-01-01

Improved formulation includes generalized Roe average and extension to three dimensions. Flux Jacobian matrices derived for use in numerical solutions of conservation-law differential equations of inviscid flows of ideal gases extended to real gases. Real-gas formulation of these matrices retains simplifying assumptions of thermodynamic and chemical equilibrium, but adds effects of vibrational excitation, dissociation, and ionization of gas molecules via general equation of state.

Bidirectional plant canopy reflection models derived from the radiation transfer equation

NASA Technical Reports Server (NTRS)

Beeth, D. R.

1975-01-01

A collection of bidirectional canopy reflection models was obtained from the solution of the radiation transfer equation for a horizontally homogeneous canopy. A phase function is derived for a collection of bidirectionally reflecting and transmitting planar elements characterized geometrically by slope and azimuth density functions. Two approaches to solving the radiation transfer equation for the canopy are presented. One approach factors the radiation transfer equation into a solvable set of three first-order linear differential equations by assuming that the radiation field within the canopy can be initially approximated by three components: uniformly diffuse downwelling, uniformly diffuse upwelling, and attenuated specular. The solution to these equations, which can be iterated to any degree of accuracy, was used to obtain overall canopy reflection from the formal solution to the radiation transfer equation. A programable solution to canopy overall bidirectional reflection is given for this approach. The special example of Lambertian leaves with constant leaf bidirectional reflection and scattering functions is considered, and a programmable solution for this example is given. The other approach to solving the radiation transfer equation, a generalized Chandrasekhar technique, is presented in the appendix.

Unsteady Subsonic and Transonic Potential Flow over Helicopter Rotor Blades

NASA Technical Reports Server (NTRS)

Isom, M. P.

1974-01-01

Differential equations and boundary conditions for a rotor blade in forward flight, with subsonic or transonic tip Mach number, are derived. A variety of limiting flow regimes determined by different limits involving blade thickness ratio, aspect ratio, advance ratio and maximum tip Mach number is discussed. The transonic problem is discussed in some detail, and in particular the conditions that make this problem quasi-steady or essentially unsteady are determined. Asymptotic forms of equations and boundary conditions that are valid in an appropriately scaled region of the tip and an azimuthal sector on the advancing side are derived. The equations are then put in a form that is valid from the blade tip inboard through the strip theory region.

Hidden simplicity of the gravity action

DOE PAGES

Cheung, Clifford; Remmen, Grant N.

2017-09-01

We derive new representations of the Einstein-Hilbert action in which graviton perturbation theory is immensely simplified. To accomplish this, we recast the Einstein-Hilbert action as a theory of purely cubic interactions among gravitons and a single auxiliary field. The corresponding equations of motion are the Einstein field equations rewritten as two coupled first-order differential equations. Since all Feynman diagrams are cubic, we are able to derive new off-shell recursion relations for tree-level graviton scattering amplitudes. With a judicious choice of gauge fixing, we then construct an especially compact form for the Einstein-Hilbert action in which all graviton interactions are simplymore » proportional to the graviton kinetic term. Our results apply to graviton perturbations about an arbitrary curved background spacetime.« less

Hidden simplicity of the gravity action

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

Cheung, Clifford; Remmen, Grant N.

We derive new representations of the Einstein-Hilbert action in which graviton perturbation theory is immensely simplified. To accomplish this, we recast the Einstein-Hilbert action as a theory of purely cubic interactions among gravitons and a single auxiliary field. The corresponding equations of motion are the Einstein field equations rewritten as two coupled first-order differential equations. Since all Feynman diagrams are cubic, we are able to derive new off-shell recursion relations for tree-level graviton scattering amplitudes. With a judicious choice of gauge fixing, we then construct an especially compact form for the Einstein-Hilbert action in which all graviton interactions are simplymore » proportional to the graviton kinetic term. Our results apply to graviton perturbations about an arbitrary curved background spacetime.« less

Equations for the determination of humidity from dewpoint and psychrometric data

NASA Technical Reports Server (NTRS)

Parish, O. O.; Putnam, T. W.

1977-01-01

A general expression based on the Claperon-Clausius differential equation that relates saturation vapor pressure, absolute temperature, and the latent heat of transformation was derived that expresses saturation vapor pressure as a function of absolute temperature. This expression was then used to derive general expressions for vapor pressure, absolute humidity, and relative humidity as functions of either dewpoint and ambient temperature or psychrometric parameters. Constants for all general expressions were then evaluated to give specific expressions in both the international system of units and U.S. customary units for temperatures above and below freezing.

Electrical and mechanical fully coupled theory and experimental verification of Rosen-type piezoelectric transformers.

PubMed

Hsu, Yu-Hsiang; Lee, Chih-Kung; Hsiao, Wen-Hsin

2005-10-01

A piezoelectric transformer is a power transfer device that converts its input and output voltage as well as current by effectively using electrical and mechanical coupling effects of piezoelectric materials. Equivalent-circuit models, which are traditionally used to analyze piezoelectric transformers, merge each mechanical resonance effect into a series of ordinary differential equations. Because of using ordinary differential equations, equivalent circuit models are insufficient to reflect the mechanical behavior of piezoelectric plates. Electromechanically, fully coupled governing equations of Rosen-type piezoelectric transformers, which are partial differential equations in nature, can be derived to address the deficiencies of the equivalent circuit models. It can be shown that the modal actuator concept can be adopted to optimize the electromechanical coupling effect of the driving section once the added spatial domain design parameters are taken into account, which are three-dimensional spatial dependencies of electromechanical properties. The maximum power transfer condition for a Rosen-type piezoelectric transformer is detailed. Experimental results, which lead us to a series of new design rules, also are presented to prove the validity and effectiveness of the theoretical predictions.

Application of ANNs approach for wave-like and heat-like equations

NASA Astrophysics Data System (ADS)

Jafarian, Ahmad; Baleanu, Dumitru

2017-12-01

Artificial neural networks are data processing systems which originate from human brain tissue studies. The remarkable abilities of these networks help us to derive desired results from complicated raw data. In this study, we intend to duplicate an efficient iterative method to the numerical solution of two famous partial differential equations, namely the wave-like and heat-like problems. It should be noted that many physical phenomena such as coupling currents in a flat multi-strand two-layer super conducting cable, non-homogeneous elastic waves in soils and earthquake stresses, are described by initial-boundary value wave and heat partial differential equations with variable coefficients. To the numerical solution of these equations, a combination of the power series method and artificial neural networks approach, is used to seek an appropriate bivariate polynomial solution of the mentioned initial-boundary value problem. Finally, several computer simulations confirmed the theoretical results and demonstrating applicability of the method.

Determining partial differential cross sections for low-energy electron photodetachment involving conical intersections using the solution of a Lippmann-Schwinger equation constructed with standard electronic structure techniques.

PubMed

Han, Seungsuk; Yarkony, David R

2011-05-07

A method for obtaining partial differential cross sections for low energy electron photodetachment in which the electronic states of the residual molecule are strongly coupled by conical intersections is reported. The method is based on the iterative solution to a Lippmann-Schwinger equation, using a zeroth order Hamiltonian consisting of the bound nonadiabatically coupled residual molecule and a free electron. The solution to the Lippmann-Schwinger equation involves only standard electronic structure techniques and a standard three-dimensional free particle Green's function quadrature for which fast techniques exist. The transition dipole moment for electron photodetachment, is a sum of matrix elements each involving one nonorthogonal orbital obtained from the solution to the Lippmann-Schwinger equation. An expression for the electron photodetachment transition dipole matrix element in terms of Dyson orbitals, which does not make the usual orthogonality assumptions, is derived.

Optical solitons, nonlinear self-adjointness and conservation laws for the cubic nonlinear Shrödinger's equation with repulsive delta potential

NASA Astrophysics Data System (ADS)

Baleanu, Dumitru; Inc, Mustafa; Aliyu, Aliyu Isa; Yusuf, Abdullahi

2017-11-01

In this paper, the complex envelope function ansatz method is used to acquire the optical solitons to the cubic nonlinear Shrödinger's equation with repulsive delta potential (δ-NLSE). The method reveals dark and bright optical solitons. The necessary constraint conditions which guarantee the existence of the solitons are also presented. We studied the δ-NLSE by analyzing a system of partial differential equations (PDEs) obtained by decomposing the equation into real and imaginary components. We derive the Lie point symmetry generators of the system and prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to establish a set of conserved vectors for the system using the general Cls theorem presented by Ibragimov. Some interesting figures for the acquired solutions are also presented.

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.

Derivation of Continuum Models from An Agent-based Cancer Model: Optimization and Sensitivity Analysis.

PubMed

Voulgarelis, Dimitrios; Velayudhan, Ajoy; Smith, Frank

2017-01-01

Agent-based models provide a formidable tool for exploring complex and emergent behaviour of biological systems as well as accurate results but with the drawback of needing a lot of computational power and time for subsequent analysis. On the other hand, equation-based models can more easily be used for complex analysis in a much shorter timescale. This paper formulates an ordinary differential equations and stochastic differential equations model to capture the behaviour of an existing agent-based model of tumour cell reprogramming and applies it to optimization of possible treatment as well as dosage sensitivity analysis. For certain values of the parameter space a close match between the equation-based and agent-based models is achieved. The need for division of labour between the two approaches is explored. Copyright© Bentham Science Publishers; For any queries, please email at epub@benthamscience.org.

Birth-jump processes and application to forest fire spotting.

PubMed

Hillen, T; Greese, B; Martin, J; de Vries, G

2015-01-01

Birth-jump models are designed to describe population models for which growth and spatial spread cannot be decoupled. A birth-jump model is a nonlinear integro-differential equation. We present two different derivations of this equation, one based on a random walk approach and the other based on a two-compartmental reaction-diffusion model. In the case that the redistribution kernels are highly concentrated, we show that the integro-differential equation can be approximated by a reaction-diffusion equation, in which the proliferation rate contributes to both the diffusion term and the reaction term. We completely solve the corresponding critical domain size problem and the minimal wave speed problem. Birth-jump models can be applied in many areas in mathematical biology. We highlight an application of our results in the context of forest fire spread through spotting. We show that spotting increases the invasion speed of a forest fire front.

Khater method for nonlinear Sharma Tasso-Olever (STO) equation of fractional order

NASA Astrophysics Data System (ADS)

Bibi, Sadaf; Mohyud-Din, Syed Tauseef; Khan, Umar; Ahmed, Naveed

In this work, we have implemented a direct method, known as Khater method to establish exact solutions of nonlinear partial differential equations of fractional order. Number of solutions provided by this method is greater than other traditional methods. Exact solutions of nonlinear fractional order Sharma Tasso-Olever (STO) equation are expressed in terms of kink, travelling wave, periodic and solitary wave solutions. Modified Riemann-Liouville derivative and Fractional complex transform have been used for compatibility with fractional order sense. Solutions have been graphically simulated for understanding the physical aspects and importance of the method. A comparative discussion between our established results and the results obtained by existing ones is also presented. Our results clearly reveal that the proposed method is an effective, powerful and straightforward technique to work out new solutions of various types of differential equations of non-integer order in the fields of applied sciences and engineering.

Molecular, metabolic, and genetic control: An introduction

NASA Astrophysics Data System (ADS)

Tyson, John J.; Mackey, Michael C.

2001-03-01

The living cell is a miniature, self-reproducing, biochemical machine. Like all machines, it has a power supply, a set of working components that carry out its necessary tasks, and control systems that ensure the proper coordination of these tasks. In this Special Issue, we focus on the molecular regulatory systems that control cell metabolism, gene expression, environmental responses, development, and reproduction. As for the control systems in human-engineered machines, these regulatory networks can be described by nonlinear dynamical equations, for example, ordinary differential equations, reaction-diffusion equations, stochastic differential equations, or cellular automata. The articles collected here illustrate (i) a range of theoretical problems presented by modern concepts of cellular regulation, (ii) some strategies for converting molecular mechanisms into dynamical systems, (iii) some useful mathematical tools for analyzing and simulating these systems, and (iv) the sort of results that derive from serious interplay between theory and experiment.

Analytical solutions for sequentially coupled one-dimensional reactive transport problems Part I: Mathematical derivations

NASA Astrophysics Data System (ADS)

Srinivasan, V.; Clement, T. P.

2008-02-01

Multi-species reactive transport equations coupled through sorption and sequential first-order reactions are commonly used to model sites contaminated with radioactive wastes, chlorinated solvents and nitrogenous species. Although researchers have been attempting to solve various forms of these reactive transport equations for over 50 years, a general closed-form analytical solution to this problem is not available in the published literature. In Part I of this two-part article, we derive a closed-form analytical solution to this problem for spatially-varying initial conditions. The proposed solution procedure employs a combination of Laplace and linear transform methods to uncouple and solve the system of partial differential equations. Two distinct solutions are derived for Dirichlet and Cauchy boundary conditions each with Bateman-type source terms. We organize and present the final solutions in a common format that represents the solutions to both boundary conditions. In addition, we provide the mathematical concepts for deriving the solution within a generic framework that can be used for solving similar transport problems.

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.

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.

A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations

NASA Astrophysics Data System (ADS)

Bhrawy, A. H.; Zaky, M. A.

2015-01-01

In this paper, we propose and analyze an efficient operational formulation of spectral tau method for multi-term time-space fractional differential equation with Dirichlet boundary conditions. The shifted Jacobi operational matrices of Riemann-Liouville fractional integral, left-sided and right-sided Caputo fractional derivatives are presented. By using these operational matrices, we propose a shifted Jacobi tau method for both temporal and spatial discretizations, which allows us to present an efficient spectral method for solving such problem. Furthermore, the error is estimated and the proposed method has reasonable convergence rates in spatial and temporal discretizations. In addition, some known spectral tau approximations can be derived as special cases from our algorithm if we suitably choose the corresponding special cases of Jacobi parameters θ and ϑ. Finally, in order to demonstrate its accuracy, we compare our method with those reported in the literature.

A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear Schrödinger equation

NASA Astrophysics Data System (ADS)

Ghanbari, Behzad; Inc, Mustafa

2018-04-01

The present paper suggests a novel technique to acquire exact solutions of nonlinear partial differential equations. The main idea of the method is to generalize the exponential rational function method. In order to examine the ability of the method, we consider the resonant nonlinear Schrödinger equation (R-NLSE). Many variants of exact soliton solutions for the equation are derived by the proposed method. Physical interpretations of some obtained solutions is also included. One can easily conclude that the new proposed method is very efficient and finds the exact solutions of the equation in a relatively easy way.

On the solution of the complex eikonal equation in acoustic VTI media: A perturbation plus optimization scheme

NASA Astrophysics Data System (ADS)

Huang, Xingguo; Sun, Jianguo; Greenhalgh, Stewart

2018-04-01

We present methods for obtaining numerical and analytic solutions of the complex eikonal equation in inhomogeneous acoustic VTI media (transversely isotropic media with a vertical symmetry axis). The key and novel point of the method for obtaining numerical solutions is to transform the problem of solving the highly nonlinear acoustic VTI eikonal equation into one of solving the relatively simple eikonal equation for the background (isotropic) medium and a system of linear partial differential equations. Specifically, to obtain the real and imaginary parts of the complex traveltime in inhomogeneous acoustic VTI media, we generalize a perturbation theory, which was developed earlier for solving the conventional real eikonal equation in inhomogeneous anisotropic media, to the complex eikonal equation in such media. After the perturbation analysis, we obtain two types of equations. One is the complex eikonal equation for the background medium and the other is a system of linearized partial differential equations for the coefficients of the corresponding complex traveltime formulas. To solve the complex eikonal equation for the background medium, we employ an optimization scheme that we developed for solving the complex eikonal equation in isotropic media. Then, to solve the system of linearized partial differential equations for the coefficients of the complex traveltime formulas, we use the finite difference method based on the fast marching strategy. Furthermore, by applying the complex source point method and the paraxial approximation, we develop the analytic solutions of the complex eikonal equation in acoustic VTI media, both for the isotropic and elliptical anisotropic background medium. Our numerical results demonstrate the effectiveness of our derivations and illustrate the influence of the beam widths and the anisotropic parameters on the complex traveltimes.

Explanation of climate and human impacts on sediment discharge change in Darwinian hydrology: Derivation of a differential equation

NASA Astrophysics Data System (ADS)

Zhang, Jianjun; Gao, Guangyao; Fu, Bojie; Zhang, Lu

2018-04-01

The assessment for impacts of climate variability and human activities on suspended sediment yield (SSY) change has long been a question of great interest. However, the sediment generation processes are sophisticated with high nonlinearity and great uncertainty, which give rise to extreme complexity for SSY change assessment in Newtonian approach. Consequently, few approaches can be simply but widely applied to decompose impacts of climatic variability and human activities on SSY change. Thus, it is an urgent need to develop advanced methods that are simple and robust. Since that the Newtonian approach is hardly achievable due to limitation of either observations or knowledge of mechanisms, there have been repeated calls to capture the hydrologic system in Darwinian approach for hydrological change prediction or explanation. As streamflow is the carrier of suspended sediment, SSY change are thus documented in changes of sediment concentrated flow and suspended sediment concentration - water discharge (C-Q) relationships. By deduced corollaries, a differential equation of sediment discharge change was derived to explicitly decompose impacts of climate variability and human activities in Darwinian hydrology. Besides, a new form of sediment rating curves was proposed and curved as C-Q relationships and probability distribution of sediment concentrated flow. River sediment flux can be revealed by this representation, which simply elucidates mechanism of SSY generation covering a range of time scales from finer than rainfall-event to long term. By the new sediment rating curves, the differential equation was partly solved using a segmentation algorithm proposed and validated in this paper, and then was submitted to water balance framework expressed by Budyko-type equation. Thus, for catchment management, hydrologists can obtain explicit explanation of how climate variation and human activities propagate through landscape and result in sediment discharge change. The differential equation is simple and robust for widely application in sediment discharge change assessment, as only discrete data of precipitation, potential evaporation and C-Q observed at gauging stations are required.

Exact traveling wave solutions of modified KdV-Zakharov-Kuznetsov equation and viscous Burgers equation.

PubMed

Islam, Md Hamidul; Khan, Kamruzzaman; Akbar, M Ali; Salam, Md Abdus

2014-01-01

Mathematical modeling of many physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear partial differential equations (NPDEs) plays a significant role in the study of nonlinear physical phenomena. In this article, we construct the traveling wave solutions of modified KDV-ZK equation and viscous Burgers equation by using an enhanced (G '/G) -expansion method. A number of traveling wave solutions in terms of unknown parameters are obtained. Derived traveling wave solutions exhibit solitary waves when special values are given to its unknown parameters. 35C07; 35C08; 35P99.

An Expansion Formula with Higher-Order Derivatives for Fractional Operators of Variable Order

PubMed Central

Almeida, Ricardo

2013-01-01

We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve differential equations, and problems of the calculus of variations that depend on fractional derivatives of Marchaud type. PMID:24319382

Fractional spectral and pseudo-spectral methods in unbounded domains: Theory and applications

NASA Astrophysics Data System (ADS)

Khosravian-Arab, Hassan; Dehghan, Mehdi; Eslahchi, M. R.

2017-06-01

This paper is intended to provide exponentially accurate Galerkin, Petrov-Galerkin and pseudo-spectral methods for fractional differential equations on a semi-infinite interval. We start our discussion by introducing two new non-classical Lagrange basis functions: NLBFs-1 and NLBFs-2 which are based on the two new families of the associated Laguerre polynomials: GALFs-1 and GALFs-2 obtained recently by the authors in [28]. With respect to the NLBFs-1 and NLBFs-2, two new non-classical interpolants based on the associated- Laguerre-Gauss and Laguerre-Gauss-Radau points are introduced and then fractional (pseudo-spectral) differentiation (and integration) matrices are derived. Convergence and stability of the new interpolants are proved in detail. Several numerical examples are considered to demonstrate the validity and applicability of the basis functions to approximate fractional derivatives (and integrals) of some functions. Moreover, the pseudo-spectral, Galerkin and Petrov-Galerkin methods are successfully applied to solve some physical ordinary differential equations of either fractional orders or integer ones. Some useful comments from the numerical point of view on Galerkin and Petrov-Galerkin methods are listed at the end.

Differential formulation of the gyrokinetic Landau operator

DOE PAGES

Hirvijoki, Eero; Brizard, Alain J.; Pfefferlé, David

2017-01-05

Subsequent to the recent rigorous derivation of an energetically consistent gyrokinetic collision operator in the so-called Landau representation, this work investigates the possibility of finding a differential formulation of the gyrokinetic Landau collision operator. It is observed that, while a differential formulation is possible in the gyrokinetic phase space, reduction of the resulting system of partial differential equations to five dimensions via gyroaveraging poses a challenge. Finally, based on the present work, it is likely that the gyrocentre analogues of the Rosenbluth–MacDonald–Judd potential functions must be kept gyroangle dependent.

Feynman path integral application on deriving black-scholes diffusion equation for european option pricing

NASA Astrophysics Data System (ADS)

Utama, Briandhika; Purqon, Acep

2016-08-01

Path Integral is a method to transform a function from its initial condition to final condition through multiplying its initial condition with the transition probability function, known as propagator. At the early development, several studies focused to apply this method for solving problems only in Quantum Mechanics. Nevertheless, Path Integral could also apply to other subjects with some modifications in the propagator function. In this study, we investigate the application of Path Integral method in financial derivatives, stock options. Black-Scholes Model (Nobel 1997) was a beginning anchor in Option Pricing study. Though this model did not successfully predict option price perfectly, especially because its sensitivity for the major changing on market, Black-Scholes Model still is a legitimate equation in pricing an option. The derivation of Black-Scholes has a high difficulty level because it is a stochastic partial differential equation. Black-Scholes equation has a similar principle with Path Integral, where in Black-Scholes the share's initial price is transformed to its final price. The Black-Scholes propagator function then derived by introducing a modified Lagrange based on Black-Scholes equation. Furthermore, we study the correlation between path integral analytical solution and Monte-Carlo numeric solution to find the similarity between this two methods.

Second-order variational equations for N-body simulations

NASA Astrophysics Data System (ADS)

Rein, Hanno; Tamayo, Daniel

2016-07-01

First-order variational equations are widely used in N-body simulations to study how nearby trajectories diverge from one another. These allow for efficient and reliable determinations of chaos indicators such as the Maximal Lyapunov characteristic Exponent (MLE) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). In this paper we lay out the theoretical framework to extend the idea of variational equations to higher order. We explicitly derive the differential equations that govern the evolution of second-order variations in the N-body problem. Going to second order opens the door to new applications, including optimization algorithms that require the first and second derivatives of the solution, like the classical Newton's method. Typically, these methods have faster convergence rates than derivative-free methods. Derivatives are also required for Riemann manifold Langevin and Hamiltonian Monte Carlo methods which provide significantly shorter correlation times than standard methods. Such improved optimization methods can be applied to anything from radial-velocity/transit-timing-variation fitting to spacecraft trajectory optimization to asteroid deflection. We provide an implementation of first- and second-order variational equations for the publicly available REBOUND integrator package. Our implementation allows the simultaneous integration of any number of first- and second-order variational equations with the high-accuracy IAS15 integrator. We also provide routines to generate consistent and accurate initial conditions without the need for finite differencing.

The mechanical and chemical equations of motion of muscle contraction

NASA Astrophysics Data System (ADS)

Shiner, J. S.; Sieniutycz, Stanislaw

1997-11-01

Up to now no formulation of muscle contraction has provided both the chemical kinetic equations for the reactions responsible for the contraction and the mechanical equation of motion for the muscle. This has most likely been due to the lack of general formalisms for nonlinear systems with chemical-nonchemical coupling valid under the far from equilibrium conditions under which muscle operates physiologically. We have recently developed such formalisms and apply them here to the formulation of muscle contraction to obtain both the chemical and the mechanical equations. The standard formulation up to now has yielded only the dynamic equations for the chemical variables and has considered these to be functions of both time and an appropriate mechanical variable. The macroscopically observable quantities were then obtained by averaging over the mechanical variable. When attempting to derive the dynamics equations for both the chemistry and mechanics this choice of variables leads to conflicting results for the mechanical equation of motion when two different general formalisms are applied. The conflict can be resolved by choosing the variables such that both the chemical variables and the mechanical variables are considered to be functions of time alone. This adds one equation to the set of differential equations to be solved but is actually a simplification of the problem, since these equations are ordinary differential equations, not the partial differential equations of the now standard formulation, and since in this choice of variables the variables themselves are the macroscopic observables the procedure of averaging over the mechanical variable is eliminated. Furthermore, the parameters occurring in the equations at this level of description should be accessible to direct experimental determination.

Pricing geometric Asian rainbow options under fractional Brownian motion

NASA Astrophysics Data System (ADS)

Wang, Lu; Zhang, Rong; Yang, Lin; Su, Yang; Ma, Feng

2018-03-01

In this paper, we explore the pricing of the assets of Asian rainbow options under the condition that the assets have self-similar and long-range dependence characteristics. Based on the principle of no arbitrage, stochastic differential equation, and partial differential equation, we obtain the pricing formula for two-asset rainbow options under fractional Brownian motion. Next, our Monte Carlo simulation experiments show that the derived pricing formula is accurate and effective. Finally, our sensitivity analysis of the influence of important parameters, such as the risk-free rate, Hurst exponent, and correlation coefficient, on the prices of Asian rainbow options further illustrate the rationality of our pricing model.

Probabilistic Density Function Method for Stochastic ODEs of Power Systems with Uncertain Power Input

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

Wang, Peng; Barajas-Solano, David A.; Constantinescu, Emil

Wind and solar power generators are commonly described by a system of stochastic ordinary differential equations (SODEs) where random input parameters represent uncertainty in wind and solar energy. The existing methods for SODEs are mostly limited to delta-correlated random parameters (white noise). Here we use the Probability Density Function (PDF) method for deriving a closed-form deterministic partial differential equation (PDE) for the joint probability density function of the SODEs describing a power generator with time-correlated power input. The resulting PDE is solved numerically. A good agreement with Monte Carlo Simulations shows accuracy of the PDF method.

Optimal control strategy for an impulsive stochastic competition system with time delays and jumps

NASA Astrophysics Data System (ADS)

Liu, Lidan; Meng, Xinzhu; Zhang, Tonghua

2017-07-01

Driven by both white and jump noises, a stochastic delayed model with two competitive species in a polluted environment is proposed and investigated. By using the comparison theorem of stochastic differential equations and limit superior theory, sufficient conditions for persistence in mean and extinction of two species are established. In addition, we obtain that the system is asymptotically stable in distribution by using ergodic method. Furthermore, the optimal harvesting effort and the maximum of expectation of sustainable yield (ESY) are derived from Hessian matrix method and optimal harvesting theory of differential equations. Finally, some numerical simulations are provided to illustrate the theoretical results.

Fractional vector calculus and fluid mechanics

NASA Astrophysics Data System (ADS)

Lazopoulos, Konstantinos A.; Lazopoulos, Anastasios K.

2017-04-01

Basic fluid mechanics equations are studied and revised under the prism of fractional continuum mechanics (FCM), a very promising research field that satisfies both experimental and theoretical demands. The geometry of the fractional differential has been clarified corrected and the geometry of the fractional tangent spaces of a manifold has been studied in Lazopoulos and Lazopoulos (Lazopoulos KA, Lazopoulos AK. Progr. Fract. Differ. Appl. 2016, 2, 85-104), providing the bases of the missing fractional differential geometry. Therefore, a lot can be contributed to fractional hydrodynamics: the basic fractional fluid equations (Navier Stokes, Euler and Bernoulli) are derived and fractional Darcy's flow in porous media is studied.

Further Development of a New, Flux-Conserving Newton Scheme for the Navier-Stokes Equations

NASA Technical Reports Server (NTRS)

Scott, James R.

1996-01-01

This paper is one of a series of papers describing the development of a new numerical approach for solving the steady Navier-Stokes equations. The key features in the current development are (1) the discrete representation of the dependent variables by way of high order polynomial expansions, (2) the retention of all derivatives in the expansions as unknowns to be explicitly solved for, (3) the automatic balancing of fluxes at cell interfaces, and (4) the discrete simulation of both the integral and differential forms of the governing equations. The main purpose of this paper is, first, to provide a systematic and rigorous derivation of the conditions that are used to simulate the differential form of the Navier-Stokes equations, and second, to extend our previously-presented internal flow scheme to external flows and nonuniform grids. Numerical results are presented for high Reynolds number flow (Re = 100,000) around a finite flat plate, and detailed comparisons are made with the Blasius flat plate solution and Goldstein wake solution. It is shown that the error in the streamwise velocity decreases like r(sup alpha)(Delta)y(exp 2), where alpha approx. 0.25 and r = delta(y)/delta(x) is the grid aspect ratio.

From quantum stochastic differential equations to Gisin-Percival state diffusion

NASA Astrophysics Data System (ADS)

Parthasarathy, K. R.; Usha Devi, A. R.

2017-08-01

Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ (L2(R+ ) ⊗(Cn⊕Cn ) ) and the Hilbert space L2(μ ) , where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion {B (t ) ,t ≥0 } , we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, J. Phys. A 167, 315 (1992)]. This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a randomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.

A stochastic hybrid systems based framework for modeling dependent failure processes

PubMed Central

Fan, Mengfei; Zeng, Zhiguo; Zio, Enrico; Kang, Rui; Chen, Ying

2017-01-01

In this paper, we develop a framework to model and analyze systems that are subject to dependent, competing degradation processes and random shocks. The degradation processes are described by stochastic differential equations, whereas transitions between the system discrete states are triggered by random shocks. The modeling is, then, based on Stochastic Hybrid Systems (SHS), whose state space is comprised of a continuous state determined by stochastic differential equations and a discrete state driven by stochastic transitions and reset maps. A set of differential equations are derived to characterize the conditional moments of the state variables. System reliability and its lower bounds are estimated from these conditional moments, using the First Order Second Moment (FOSM) method and Markov inequality, respectively. The developed framework is applied to model three dependent failure processes from literature and a comparison is made to Monte Carlo simulations. The results demonstrate that the developed framework is able to yield an accurate estimation of reliability with less computational costs compared to traditional Monte Carlo-based methods. PMID:28231313

A stochastic hybrid systems based framework for modeling dependent failure processes.

PubMed

Fan, Mengfei; Zeng, Zhiguo; Zio, Enrico; Kang, Rui; Chen, Ying

2017-01-01

In this paper, we develop a framework to model and analyze systems that are subject to dependent, competing degradation processes and random shocks. The degradation processes are described by stochastic differential equations, whereas transitions between the system discrete states are triggered by random shocks. The modeling is, then, based on Stochastic Hybrid Systems (SHS), whose state space is comprised of a continuous state determined by stochastic differential equations and a discrete state driven by stochastic transitions and reset maps. A set of differential equations are derived to characterize the conditional moments of the state variables. System reliability and its lower bounds are estimated from these conditional moments, using the First Order Second Moment (FOSM) method and Markov inequality, respectively. The developed framework is applied to model three dependent failure processes from literature and a comparison is made to Monte Carlo simulations. The results demonstrate that the developed framework is able to yield an accurate estimation of reliability with less computational costs compared to traditional Monte Carlo-based methods.

Modelling Evolutionary Algorithms with Stochastic Differential Equations.

PubMed

Heredia, Jorge Pérez

2017-11-20

There has been renewed interest in modelling the behaviour of evolutionary algorithms (EAs) by more traditional mathematical objects, such as ordinary differential equations or Markov chains. The advantage is that the analysis becomes greatly facilitated due to the existence of well established methods. However, this typically comes at the cost of disregarding information about the process. Here, we introduce the use of stochastic differential equations (SDEs) for the study of EAs. SDEs can produce simple analytical results for the dynamics of stochastic processes, unlike Markov chains which can produce rigorous but unwieldy expressions about the dynamics. On the other hand, unlike ordinary differential equations (ODEs), they do not discard information about the stochasticity of the process. We show that these are especially suitable for the analysis of fixed budget scenarios and present analogues of the additive and multiplicative drift theorems from runtime analysis. In addition, we derive a new more general multiplicative drift theorem that also covers non-elitist EAs. This theorem simultaneously allows for positive and negative results, providing information on the algorithm's progress even when the problem cannot be optimised efficiently. Finally, we provide results for some well-known heuristics namely Random Walk (RW), Random Local Search (RLS), the (1+1) EA, the Metropolis Algorithm (MA), and the Strong Selection Weak Mutation (SSWM) algorithm.

Hidden symmetry in the presence of fluxes

NASA Astrophysics Data System (ADS)

Kubizňák, David; Warnick, Claude M.; Krtouš, Pavel

2011-03-01

We derive the most general first-order symmetry operator for the Dirac equation coupled to arbitrary fluxes. Such an operator is given in terms of an inhomogeneous form ω which is a solution to a coupled system of first-order partial differential equations which we call the generalized conformal Killing-Yano system. Except trivial fluxes, solutions of this system are subject to additional constraints. We discuss various special cases of physical interest. In particular, we demonstrate that in the case of a Dirac operator coupled to the skew symmetric torsion and U(1) field, the system of generalized conformal Killing-Yano equations decouples into the homogeneous conformal Killing-Yano equations with torsion introduced in D. Kubiznak et al. (2009) [8] and the symmetry operator is essentially the one derived in T. Houri et al. (2010) [9]. We also discuss the Dirac field coupled to a scalar potential and in the presence of 5-form and 7-form fluxes.

Continual approach at T=0 in the mean field theory of incommensurate magnetic states in the frustrated Heisenberg ferromagnet with an easy axis anisotropy

NASA Astrophysics Data System (ADS)

Martynov, S. N.; Tugarinov, V. I.; Martynov, A. S.

2017-10-01

The algorithm of approximate solution was developed for the differential equation describing the anharmonical change of the spin orientation angle in the model of ferromagnet with the exchange competition between nearest and next nearest magnetic neighbors and the easy axis exchange anisotropy. The equation was obtained from the collinearity constraint on the discrete lattice. In the low anharmonicity approximation the equation is resulted to an autonomous form and is integrated in quadratures. The obvious dependence of the angle velocity and second derivative of angle from angle and initial condition was derived by expanding the first integral of the equation in the Taylor series in vicinity of initial condition. The ground state of the soliton solutions was calculated by a numerical minimization of the energy integral. The evaluation of the used approximation was made for a triple point of the phase diagram.

Exact closed-form solution of the hyperbolic equation of string vibrations with material relaxation properties taken into account

NASA Astrophysics Data System (ADS)

Kudinov, I. V.; Kudinov, V. A.

2014-09-01

The differential equation of damped string vibrations was obtained with the finite speed of extension and strain propagation in the Hooke's law formula taken into account. In contrast to the well-known equations, the obtained equation contains the first and third time derivatives of the displacement and the mixed derivative with respect to the space and time variables. Separation of variables was used to obtain its exact closed-form solution, whose analysis showed that, for large values of the relaxation coefficient, the string return to the initial state after its escape from equilibrium is accompanied by high-frequency low-amplitude damped vibrations, which occur on the initial time interval only in the region of positive displacements. And in the limit, for some large values of the relaxation coefficient, the string return to the initial state occurs practically without any oscillatory process.

Nonlinear analysis of 0-3 polarized PLZT microplate based on the new modified couple stress theory

NASA Astrophysics Data System (ADS)

Wang, Liming; Zheng, Shijie

2018-02-01

In this study, based on the new modified couple stress theory, the size- dependent model for nonlinear bending analysis of a pure 0-3 polarized PLZT plate is developed for the first time. The equilibrium equations are derived from a variational formulation based on the potential energy principle and the new modified couple stress theory. The Galerkin method is adopted to derive the nonlinear algebraic equations from governing differential equations. And then the nonlinear algebraic equations are solved by using Newton-Raphson method. After simplification, the new model includes only a material length scale parameter. In addition, numerical examples are carried out to study the effect of material length scale parameter on the nonlinear bending of a simply supported pure 0-3 polarized PLZT plate subjected to light illumination and uniform distributed load. The results indicate the new model is able to capture the size effect and geometric nonlinearity.

High-order asynchrony-tolerant finite difference schemes for partial differential equations

NASA Astrophysics Data System (ADS)

Aditya, Konduri; Donzis, Diego A.

2017-12-01

Synchronizations of processing elements (PEs) in massively parallel simulations, which arise due to communication or load imbalances between PEs, significantly affect the scalability of scientific applications. We have recently proposed a method based on finite-difference schemes to solve partial differential equations in an asynchronous fashion - synchronization between PEs is relaxed at a mathematical level. While standard schemes can maintain their stability in the presence of asynchrony, their accuracy is drastically affected. In this work, we present a general methodology to derive asynchrony-tolerant (AT) finite difference schemes of arbitrary order of accuracy, which can maintain their accuracy when synchronizations are relaxed. We show that there are several choices available in selecting a stencil to derive these schemes and discuss their effect on numerical and computational performance. We provide a simple classification of schemes based on the stencil and derive schemes that are representative of different classes. Their numerical error is rigorously analyzed within a statistical framework to obtain the overall accuracy of the solution. Results from numerical experiments are used to validate the performance of the schemes.

Theory and observation of electromagnetic ion cyclotron triggered emissions in the magnetosphere

NASA Astrophysics Data System (ADS)

Omura, Yoshiharu; Pickett, Jolene; Grison, Benjamin; Santolik, Ondrej; Dandouras, Iannis; Engebretson, Mark; Décréau, Pierrette M. E.; Masson, Arnaud

2010-07-01

We develop a nonlinear wave growth theory of electromagnetic ion cyclotron (EMIC) triggered emissions observed in the inner magnetosphere. We first derive the basic wave equations from Maxwell's equations and the momentum equations for the electrons and ions. We then obtain equations that describe the nonlinear dynamics of resonant protons interacting with an EMIC wave. The frequency sweep rate of the wave plays an important role in forming the resonant current that controls the wave growth. Assuming an optimum condition for the maximum growth rate as an absolute instability at the magnetic equator and a self-sustaining growth condition for the wave propagating from the magnetic equator, we obtain a set of ordinary differential equations that describe the nonlinear evolution of a rising tone emission generated at the magnetic equator. Using the physical parameters inferred from the wave, particle, and magnetic field data measured by the Cluster spacecraft, we determine the dispersion relation for the EMIC waves. Integrating the differential equations numerically, we obtain a solution for the time variation of the amplitude and frequency of a rising tone emission at the equator. Assuming saturation of the wave amplitude, as is found in the observations, we find good agreement between the numerical solutions and the wave spectrum of the EMIC triggered emissions.

Soliton evolution and radiation loss for the sine-Gordon equation.

PubMed

Smyth, N F; Worthy, A L

1999-08-01

An approximate method for describing the evolution of solitonlike initial conditions to solitons for the sine-Gordon equation is developed. This method is based on using a solitonlike pulse with variable parameters in an averaged Lagrangian for the sine-Gordon equation. This averaged Lagrangian is then used to determine ordinary differential equations governing the evolution of the pulse parameters. The pulse evolves to a steady soliton by shedding dispersive radiation. The effect of this radiation is determined by examining the linearized sine-Gordon equation and loss terms are added to the variational equations derived from the averaged Lagrangian by using the momentum and energy conservation equations for the sine-Gordon equation. Solutions of the resulting approximate equations, which include loss, are found to be in good agreement with full numerical solutions of the sine-Gordon equation.

Conformal dynamics of precursors to fracture

NASA Astrophysics Data System (ADS)

Barra, F.; Herrera, M.; Procaccia, I.

2003-09-01

An exact integro-differential equation for the conformal map from the unit circle to the boundary of an evolving cavity in a stressed 2-dimensional solid is derived. This equation provides an accurate description of the dynamics of precursors to fracture when surface diffusion is important. The solution predicts the creation of sharp grooves that eventually lead to material failure via rapid fracture. Solutions of the new equation are demonstrated for the dynamics of an elliptical cavity and the stability of a circular cavity under biaxial stress, including the effects of surface stress.

Action principle for Coulomb collisions in plasmas

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

Hirvijoki, Eero

In this study, an action principle for Coulomb collisions in plasmas is proposed. Although no natural Lagrangian exists for the Landau-Fokker-Planck equation, an Eulerian variational formulation is found considering the system of partial differential equations that couple the distribution function and the Rosenbluth-MacDonald-Judd potentials. Conservation laws are derived after generalizing the energy-momentum stress tensor for second order Lagrangians and, in the case of a test-particle population in a given plasma background, the action principle is shown to correspond to the Langevin equation for individual particles.

Towards an exact relativistic theory of Earth's geoid undulation

NASA Astrophysics Data System (ADS)

Kopeikin, Sergei M.; Mazurova, Elena M.; Karpik, Alexander P.

2015-08-01

The present paper extends the Newtonian concept of the geoid in classic geodesy towards the realm of general relativity by utilizing the covariant geometric methods of the perturbation theory of curved manifolds. It yields a covariant definition of the anomalous (disturbing) gravity potential and formulates differential equation for it in the form of a covariant Laplace equation. The paper also derives the Bruns equation for calculation of geoid's height with full account for relativistic effects beyond the Newtonian approximation. A brief discussion of the relativistic Bruns formula is provided.

Action principle for Coulomb collisions in plasmas

DOE PAGES

Hirvijoki, Eero

2016-09-14

In this study, an action principle for Coulomb collisions in plasmas is proposed. Although no natural Lagrangian exists for the Landau-Fokker-Planck equation, an Eulerian variational formulation is found considering the system of partial differential equations that couple the distribution function and the Rosenbluth-MacDonald-Judd potentials. Conservation laws are derived after generalizing the energy-momentum stress tensor for second order Lagrangians and, in the case of a test-particle population in a given plasma background, the action principle is shown to correspond to the Langevin equation for individual particles.

Time-fractional Cahn-Allen and time-fractional Klein-Gordon equations: Lie symmetry analysis, explicit solutions and convergence analysis

NASA Astrophysics Data System (ADS)

Inc, Mustafa; Yusuf, Abdullahi; Isa Aliyu, Aliyu; Baleanu, Dumitru

2018-03-01

This research analyzes the symmetry analysis, explicit solutions and convergence analysis to the time fractional Cahn-Allen (CA) and time-fractional Klein-Gordon (KG) equations with Riemann-Liouville (RL) derivative. The time fractional CA and time fractional KG are reduced to respective nonlinear ordinary differential equation of fractional order. We solve the reduced fractional ODEs using an explicit power series method. The convergence analysis for the obtained explicit solutions are investigated. Some figures for the obtained explicit solutions are also presented.

Calculation of the flow field including boundary layer effects for supersonic mixed compression inlets at angles of attack

NASA Technical Reports Server (NTRS)

Vadyak, J.; Hoffman, J. D.

1982-01-01

The flow field in supersonic mixed compression aircraft inlets at angle of attack is calculated. A zonal modeling technique is employed to obtain the solution which divides the flow field into different computational regions. The computational regions consist of a supersonic core flow, boundary layer flows adjacent to both the forebody/centerbody and cowl contours, and flow in the shock wave boundary layer interaction regions. The zonal modeling analysis is described and some computational results are presented. The governing equations for the supersonic core flow form a hyperbolic system of partial differential equations. The equations for the characteristic surfaces and the compatibility equations applicable along these surfaces are derived. The characteristic surfaces are the stream surfaces, which are surfaces composed of streamlines, and the wave surfaces, which are surfaces tangent to a Mach conoid. The compatibility equations are expressed as directional derivatives along streamlines and bicharacteristics, which are the lines of tangency between a wave surface and a Mach conoid.

The continuous adjoint approach to the k-ε turbulence model for shape optimization and optimal active control of turbulent flows

NASA Astrophysics Data System (ADS)

Papoutsis-Kiachagias, E. M.; Zymaris, A. S.; Kavvadias, I. S.; Papadimitriou, D. I.; Giannakoglou, K. C.

2015-03-01

The continuous adjoint to the incompressible Reynolds-averaged Navier-Stokes equations coupled with the low Reynolds number Launder-Sharma k-ε turbulence model is presented. Both shape and active flow control optimization problems in fluid mechanics are considered, aiming at minimum viscous losses. In contrast to the frequently used assumption of frozen turbulence, the adjoint to the turbulence model equations together with appropriate boundary conditions are derived, discretized and solved. This is the first time that the adjoint equations to the Launder-Sharma k-ε model have been derived. Compared to the formulation that neglects turbulence variations, the impact of additional terms and equations is evaluated. Sensitivities computed using direct differentiation and/or finite differences are used for comparative purposes. To demonstrate the need for formulating and solving the adjoint to the turbulence model equations, instead of merely relying upon the 'frozen turbulence assumption', the gain in the optimization turnaround time offered by the proposed method is quantified.

New envelope solitons for Gerdjikov-Ivanov model in nonlinear fiber optics

NASA Astrophysics Data System (ADS)

Triki, Houria; Alqahtani, Rubayyi T.; Zhou, Qin; Biswas, Anjan

2017-11-01

Exact soliton solutions in a class of derivative nonlinear Schrödinger equations including a pure quintic nonlinearity are investigated. By means of the coupled amplitude-phase formulation, we derive a nonlinear differential equation describing the evolution of the wave amplitude in the non-Kerr quintic media. The resulting amplitude equation is then solved to get exact analytical chirped bright, kink, antikink, and singular soliton solutions for the model. It is also shown that the nonlinear chirp associated with these solitons is crucially dependent on the wave intensity and related to self-steepening and group velocity dispersion parameters. Parametric conditions on physical parameters for the existence of chirped solitons are also presented. These localized structures exist due to a balance among quintic nonlinearity, group velocity dispersion, and self-steepening effects.

Closed form solutions of two time fractional nonlinear wave equations

NASA Astrophysics Data System (ADS)

Akbar, M. Ali; Ali, Norhashidah Hj. Mohd.; Roy, Ripan

2018-06-01

In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized (G‧ / G) -expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics.

Approximate solution of space and time fractional higher order phase field equation

NASA Astrophysics Data System (ADS)

Shamseldeen, S.

2018-03-01

This paper is concerned with a class of space and time fractional partial differential equation (STFDE) with Riesz derivative in space and Caputo in time. The proposed STFDE is considered as a generalization of a sixth-order partial phase field equation. We describe the application of the optimal homotopy analysis method (OHAM) to obtain an approximate solution for the suggested fractional initial value problem. An averaged-squared residual error function is defined and used to determine the optimal convergence control parameter. Two numerical examples are studied, considering periodic and non-periodic initial conditions, to justify the efficiency and the accuracy of the adopted iterative approach. The dependence of the solution on the order of the fractional derivative in space and time and model parameters is investigated.

General PFG signal attenuation expressions for anisotropic anomalous diffusion by modified-Bloch equations

NASA Astrophysics Data System (ADS)

Lin, Guoxing

2018-05-01

Anomalous diffusion exists widely in polymer and biological systems. Pulsed-field gradient (PFG) anomalous diffusion is complicated, especially in the anisotropic case where limited research has been reported. A general PFG signal attenuation expression, including the finite gradient pulse (FGPW) effect for free general anisotropic fractional diffusion { 0 < α , β ≤ 2 } based on the fractional derivative, has not been obtained, where α and β are time and space derivative orders. It is essential to derive a general PFG signal attenuation expression including the FGPW effect for PFG anisotropic anomalous diffusion research. In this paper, two recently developed modified-Bloch equations, the fractal differential modified-Bloch equation and the fractional integral modified-Bloch equation, were extended to obtain general PFG signal attenuation expressions for anisotropic anomalous diffusion. Various cases of PFG anisotropic anomalous diffusion were investigated, including coupled and uncoupled anisotropic anomalous diffusion. The continuous-time random walk (CTRW) simulation was also carried out to support the theoretical results. The theory and the CTRW simulation agree with each other. The obtained signal attenuation expressions and the three-dimensional fractional modified-Bloch equations are important for analyzing PFG anisotropic anomalous diffusion in NMR and MRI.

Finite Temperature Densities via the S-Function Method with Application to Electron Screening in Plasmas

NASA Astrophysics Data System (ADS)

Watrous, Mitchell James

1997-12-01

A new approach to the Green's-function method for the calculation of equilibrium densities within the finite temperature, Kohn-Sham formulation of density functional theory is presented, which extends the method to all temperatures. The contour of integration in the complex energy plane is chosen such that the density is given by a sum of Green's function differences evaluated at the Matsubara frequencies, rather than by the calculation and summation of Kohn-Sham single-particle wave functions. The Green's functions are written in terms of their spectral representation and are calculated as the solutions of their defining differential equations. These differential equations are boundary value problems as opposed to the standard eigenvalue problems. For large values of the complex energy, the differential equations are further simplified from second to first-order by writing the Green's functions in terms of logarithmic derivatives. An asymptotic expression for the Green's functions is derived, which allows the sum over Matsubara poles to be approximated. The method is applied to the screening of nuclei by electrons in finite temperature plasmas. To demonstrate the method's utility, and to illustrate its advantages, the results of previous wave function type calculations for protons and neon nuclei are reproduced. The method is also used to formulate a new screening model for fusion reactions in the solar core, and the predicted reaction rate enhancements factors are compared with existing models.

Working With the Wave Equation in Aeroacoustics: The Pleasures of Generalized Functions

NASA Technical Reports Server (NTRS)

Farassat, F.; Brentner, Kenneth S.; Dunn, mark H.

2007-01-01

The theme of this paper is the applications of generalized function (GF) theory to the wave equation in aeroacoustics. We start with a tutorial on GFs with particular emphasis on viewing functions as continuous linear functionals. We next define operations on GFs. The operation of interest to us in this paper is generalized differentiation. We give many applications of generalized differentiation, particularly for the wave equation. We discuss the use of GFs in finding Green s function and some subtleties that only GF theory can clarify without ambiguities. We show how the knowledge of the Green s function of an operator L in a given domain D can allow us to solve a whole range of problems with operator L for domains situated within D by the imbedding method. We will show how we can use the imbedding method to find the Kirchhoff formulas for stationary and moving surfaces with ease and elegance without the use of the four-dimensional Green s theorem, which is commonly done. Other subjects covered are why the derivatives in conservation laws should be viewed as generalized derivatives and what are the consequences of doing this. In particular we show how we can imbed a problem in a larger domain for the identical differential equation for which the Green s function is known. The primary purpose of this paper is to convince the readers that GF theory is absolutely essential in aeroacoustics because of its powerful operational properties. Furthermore, learning the subject and using it can be fun.

Variational principles for stochastic fluid dynamics

PubMed Central

Holm, Darryl D.

2015-01-01

This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The paper proceeds by taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their Itô representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models. As with deterministic ideal flows, motion along the stochastic Stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows. However, these Stratonovich properties are not apparent in the equivalent Itô representation, because they are disguised by the quadratic covariation drift term arising in the Stratonovich to Itô transformation. This term is a geometric generalization of the quadratic covariation drift term already found for scalar densities in Stratonovich's famous 1966 paper. The paper also derives motion equations for two examples of stochastic geophysical fluid dynamics; namely, the Euler–Boussinesq and quasi-geostropic approximations. PMID:27547083

Oblique scattering from radially inhomogeneous dielectric cylinders: An exact Volterra integral equation formulation

NASA Astrophysics Data System (ADS)

Tsalamengas, John L.

2018-07-01

We study plane-wave electromagnetic scattering by radially and strongly inhomogeneous dielectric cylinders at oblique incidence. The method of analysis relies on an exact reformulation of the underlying field equations as a first-order 4 × 4 system of differential equations and on the ability to restate the associated initial-value problem in the form of a system of coupled linear Volterra integral equations of the second kind. The integral equations so derived are discretized via a sophisticated variant of the Nyström method. The proposed method yields results accurate up to machine precision without relying on approximations. Numerical results and case studies ably demonstrate the efficiency and high accuracy of the algorithms.

The probability density function (PDF) of Lagrangian Turbulence

NASA Astrophysics Data System (ADS)

Birnir, B.

2012-12-01

The statistical theory of Lagrangian turbulence is derived from the stochastic Navier-Stokes equation. Assuming that the noise in fully-developed turbulence is a generic noise determined by the general theorems in probability, the central limit theorem and the large deviation principle, we are able to formulate and solve the Kolmogorov-Hopf equation for the invariant measure of the stochastic Navier-Stokes equations. The intermittency corrections to the scaling exponents of the structure functions require a multiplicative (multipling the fluid velocity) noise in the stochastic Navier-Stokes equation. We let this multiplicative noise, in the equation, consists of a simple (Poisson) jump process and then show how the Feynmann-Kac formula produces the log-Poissonian processes, found by She and Leveque, Waymire and Dubrulle. These log-Poissonian processes give the intermittency corrections that agree with modern direct Navier-Stokes simulations (DNS) and experiments. The probability density function (PDF) plays a key role when direct Navier-Stokes simulations or experimental results are compared to theory. The statistical theory of turbulence is determined, including the scaling of the structure functions of turbulence, by the invariant measure of the Navier-Stokes equation and the PDFs for the various statistics (one-point, two-point, N-point) can be obtained by taking the trace of the corresponding invariant measures. Hopf derived in 1952 a functional equation for the characteristic function (Fourier transform) of the invariant measure. In distinction to the nonlinear Navier-Stokes equation, this is a linear functional differential equation. The PDFs obtained from the invariant measures for the velocity differences (two-point statistics) are shown to be the four parameter generalized hyperbolic distributions, found by Barndorff-Nilsen. These PDF have heavy tails and a convex peak at the origin. A suitable projection of the Kolmogorov-Hopf equations is the differential equation determining the generalized hyperbolic distributions. Then we compare these PDFs with DNS results and experimental data.

Prolongation structures of nonlinear evolution equations. II

NASA Technical Reports Server (NTRS)

Estabrook, F. B.; Wahlquist, H. D.

1976-01-01

The prolongation structure of a closed ideal of exterior differential forms is further discussed, and its use illustrated by application to an ideal (in six dimensions) representing the cubically nonlinear Schroedinger equation. The prolongation structure in this case is explicitly given, and recurrence relations derived which support the conjecture that the structure is open - i.e., does not terminate as a set of structure relations of a finite-dimensional Lie group. We introduce the use of multiple pseudopotentials to generate multiple Baecklund transformation, and derive the double Baecklund transformation. This symmetric transformation concisely expresses the (usually conjectured) theorem of permutability, which must consequently apply to all solutions irrespective of asymptotic constraints.

Mixed finite-difference scheme for free vibration analysis of noncircular cylinders

NASA Technical Reports Server (NTRS)

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

1973-01-01

A mixed finite-difference scheme is presented for the free-vibration analysis of simply supported closed noncircular cylindrical shells. The problem is formulated in terms of eight first-order differential equations in the circumferential coordinate which possess a symmetric coefficient matrix and are free of the derivatives of the elastic and geometric characteristics of the shell. In the finite-difference discretization, two interlacing grids are used for the different fundamental unknowns in such a way as to avoid averaging in the difference-quotient expressions used for the first derivative. The resulting finite-difference equations are symmetric. The inverse-power method is used for obtaining the eigenvalues and eigenvectors.

General equation for the differential pathlength factor of the frontal human head depending on wavelength and age.

PubMed

Scholkmann, Felix; Wolf, Martin

2013-10-01

Continuous-wave near-infrared spectroscopy and near-infrared imaging enable the measurement of relative concentration changes in oxy- and deoxyhemoglobin and thus hemodynamics and oxygenation. The accuracy of determined changes depends mainly on the modeling of the light transport through the probed tissue. Due to the highly scattering nature of tissue, the light path is longer than the source-detector separation (d). This is incorporated in modeling by multiplying d by a differential pathlength factor (DPF) which depends on several factors such as wavelength, age of the subject, and type of tissue. In the present work, we derive a general DPF equation for the frontal human head, incorporating dependency on wavelength and age, based on published data. We validated the equation using different data sets of experimentally determined DPFs from six independent studies.

Simulation of elastic wave propagation using cellular automata and peridynamics, and comparison with experiments

DOE PAGES

Nishawala, Vinesh V.; Ostoja-Starzewski, Martin; Leamy, Michael J.; ...

2015-09-10

Peridynamics is a non-local continuum mechanics formulation that can handle spatial discontinuities as the governing equations are integro-differential equations which do not involve gradients such as strains and deformation rates. This paper employs bond-based peridynamics. Cellular Automata is a local computational method which, in its rectangular variant on interior domains, is mathematically equivalent to the central difference finite difference method. However, cellular automata does not require the derivation of the governing partial differential equations and provides for common boundary conditions based on physical reasoning. Both methodologies are used to solve a half-space subjected to a normal load, known as Lamb’smore » Problem. The results are compared with theoretical solution from classical elasticity and experimental results. Furthermore, this paper is used to validate our implementation of these methods.« less

Quantum Computer Circuit Analysis and Design

DTIC Science & Technology

2009-02-01

is a first order nonlinear differential matrix equation of the Lax type. This report gives derivations of the Levi - Civita connection, Riemann...directions on the manifold not easily simulated by local gates. In this way, basic differential geometric concepts such as the Levi - Civita connection...and two - body terms, and Q(H) contains more than two - body terms. Thus ),()( HQHPH  (1) in which P and Q are superoperators (matrices) acting on

Dynamics and control for Constrained Multibody Systems modeled with Maggi's equation: Application to Differential Mobile Robots Partll

NASA Astrophysics Data System (ADS)

Amengonu, Yawo H.; Kakad, Yogendra P.

2014-07-01

Quasivelocity techniques were applied to derive the dynamics of a Differential Wheeled Mobile Robot (DWMR) in the companion paper. The present paper formulates a control system design for trajectory tracking of this class of robots. The method develops a feedback linearization technique for the nonlinear system using dynamic extension algorithm. The effectiveness of the nonlinear controller is illustrated with simulation example.

Control of Stirling engine. Simplified, compressible model

NASA Astrophysics Data System (ADS)

Plotnikov, P. I.; Sokołowski, J.; Żochowski, A.

2016-06-01

A one-dimensional free boundary problem on a motion of a heavy piston in a tube filled with viscous gas is considered. The system of governing equations and boundary conditions is derived. The obtained system of differential equations can be regarded as a mathematical model of an exterior combustion engine. The existence of a weak solution to this model is proved. The problem of maximization of the total work of the engine is considered.

On the renewal risk model under a threshold strategy

NASA Astrophysics Data System (ADS)

Dong, Yinghui; Wang, Guojing; Yuen, Kam C.

2009-08-01

In this paper, we consider the renewal risk process under a threshold dividend payment strategy. For this model, the expected discounted dividend payments and the Gerber-Shiu expected discounted penalty function are investigated. Integral equations, integro-differential equations and some closed form expressions for them are derived. When the claims are exponentially distributed, it is verified that the expected penalty of the deficit at ruin is proportional to the ruin probability.

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

Lie symmetry analysis and reduction for exact solution of (2+1)-dimensional Bogoyavlensky-Konopelchenko equation by geometric approach

NASA Astrophysics Data System (ADS)

Ray, S. Saha

2018-04-01

In this paper, the symmetry analysis and similarity reduction of the (2+1)-dimensional Bogoyavlensky-Konopelchenko (B-K) equation are investigated by means of the geometric approach of an invariance group, which is equivalent to the classical Lie symmetry method. Using the extended Harrison and Estabrook’s differential forms approach, the infinitesimal generators for (2+1)-dimensional B-K equation are obtained. Firstly, the vector field associated with the Lie group of transformation is derived. Then the symmetry reduction and the corresponding explicit exact solution of (2+1)-dimensional B-K equation is obtained.

High-frequency sound waves to eliminate a horizon in the mixmaster universe.

NASA Technical Reports Server (NTRS)

Chitre, D. M.

1972-01-01

From the linear wave equation for small-amplitude sound waves in a curved space-time, there is derived a geodesiclike differential equation for sound rays to describe the motion of wave packets. These equations are applied in the generic, nonrotating, homogeneous closed-model universe (the 'mixmaster universe,' Bianchi type IX). As for light rays described by Doroshkevich and Novikov (DN), these sound rays can circumnavigate the universe near the singularity to remove particle horizons only for a small class of these models and in special directions. Although these results parallel those of DN, different Hamiltonian methods are used for treating the Einstein equations.

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.

An investigation of the accuracy of the Merkel equation for evaporative cooling tower calculations. Waste heat management

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

Yadigaroglu, G.; Pastor, E.J.

1974-01-01

The exact differential equations governing heat and mass transfer and air flow in an evaporative, natural-draft cooling tower are presented. The Merkel equation is then derived starting from this exact formulation and showing all the approximations involved. The Merkel formulation lumps the sensible and the latent heat transfer together and considers a single enthalpy-difference driving force for the total heat transfer. The effect of the approximations inherent in the Merkel equation is investigated and analyzed by a series of parametric numerical calculations of cooling tower performance under various ambient conditions and load conditions.

Constructing analytic solutions on the Tricomi equation

NASA Astrophysics Data System (ADS)

Ghiasi, Emran Khoshrouye; Saleh, Reza

2018-04-01

In this paper, homotopy analysis method (HAM) and variational iteration method (VIM) are utilized to derive the approximate solutions of the Tricomi equation. Afterwards, the HAM is optimized to accelerate the convergence of the series solution by minimizing its square residual error at any order of the approximation. It is found that effect of the optimal values of auxiliary parameter on the convergence of the series solution is not negligible. Furthermore, the present results are found to agree well with those obtained through a closed-form equation available in the literature. To conclude, it is seen that the two are effective to achieve the solution of the partial differential equations.

Traveling waves and conservation laws for highly nonlinear wave equations modeling Hertz chains

NASA Astrophysics Data System (ADS)

Przedborski, Michelle; Anco, Stephen C.

2017-09-01

A highly nonlinear, fourth-order wave equation that models the continuum theory of long wavelength pulses in weakly compressed, homogeneous, discrete chains with a general power-law contact interaction is studied. For this wave equation, all solitary wave solutions and all nonlinear periodic wave solutions, along with all conservation laws, are derived. The solutions are explicitly parameterized in terms of the asymptotic value of the wave amplitude in the case of solitary waves and the peak of the wave amplitude in the case of nonlinear periodic waves. All cases in which the solution expressions can be stated in an explicit analytic form using elementary functions are worked out. In these cases, explicit expressions for the total energy and total momentum for all solutions are obtained as well. The derivation of the solutions uses the conservation laws combined with an energy analysis argument to reduce the wave equation directly to a separable first-order differential equation that determines the wave amplitude in terms of the traveling wave variable. This method can be applied more generally to other highly nonlinear wave equations.

Numerical studies of the thermal design sensitivity calculation for a reaction-diffusion system with discontinuous derivatives

NASA Technical Reports Server (NTRS)

Hou, Jean W.; Sheen, Jeen S.

1987-01-01

The aim of this study is to find a reliable numerical algorithm to calculate thermal design sensitivities of a transient problem with discontinuous derivatives. The thermal system of interest is a transient heat conduction problem related to the curing process of a composite laminate. A logical function which can smoothly approximate the discontinuity is introduced to modify the system equation. Two commonly used methods, the adjoint variable method and the direct differentiation method, are then applied to find the design derivatives of the modified system. The comparisons of numerical results obtained by these two methods demonstrate that the direct differentiation method is a better choice to be used in calculating thermal design sensitivity.

A structure-preserving method for a class of nonlinear dissipative wave equations with Riesz space-fractional derivatives

NASA Astrophysics Data System (ADS)

Macías-Díaz, J. E.

2017-12-01

In this manuscript, we consider an initial-boundary-value problem governed by a (1 + 1)-dimensional hyperbolic partial differential equation with constant damping that generalizes many nonlinear wave equations from mathematical physics. The model considers the presence of a spatial Laplacian of fractional order which is defined in terms of Riesz fractional derivatives, as well as the inclusion of a generic continuously differentiable potential. It is known that the undamped regime has an associated positive energy functional, and we show here that it is preserved throughout time under suitable boundary conditions. To approximate the solutions of this model, we propose a finite-difference discretization based on fractional centered differences. Some discrete quantities are proposed in this work to estimate the energy functional, and we show that the numerical method is capable of conserving the discrete energy under the same boundary conditions for which the continuous model is conservative. Moreover, we establish suitable computational constraints under which the discrete energy of the system is positive. The method is consistent of second order, and is both stable and convergent. The numerical simulations shown here illustrate the most important features of our numerical methodology.

A new flux-conserving numerical scheme for the steady, incompressible Navier-Stokes equations

NASA Technical Reports Server (NTRS)

Scott, James R.

1994-01-01

This paper is concerned with the continued development of a new numerical method, the space-time solution element (STS) method, for solving conservation laws. The present work focuses on the two-dimensional, steady, incompressible Navier-Stokes equations. Using first an integral approach, and then a differential approach, the discrete flux conservation equations presented in a recent paper are rederived. Here a simpler method for determining the flux expressions at cell interfaces is given; a systematic and rigorous derivation of the conditions used to simulate the differential form of the governing conservation law(s) is provided; necessary and sufficient conditions for a discrete approximation to satisfy a conservation law in E2 are derived; and an estimate of the local truncation error is given. A specific scheme is then constructed for the solution of the thin airfoil boundary layer problem. Numerical results are presented which demonstrate the ability of the scheme to accurately resolve the developing boundary layer and wake regions using grids which are much coarser than those employed by other numerical methods. It is shown that ten cells in the cross-stream direction are sufficient to accurately resolve the developing airfoil boundary layer.

Derivation and application of the reciprocity relations for radiative transfer with internal illumination

NASA Technical Reports Server (NTRS)

Cogley, A. C.

1975-01-01

A Green's function formulation is used to derive basic reciprocity relations for planar radiative transfer in a general medium with internal illumination. Reciprocity (or functional symmetry) allows an explicit and generalized development of the equivalence between source and probability functions. Assuming similar symmetry in three-dimensional space, a general relationship is derived between planar-source intensity and point-source total directional energy. These quantities are expressed in terms of standard (universal) functions associated with the planar medium, while all results are derived from the differential equation of radiative transfer.

Theoretical predictions of latitude dependencies in the solar wind

NASA Technical Reports Server (NTRS)

Winge, C. R., Jr.; Coleman, P. J., Jr.

1974-01-01

Results are presented which were obtained with the Winge-Coleman model for theoretical predictions of latitudinal dependencies in the solar wind. A first-order expansion is described which allows analysis of first-order latitudinal variations in the coronal boundary conditions and results in a second-order partial differential equation for the perturbation stream function. Latitudinal dependencies are analytically separated out in the form of Legendre polynomials and their derivative, and are reduced to the solution of radial differential equations. This analysis is shown to supply an estimate of how large the coronal variation in latitude must be to produce an 11 km/sec/deg gradient in the radial velocity of the solar wind, assuming steady-state processes.

A Novel Approach to Solve Linearized Stellar Pulsation Equations

NASA Astrophysics Data System (ADS)

Bard, Christopher; Teitler, S.

2011-01-01

We present a new approach to modeling linearized, non-radial pulsations in differentially rotating, massive stars. As a first step in this direction, we consider adiabatic pulsations and adopt the Cowling approximation that perturbations of the gravitational potential and its radial derivative are negligible. The angular dependence of the pulsation modes is expressed as a series expansion of associated Legendre polynomials; the resulting coupled system of differential equations is then solved by finding the eigenfrequencies at which the determinant of a characteristic matrix vanishes. Our method improves on previous treatments by removing the requirement that an arbitrary normalization be applied to the eigenfunctions; this brings the benefit of improved numerical robustness.

A refined analysis of composite laminates. [theory of statics and dynamics

NASA Technical Reports Server (NTRS)

Srinivas, S.

1973-01-01

The purpose of this paper is to develop a sufficiently accurate analysis, which is much simpler than exact three-dimensional analysis, for statics and dynamics of composite laminates. The governing differential equations and boundary conditions are derived by following a variational approach. The displacements are assumed piecewise linear across the thickness and the effects of transverse shear deformations and rotary inertia are included. A procedure for obtaining the general solution of the above governing differential equations in the form of hyperbolic-trigonometric series is given. The accuracy of the present theory is assessed by obtaining results for free vibrations and flexure of simply supported rectangular laminates and comparing them with results from exact three-dimensional analysis.

Hypergeometric Series Solution to a Class of Second-Order Boundary Value Problems via Laplace Transform with Applications to Nanofluids

NASA Astrophysics Data System (ADS)

Ebaid, Abdelhalim; Wazwaz, Abdul-Majid; Alali, Elham; Masaedeh, Basem S.

2017-03-01

Very recently, it was observed that the temperature of nanofluids is finally governed by second-order ordinary differential equations with variable coefficients of exponential orders. Such coefficients were then transformed to polynomials type by using new independent variables. In this paper, a class of second-order ordinary differential equations with variable coefficients of polynomials type has been solved analytically. The analytical solution is expressed in terms of a hypergeometric function with generalized parameters. Moreover, applications of the present results have been applied on some selected nanofluids problems in the literature. The exact solutions in the literature were derived as special cases of our generalized analytical solution.

Analytical pricing of geometric Asian power options on an underlying driven by a mixed fractional Brownian motion

NASA Astrophysics Data System (ADS)

Zhang, Wei-Guo; Li, Zhe; Liu, Yong-Jun

2018-01-01

In this paper, we study the pricing problem of the continuously monitored fixed and floating strike geometric Asian power options in a mixed fractional Brownian motion environment. First, we derive both closed-form solutions and mixed fractional partial differential equations for fixed and floating strike geometric Asian power options based on delta-hedging strategy and partial differential equation method. Second, we present the lower and upper bounds of the prices of fixed and floating strike geometric Asian power options under the assumption that both risk-free interest rate and volatility are interval numbers. Finally, numerical studies are performed to illustrate the performance of our proposed pricing model.

Second-order oriented partial-differential equations for denoising in electronic-speckle-pattern interferometry fringes.

PubMed

Tang, Chen; Han, Lin; Ren, Hongwei; Zhou, Dongjian; Chang, Yiming; Wang, Xiaohang; Cui, Xiaolong

2008-10-01

We derive the second-order oriented partial-differential equations (PDEs) for denoising in electronic-speckle-pattern interferometry fringe patterns from two points of view. The first is based on variational methods, and the second is based on controlling diffusion direction. Our oriented PDE models make the diffusion along only the fringe orientation. The main advantage of our filtering method, based on oriented PDE models, is that it is very easy to implement compared with the published filtering methods along the fringe orientation. We demonstrate the performance of our oriented PDE models via application to two computer-simulated and experimentally obtained speckle fringes and compare with related PDE models.

Optimal control problems with mixed control-phase variable equality and inequality constraints

NASA Technical Reports Server (NTRS)

Makowski, K.; Neustad, L. W.

1974-01-01

In this paper, necessary conditions are obtained for optimal control problems containing equality constraints defined in terms of functions of the control and phase variables. The control system is assumed to be characterized by an ordinary differential equation, and more conventional constraints, including phase inequality constraints, are also assumed to be present. Because the first-mentioned equality constraint must be satisfied for all t (the independent variable of the differential equation) belonging to an arbitrary (prescribed) measurable set, this problem gives rise to infinite-dimensional equality constraints. To obtain the necessary conditions, which are in the form of a maximum principle, an implicit-function-type theorem in Banach spaces is derived.

PRODUCTION OF SOUND BY UNSTEADY THROTTLING OF FLOW INTO A RESONANT CAVITY, WITH APPLICATION TO VOICED SPEECH

PubMed Central

Howe, M. S.; McGowan, R. S.

2011-01-01

An analysis is made of the sound generated by the time-dependent throttling of a nominally steady stream of air through a small orifice into a flow-through resonant cavity. This is exemplified by the production of voiced speech, where air from the lungs enters the vocal tract through the glottis at a time variable volume flow rate Q(t) controlled by oscillations of the glottis cross-section. Voicing theory has hitherto determined Q from a heuristic, reduced complexity ‘Fant’ differential equation (G. Fant, Acoustic Theory of Speech Production, 1960). A new self-consistent, integro-differential form of this equation is derived in this paper using the theory of aerodynamic sound, with full account taken of the back-reaction of the resonant tract on the glottal flux Q. The theory involves an aeroacoustic Green’s function (G) for flow-surface interactions in a time-dependent glottis, so making the problem non-self-adjoint. In complex problems of this type it is not usually possible to obtain G in an explicit analytic form. The principal objective of the paper is to show how the Fant equation can still be derived in such cases from a consideration of the equation of aerodynamic sound and from the adjoint of the equation governing G in the neighbourhood of the ‘throttle’. The theory is illustrated by application to the canonical problem of throttled flow into a Helmholtz resonator. PMID:21666824

Differential form representation of stochastic electromagnetic fields

NASA Astrophysics Data System (ADS)

Haider, Michael; Russer, Johannes A.

2017-09-01

In this work, we revisit the theory of stochastic electromagnetic fields using exterior differential forms. We present a short overview as well as a brief introduction to the application of differential forms in electromagnetic theory. Within the framework of exterior calculus we derive equations for the second order moments, describing stochastic electromagnetic fields. Since the resulting objects are continuous quantities in space, a discretization scheme based on the Method of Moments (MoM) is introduced for numerical treatment. The MoM is applied in such a way, that the notation of exterior calculus is maintained while we still arrive at the same set of algebraic equations as obtained for the case of formulating the theory using the traditional notation of vector calculus. We conclude with an analytic calculation of the radiated electric field of two Hertzian dipole, excited by uncorrelated random currents.

For numerical differentiation, dimensionality can be a blessing!

NASA Astrophysics Data System (ADS)

Anderssen, Robert S.; Hegland, Markus

Finite difference methods, such as the mid-point rule, have been applied successfully to the numerical solution of ordinary and partial differential equations. If such formulas are applied to observational data, in order to determine derivatives, the results can be disastrous. The reason for this is that measurement errors, and even rounding errors in computer approximations, are strongly amplified in the differentiation process, especially if small step-sizes are chosen and higher derivatives are required. A number of authors have examined the use of various forms of averaging which allows the stable computation of low order derivatives from observational data. The size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size h. In this paper, it is initially shown how first (and higher) order single-variate numerical differentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding differentiation of one-dimensional data. The result is then extended to the multivariate differentiation of higher dimensional data. The nature of the trade-off between convergence and stability is explicitly characterized, and the complexity of various implementations is examined.

Modified equations, rational solutions, and the Painleve property for the Kadomtsev--Petviashvili and Hirota--Satsuma equations

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

Weiss, J.

1985-09-01

We propose a method for finding the Lax pairs and rational solutions of integrable partial differential equations. That is, when an equation possesses the Painleve property, a Baecklund transformation is defined in terms of an expansion about the singular manifold. This Baecklund transformation obtains (1) a type of modified equation that is formulated in terms of Schwarzian derivatives and (2) a Miura transformation from the modified to the original equation. By linearizing the (Ricati-type) Miura transformation the Lax pair is found. On the other hand, consideration of the (distinct) Baecklund transformations of the modified equations provides a method for themore » iterative construction of rational solutions. This also obtains the Lax pairs for the modified equations. In this paper we apply this method to the Kadomtsev--Petviashvili equation and the Hirota--Satsuma equations.« less

Mixed Effects Modeling Using Stochastic Differential Equations: Illustrated by Pharmacokinetic Data of Nicotinic Acid in Obese Zucker Rats.

PubMed

Leander, Jacob; Almquist, Joachim; Ahlström, Christine; Gabrielsson, Johan; Jirstrand, Mats

2015-05-01

Inclusion of stochastic differential equations in mixed effects models provides means to quantify and distinguish three sources of variability in data. In addition to the two commonly encountered sources, measurement error and interindividual variability, we also consider uncertainty in the dynamical model itself. To this end, we extend the ordinary differential equation setting used in nonlinear mixed effects models to include stochastic differential equations. The approximate population likelihood is derived using the first-order conditional estimation with interaction method and extended Kalman filtering. To illustrate the application of the stochastic differential mixed effects model, two pharmacokinetic models are considered. First, we use a stochastic one-compartmental model with first-order input and nonlinear elimination to generate synthetic data in a simulated study. We show that by using the proposed method, the three sources of variability can be successfully separated. If the stochastic part is neglected, the parameter estimates become biased, and the measurement error variance is significantly overestimated. Second, we consider an extension to a stochastic pharmacokinetic model in a preclinical study of nicotinic acid kinetics in obese Zucker rats. The parameter estimates are compared between a deterministic and a stochastic NiAc disposition model, respectively. Discrepancies between model predictions and observations, previously described as measurement noise only, are now separated into a comparatively lower level of measurement noise and a significant uncertainty in model dynamics. These examples demonstrate that stochastic differential mixed effects models are useful tools for identifying incomplete or inaccurate model dynamics and for reducing potential bias in parameter estimates due to such model deficiencies.

Long-time asymptotic solution structure of Camassa-Holm equation subject to an initial condition with non-zero reflection coefficient of the scattering data

NASA Astrophysics Data System (ADS)

Chang, Chueh-Hsin; Yu, Ching-Hao; Sheu, Tony Wen-Hann

2016-10-01

In this article, we numerically revisit the long-time solution behavior of the Camassa-Holm equation ut - uxxt + 2ux + 3uux = 2uxuxx + uuxxx. The finite difference solution of this integrable equation is sought subject to the newly derived initial condition with Delta-function potential. Our underlying strategy of deriving a numerical phase accurate finite difference scheme in time domain is to reduce the numerical dispersion error through minimization of the derived discrepancy between the numerical and exact modified wavenumbers. Additionally, to achieve the goal of conserving Hamiltonians in the completely integrable equation of current interest, a symplecticity-preserving time-stepping scheme is developed. Based on the solutions computed from the temporally symplecticity-preserving and the spatially wavenumber-preserving schemes, the long-time asymptotic CH solution characters can be accurately depicted in distinct regions of the space-time domain featuring with their own quantitatively very different solution behaviors. We also aim to numerically confirm that in the two transition zones their long-time asymptotics can indeed be described in terms of the theoretically derived Painlevé transcendents. Another attempt of this study is to numerically exhibit a close connection between the presently predicted finite-difference solution and the solution of the Painlevé ordinary differential equation of type II in two different transition zones.

Flap-lag-torsional dynamic modelling of rotor blades in hover and in forward flight, including the effect of cubic nonlinearities

NASA Technical Reports Server (NTRS)

Crespodasilva, M. R. M.

1981-01-01

The differential equations of motion, and boundary conditions, describing the flap-lead/lag-torsional motion of a flexible rotor blade with a precone angle and a variable pitch angle, which incorporates a pretwist, are derived via Hamilton's principle. The meaning of inextensionality is discussed. The equations are reduced to a set of three integro partial differential equations by elimination of the extension variable. The generalized aerodynamic forces are modelled using Greenberg's extension of Theodorsen's strip theory. The equations of motion are systematically expanded into polynomial nonlinearities with the objective of retaining all terms up to third degree. The blade is modeled as a long, slender, of isotropic Hookean materials. Offsets from the blade's elastic axis through its shear center and the axes for the mass, area and aerodynamic centers, radial nonuniformaties of the blade's stiffnesses and cross section properties are considered and the effect of warp of the cross section is included in the formulation.

Maximum Principle for General Controlled Systems Driven by Fractional Brownian Motions

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

Han Yuecai; Hu Yaozhong; Song Jian, E-mail: jsong2@math.rutgers.edu

2013-04-15

We obtain a maximum principle for stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter H>1/2). This maximum principle specifies a system of equations that the optimal control must satisfy (necessary condition for the optimal control). This system of equations consists of a backward stochastic differential equation driven by both fractional Brownian motions and the corresponding underlying standard Brownian motions. In addition to this backward equation, the maximum principle also involves the Malliavin derivatives. Our approach is to use conditioning and Malliavin calculus. To arrive at our maximum principle we need tomore » develop some new results of stochastic analysis of the controlled systems driven by fractional Brownian motions via fractional calculus. Our approach of conditioning and Malliavin calculus is also applied to classical system driven by standard Brownian motions while the controller has only partial information. As a straightforward consequence, the classical maximum principle is also deduced in this more natural and simpler way.« less

Evolution of magnetic field and atmospheric response. I - Three-dimensional formulation by the method of projected characteristics. II - Formulation of proper boundary equations. [stellar magnetohydrodynamics

NASA Technical Reports Server (NTRS)

Nakagawa, Y.

1981-01-01

The method described as the method of nearcharacteristics by Nakagawa (1980) is renamed the method of projected characteristics. Making full use of properties of the projected characteristics, a new and simpler formulation is developed. As a result, the formulation for the examination of the general three-dimensional problems is presented. It is noted that since in practice numerical solutions must be obtained, the final formulation is given in the form of difference equations. The possibility of including effects of viscous and ohmic dissipations in the formulation is considered, and the physical interpretation is discussed. A systematic manner is then presented for deriving physically self-consistent, time-dependent boundary equations for MHD initial boundary problems. It is demonstrated that the full use of the compatibility equations (differential equations relating variations at two spatial locations and times) is required in determining the time-dependent boundary conditions. In order to provide a clear physical picture as an example, the evolution of axisymmetric global magnetic field by photospheric differential rotation is considered.

Volume integral equation for electromagnetic scattering: Rigorous derivation and analysis for a set of multilayered particles with piecewise-smooth boundaries in a passive host medium

NASA Astrophysics Data System (ADS)

Yurkin, Maxim A.; Mishchenko, Michael I.

2018-04-01

We present a general derivation of the frequency-domain volume integral equation (VIE) for the electric field inside a nonmagnetic scattering object from the differential Maxwell equations, transmission boundary conditions, radiation condition at infinity, and locally-finite-energy condition. The derivation applies to an arbitrary spatially finite group of particles made of isotropic materials and embedded in a passive host medium, including those with edges, corners, and intersecting internal interfaces. This is a substantially more general type of scatterer than in all previous derivations. We explicitly treat the strong singularity of the integral kernel, but keep the entire discussion accessible to the applied scattering community. We also consider the known results on the existence and uniqueness of VIE solution and conjecture a general sufficient condition for that. Finally, we discuss an alternative way of deriving the VIE for an arbitrary object by means of a continuous transformation of the everywhere smooth refractive-index function into a discontinuous one. Overall, the paper examines and pushes forward the state-of-the-art understanding of various analytical aspects of the VIE.

Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics.

PubMed

Cotter, C J; Gottwald, G A; Holm, D D

2017-09-01

In Holm (Holm 2015 Proc. R. Soc. A 471 , 20140963. (doi:10.1098/rspa.2014.0963)), stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small-scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby obtaining stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centring condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.

Dynamic Noise and its Role in Understanding Epidemiological Processes

NASA Astrophysics Data System (ADS)

Stollenwerk, Nico; Aguiar, Maíra

2010-09-01

We investigate the role of dynamic noise in understanding epidemiological systems, such as influenza or dengue fever by deriving stochastic ordinary differential equations from markov processes for discrete populations. This approach allows for an easy analysis of dynamical noise transitions between co-existing attractors.

Response to Oud & Folmer: Randomness and Residuals

ERIC Educational Resources Information Center

Steele, Joel S.; Ferrer, Emilio

2011-01-01

This article presents our response to Oud and Folmer's "Modeling Oscillation, Approximately or Exactly?" (2011), which criticizes aspects of our article, "Latent Differential Equation Modeling of Self-Regulatory and Coregulatory Affective Processes" (2011). In this response, we present a conceptual explanation of the derivative-based estimation…

An Exactly Solvable Model for the Spread of Disease

ERIC Educational Resources Information Center

Mickens, Ronald E.

2012-01-01

We present a new SIR epidemiological model whose exact analytical solution can be calculated. In this model, unlike previous models, the infective population becomes zero at a finite time. Remarkably, these results can be derived from only an elementary knowledge of differential equations.

On the Benefits of Exposing Mathematics Majors to the Rayleigh-Ritz Procedure

ERIC Educational Resources Information Center

Savoye, Philippe

2011-01-01

In this article, I summarize how I present the Rayleigh-Ritz procedure, in a second-semester differential equations course that I teach, and describe some benefits that I believe my students have derived from exposure to this topic. (Contains 3 figures.)

Exponential-fitted methods for integrating stiff systems of ordinary differential equations: Applications to homogeneous gas-phase chemical kinetics

NASA Technical Reports Server (NTRS)

Pratt, D. T.

1984-01-01

Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.

Analyzing a stochastic time series obeying a second-order differential equation.

PubMed

Lehle, B; Peinke, J

2015-06-01

The stochastic properties of a Langevin-type Markov process can be extracted from a given time series by a Markov analysis. Also processes that obey a stochastically forced second-order differential equation can be analyzed this way by employing a particular embedding approach: To obtain a Markovian process in 2N dimensions from a non-Markovian signal in N dimensions, the system is described in a phase space that is extended by the temporal derivative of the signal. For a discrete time series, however, this derivative can only be calculated by a differencing scheme, which introduces an error. If the effects of this error are not accounted for, this leads to systematic errors in the estimation of the drift and diffusion functions of the process. In this paper we will analyze these errors and we will propose an approach that correctly accounts for them. This approach allows an accurate parameter estimation and, additionally, is able to cope with weak measurement noise, which may be superimposed to a given time series.

Fisher-Wright model with deterministic seed bank and selection.

PubMed

Koopmann, Bendix; Müller, Johannes; Tellier, Aurélien; Živković, Daniel

2017-04-01

Seed banks are common characteristics to many plant species, which allow storage of genetic diversity in the soil as dormant seeds for various periods of time. We investigate an above-ground population following a Fisher-Wright model with selection coupled with a deterministic seed bank assuming the length of the seed bank is kept constant and the number of seeds is large. To assess the combined impact of seed banks and selection on genetic diversity, we derive a general diffusion model. The applied techniques outline a path of approximating a stochastic delay differential equation by an appropriately rescaled stochastic differential equation. We compute the equilibrium solution of the site-frequency spectrum and derive the times to fixation of an allele with and without selection. Finally, it is demonstrated that seed banks enhance the effect of selection onto the site-frequency spectrum while slowing down the time until the mutation-selection equilibrium is reached. Copyright © 2016 Elsevier Inc. All rights reserved.

[Penetration of external thermal perturbations into homeothermic organisms, part I (author's transl)].

PubMed

Theves, B

1978-03-20

The general importance of the mean surface curvature for heat conduction problems is explained and a special symmetry with constant mean curvature on the isothermal surfaces is defined. The applicability for the body shapes of homeothermic organisms is demonstrated and the partial differential equation of heat conduction for this case is derived. The definition: heat release = real heat production + convective pseudoproduction eliminates the term of convective heat transfer through the blood stream and allows the reduction to a mere heat conduction problem. Formulas for the heat loss to the environment and for steady state temperature profiles are given. In case of sudden change of heat loss the partial differential equation is solved and a formula is derived, using dimensionless coordinates of time and distance. The mean surface curvature has strongest influence to the interior temperature field. The solution shows clearly the importance of thermal inertia of the homeothermic organism, for the external temperature wave penetrates into the body with a long phase displacement in time.

Generalized wave operators, weighted Killing fields, and perturbations of higher dimensional spacetimes

NASA Astrophysics Data System (ADS)

Araneda, Bernardo

2018-04-01

We present weighted covariant derivatives and wave operators for perturbations of certain algebraically special Einstein spacetimes in arbitrary dimensions, under which the Teukolsky and related equations become weighted wave equations. We show that the higher dimensional generalization of the principal null directions are weighted conformal Killing vectors with respect to the modified covariant derivative. We also introduce a modified Laplace–de Rham-like operator acting on tensor-valued differential forms, and show that the wave-like equations are, at the linear level, appropriate projections off shell of this operator acting on the curvature tensor; the projection tensors being made out of weighted conformal Killing–Yano tensors. We give off shell operator identities that map the Einstein and Maxwell equations into weighted scalar equations, and using adjoint operators we construct solutions of the original field equations in a compact form from solutions of the wave-like equations. We study the extreme and zero boost weight cases; extreme boost corresponding to perturbations of Kundt spacetimes (which includes near horizon geometries of extreme black holes), and zero boost to static black holes in arbitrary dimensions. In 4D our results apply to Einstein spacetimes of Petrov type D and make use of weighted Killing spinors.

An analytical model of a curved beam with a T shaped cross section

NASA Astrophysics Data System (ADS)

Hull, Andrew J.; Perez, Daniel; Cox, Donald L.

2018-03-01

This paper derives a comprehensive analytical dynamic model of a closed circular beam that has a T shaped cross section. The new model includes in-plane and out-of-plane vibrations derived using continuous media expressions which produces results that have a valid frequency range above those available from traditional lumped parameter models. The web is modeled using two-dimensional elasticity equations for in-plane motion and the classical flexural plate equation for out-of-plane motion. The flange is modeled using two sets of Donnell shell equations: one for the left side of the flange and one for the right side of the flange. The governing differential equations are solved with unknown wave propagation coefficients multiplied by spatial domain and time domain functions which are inserted into equilibrium and continuity equations at the intersection of the web and flange and into boundary conditions at the edges of the system resulting in 24 algebraic equations. These equations are solved to yield the wave propagation coefficients and this produces a solution to the displacement field in all three dimensions. An example problem is formulated and compared to results from finite element analysis.

Emergence and analysis of Kuramoto-Sakaguchi-like models as an effective description for the dynamics of coupled Wien-bridge oscillators.

PubMed

English, L Q; Mertens, David; Abdoulkary, Saidou; Fritz, C B; Skowronski, K; Kevrekidis, P G

2016-12-01

We derive the Kuramoto-Sakaguchi model from the basic circuit equations governing two coupled Wien-bridge oscillators. A Wien-bridge oscillator is a particular realization of a tunable autonomous oscillator that makes use of frequency filtering (via an RC bandpass filter) and positive feedback (via an operational amplifier). In the past few years, such oscillators have started to be utilized in synchronization studies. We first show that the Wien-bridge circuit equations can be cast in the form of a coupled pair of van der Pol equations. Subsequently, by applying the method of multiple time scales, we derive the differential equations that govern the slow evolution of the oscillator phases and amplitudes. These equations are directly reminiscent of the Kuramoto-Sakaguchi-type models for the study of synchronization. We analyze the resulting system in terms of the existence and stability of various coupled oscillator solutions and explain on that basis how their synchronization emerges. The phase-amplitude equations are also compared numerically to the original circuit equations and good agreement is found. Finally, we report on experimental measurements of two coupled Wien-bridge oscillators and relate the results to the theoretical predictions.

Global differential geometry: An introduction for control engineers

NASA Technical Reports Server (NTRS)

Doolin, B. F.; Martin, C. F.

1982-01-01

The basic concepts and terminology of modern global differential geometry are discussed as an introduction to the Lie theory of differential equations and to the role of Grassmannians in control systems analysis. To reach these topics, the fundamental notions of manifolds, tangent spaces, vector fields, and Lie algebras are discussed and exemplified. An appendix reviews such concepts needed for vector calculus as open and closed sets, compactness, continuity, and derivative. Although the content is mathematical, this is not a mathematical treatise but rather a text for engineers to understand geometric and nonlinear control.

Dressed soliton in quantum dusty pair-ion plasma

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

Chatterjee, Prasanta; Muniandy, S. V.; Wong, C. S.

Nonlinear propagation of a quantum ion-acoustic dressed soliton is studied in a dusty pair-ion plasma. The Korteweg-de Vries (KdV) equation is derived using reductive perturbation technique. A higher order inhomogeneous differential equation is obtained for the higher order correction. The expression for a dressed soliton is calculated using a renormalization method. The expressions for higher order correction are determined using a series solution technique developed by Chatterjee et al. [Phys. Plasmas 16, 072102 (2009)].

Theoretical investigations on plasma processes in the Kaufman thruster

NASA Technical Reports Server (NTRS)

Wilhelm, H. E.

1973-01-01

The lateral neutralization of ion beams is treated by standard mathematical methods for first order, nonlinear partial differential equations. A closed form analytical solution is derived for the transient lateral beam neutralization for electron injection by means of a von Mises transformation. A nonlinear theory of the longitudinal ion beam neutralization is developed using the von Mises transformation. By means of the Lenard-Balescu equation, the intercomponent momentum transfer between stable, collisionless electron and ion components is calculated.

Hankel-Bessel laser beams.

PubMed

Kotlyar, Victor V; Kovalev, Alexey A; Soifer, Victor A

2012-05-01

An analytical solution of the scalar Helmholtz equation to describe the propagation of a laser light beam in the positive direction of the optical axis is derived. The complex amplitude of such a beam is found to be in direct proportion to the product of two linearly independent solutions of Kummer's differential equation. Relationships for a particular case of such beams-namely, the Hankel-Bessel (HB) beams-are deduced. The focusing of the HB beams is studied. © 2012 Optical Society of America

Differential morphology and image processing.

PubMed

Maragos, P

1996-01-01

Image processing via mathematical morphology has traditionally used geometry to intuitively understand morphological signal operators and set or lattice algebra to analyze them in the space domain. We provide a unified view and analytic tools for morphological image processing that is based on ideas from differential calculus and dynamical systems. This includes ideas on using partial differential or difference equations (PDEs) to model distance propagation or nonlinear multiscale processes in images. We briefly review some nonlinear difference equations that implement discrete distance transforms and relate them to numerical solutions of the eikonal equation of optics. We also review some nonlinear PDEs that model the evolution of multiscale morphological operators and use morphological derivatives. Among the new ideas presented, we develop some general 2-D max/min-sum difference equations that model the space dynamics of 2-D morphological systems (including the distance computations) and some nonlinear signal transforms, called slope transforms, that can analyze these systems in a transform domain in ways conceptually similar to the application of Fourier transforms to linear systems. Thus, distance transforms are shown to be bandpass slope filters. We view the analysis of the multiscale morphological PDEs and of the eikonal PDE solved via weighted distance transforms as a unified area in nonlinear image processing, which we call differential morphology, and briefly discuss its potential applications to image processing and computer vision.

New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.; Manafian, Jalil

2018-03-01

This paper examines the effectiveness of an integration scheme which called the extended trial equation method (ETEM) in exactly solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the longitudinal wave equation (LWE) that arises in mathematical physics with dispersion caused by the transverse Poisson's effect in a magneto-electro-elastic (MEE) circular rod, which a series of exact traveling wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of the longitudinal wave equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this longitudinal wave equation. Many other such types of nonlinear equations arising in non-destructive evaluation of structures made of the advanced MEE material can also be solved by this method.

Nonlinear storage models of unconfined flow through a shallow aquifer on an inclined base and their quasi-steady flow application

NASA Astrophysics Data System (ADS)

Varvaris, Ioannis; Gravanis, Elias; Koussis, Antonis; Akylas, Evangelos

2013-04-01

Hillslope processes involving flow through an inclined shallow aquifer range from subsurface stormflow to stream base flow (drought flow, or groundwater recession flow). In the case of recharge, the infiltrating water moves vertically as unsaturated flow until it reaches the saturated groundwater, where the flow is approximately parallel to the base of the aquifer. Boussinesq used the Dupuit-Forchheimer (D-F) hydraulic theory to formulate unconfined groundwater flow through a soil layer resting on an impervious inclined bed, deriving a nonlinear equation for the flow rate that consists of a linear gravity-driven component and a quadratic pressure-gradient component. Inserting that flow rate equation into the differential storage balance equation (volume conservation) Boussinesq obtained a nonlinear second-order partial differential equation for the depth. So far however, only few special solutions have been advanced for that governing equation. The nonlinearity of the equation of Boussinesq is the major obstacle to deriving a general analytical solution for the depth profile of unconfined flow on a sloping base with recharge (from which the discharges could be then determined). Henderson and Wooding (1964) were able to obtain an exact analytical solution for steady unconfined flow on a sloping base, with recharge, and their work deserves special note in the realm of solutions of the nonlinear equation of Boussinesq. However, the absence of a general solution for the transient case, which is of practical interest to hydrologists, has been the motivation for developing approximate solutions of the non-linear equation of Boussinesq. In this work, we derive the aquifer storage function by integrating analytically over the aquifer base the depth profiles resulting from the complete nonlinear Boussinesq equation for steady flow. This storage function consists of a linear and a nonlinear outflow-dependent term. Then, we use this physics-based storage function in the transient storage balance over the hillslope, obtaining analytical solutions of the outflow and the storage, for recharge and drainage, via a quasi-steady flow calculation. The hydraulically derived storage model is thus embedded in a quasi-steady approximation of transient unconfined flow in sloping aquifers. We generalise this hydrologic model of groundwater flow by modifying the storage function to be the weighted sum of the linear and the nonlinear storage terms, determining the weighting factor objectively from a known integral quantity of the flow (either an initial volume of water stored in the aquifer or a drained water volume). We demonstrate the validity of this model through comparisons with experimental data and simulation results.

Universal shocks in the Wishart random-matrix ensemble.

PubMed

Blaizot, Jean-Paul; Nowak, Maciej A; Warchoł, Piotr

2013-05-01

We show that the derivative of the logarithm of the average characteristic polynomial of a diffusing Wishart matrix obeys an exact partial differential equation valid for an arbitrary value of N, the size of the matrix. In the large N limit, this equation generalizes the simple inviscid Burgers equation that has been obtained earlier for Hermitian or unitary matrices. The solution, through the method of characteristics, presents singularities that we relate to the precursors of shock formation in the Burgers equation. The finite N effects appear as a viscosity term in the Burgers equation. Using a scaling analysis of the complete equation for the characteristic polynomial, in the vicinity of the shocks, we recover in a simple way the universal Bessel oscillations (so-called hard-edge singularities) familiar in random-matrix theory.

Soliton-type solutions for two models in mathematical physics

NASA Astrophysics Data System (ADS)

Al-Ghafri, K. S.

2018-04-01

In this paper, the generalised Klein-Gordon and Kadomtsov-Petviashvili Benjamin-Bona-Mahony equations with power law nonlinearity are investigated. Our study is based on reducing the form of both equations to a first-order ordinary differential equation having the travelling wave solutions. Subsequently, soliton-type solutions such as compacton and solitary pattern solutions are obtained analytically. Additionally, the peaked soliton has been derived where it exists under a specific restrictions. In addition to the soliton solutions, the mathematical method which is exploited in this work also creates a few amount of travelling wave solutions.

Solitons in thin-film ferroelectric material

NASA Astrophysics Data System (ADS)

Boudoue Hubert, Malwe; Justin, Mibaile; Kudryashov, Nikolai A.; Betchewe, Gambo; Douvagai; Doka, Serge Y.

2018-07-01

Through the Landau–Ginzburg–Devonshire mean field theory, the equation governing the behavior of the polarization field in ferroelectric material is derived. Ferroelectric material is subjected to a standing electric field which inhibits remanent polarization and facilitates the access to the instantaneous polarization. Some transformations turn the equation into a well-known ordinary differential equation. As a result, dark soliton and cnoidal waves, which have not yet been observed in ferroelectrics, are obtained. Also, a bright soliton is found. It exists in a given range of temperatures and has an amplitude and a width which vary inversely with temperature.

The first boundary-value problem for a fractional diffusion-wave equation in a non-cylindrical domain

NASA Astrophysics Data System (ADS)

Pskhu, A. V.

2017-12-01

We solve the first boundary-value problem in a non-cylindrical domain for a diffusion-wave equation with the Dzhrbashyan- Nersesyan operator of fractional differentiation with respect to the time variable. We prove an existence and uniqueness theorem for this problem, and construct a representation of the solution. We show that a sufficient condition for unique solubility is the condition of Hölder smoothness for the lateral boundary of the domain. The corresponding results for equations with Riemann- Liouville and Caputo derivatives are particular cases of results obtained here.

Exact analytic solutions for a global equation of plant cell growth.

PubMed

Pietruszka, Mariusz

2010-05-21

A generalization of the Lockhart equation for plant cell expansion in isotropic case is presented. The goal is to account for the temporal variation in the wall mechanical properties--in this case by making the wall extensibility a time dependent parameter. We introduce a time-differential equation describing the plant growth process with some key biophysical aspects considered. The aim of this work was to improve prior modeling efforts by taking into account the dynamic character of the plant cell wall with characteristics reminiscent of damped (aperiodic) motion. The equations selected to encapsulate the time evolution of the wall extensibility offer a new insight into the control of cell wall expansion. We find that the solutions to the time dependent second order differential equation reproduce much of the known experimental data for long- and short-time scales. Additionally, in order to support the biomechanical approach, a new growth equation based on the action of expansin proteins is proposed. Remarkably, both methods independently converge to the same kind, sigmoid-shaped, growth description functional V(t) proportional, exp(-exp(-t)), properly describing the volumetric growth and, consequently, growth rate as its time derivative. Copyright (c) 2010 Elsevier Ltd. All rights reserved.

Dynamic equations for an isotropic spherical shell using the power series method and surface differential operators

NASA Astrophysics Data System (ADS)

Okhovat, Reza; Boström, Anders

2017-04-01

Dynamic equations for an isotropic spherical shell are derived by using a series expansion technique. The displacement field is split into a scalar (radial) part and a vector (tangential) part. Surface differential operators are introduced to decrease the length of all equations. The starting point is a power series expansion of the displacement components in the thickness coordinate relative to the mid-surface of the shell. By using the expansions of the displacement components, the three-dimensional elastodynamic equations yield a set of recursion relations among the expansion functions that can be used to eliminate all but the four of lowest order and to express higher order expansion functions in terms of those of lowest orders. Applying the boundary conditions on the surfaces of the spherical shell and eliminating all but the four lowest order expansion functions give the shell equations as a power series in the shell thickness. After lengthy manipulations, the final four shell equations are obtained in a relatively compact form which are given to second order in shell thickness explicitly. The eigenfrequencies are compared to exact three-dimensional theory with excellent agreement and to membrane theory.

The explicit computation of integration algorithms and first integrals for ordinary differential equations with polynomials coefficients using trees

NASA Technical Reports Server (NTRS)

Crouch, P. E.; Grossman, Robert

1992-01-01

This note is concerned with the explicit symbolic computation of expressions involving differential operators and their actions on functions. The derivation of specialized numerical algorithms, the explicit symbolic computation of integrals of motion, and the explicit computation of normal forms for nonlinear systems all require such computations. More precisely, if R = k(x(sub 1),...,x(sub N)), where k = R or C, F denotes a differential operator with coefficients from R, and g member of R, we describe data structures and algorithms for efficiently computing g. The basic idea is to impose a multiplicative structure on the vector space with basis the set of finite rooted trees and whose nodes are labeled with the coefficients of the differential operators. Cancellations of two trees with r + 1 nodes translates into cancellation of O(N(exp r)) expressions involving the coefficient functions and their derivatives.

Periodic differential equations with self-adjoint monodromy operator

NASA Astrophysics Data System (ADS)

Yudovich, V. I.

2001-04-01

A linear differential equation \\dot u=A(t)u with p-periodic (generally speaking, unbounded) operator coefficient in a Euclidean or a Hilbert space \\mathbb H is considered. It is proved under natural constraints that the monodromy operator U_p is self-adjoint and strictly positive if A^*(-t)=A(t) for all t\\in\\mathbb R.It is shown that Hamiltonian systems in the class under consideration are usually unstable and, if they are stable, then the operator U_p reduces to the identity and all solutions are p-periodic.For higher frequencies averaged equations are derived. Remarkably, high-frequency modulation may double the number of critical values.General results are applied to rotational flows with cylindrical components of the velocity a_r=a_z=0, a_\\theta=\\lambda c(t)r^\\beta, \\beta<-1, c(t) is an even p-periodic function, and also to several problems of free gravitational convection of fluids in periodic fields.

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.

A finite difference method for the solution of the transonic flow around harmonically oscillating wings

NASA Technical Reports Server (NTRS)

Ehlers, E. F.

1974-01-01

A finite difference method for the solution of the transonic flow about a harmonically oscillating wing is presented. The partial differential equation for the unsteady transonic flow was linearized by dividing the flow into separate steady and unsteady perturbation velocity potentials and by assuming small amplitudes of harmonic oscillation. The resulting linear differential equation is of mixed type, being elliptic or hyperbolic whereever the steady flow equation is elliptic or hyperbolic. Central differences were used for all derivatives except at supersonic points where backward differencing was used for the streamwise direction. Detailed formulas and procedures are described in sufficient detail for programming on high speed computers. To test the method, the problem of the oscillating flap on a NACA 64A006 airfoil was programmed. The numerical procedure was found to be stable and convergent even in regions of local supersonic flow with shocks.

Nonlinear GARCH model and 1 / f noise

NASA Astrophysics Data System (ADS)

Kononovicius, A.; Ruseckas, J.

2015-06-01

Auto-regressive conditionally heteroskedastic (ARCH) family models are still used, by practitioners in business and economic policy making, as a conditional volatility forecasting models. Furthermore ARCH models still are attracting an interest of the researchers. In this contribution we consider the well known GARCH(1,1) process and its nonlinear modifications, reminiscent of NGARCH model. We investigate the possibility to reproduce power law statistics, probability density function and power spectral density, using ARCH family models. For this purpose we derive stochastic differential equations from the GARCH processes in consideration. We find the obtained equations to be similar to a general class of stochastic differential equations known to reproduce power law statistics. We show that linear GARCH(1,1) process has power law distribution, but its power spectral density is Brownian noise-like. However, the nonlinear modifications exhibit both power law distribution and power spectral density of the 1 /fβ form, including 1 / f noise.

Lagrangian averaging, nonlinear waves, and shock regularization

NASA Astrophysics Data System (ADS)

Bhat, Harish S.

In this thesis, we explore various models for the flow of a compressible fluid as well as model equations for shock formation, one of the main features of compressible fluid flows. We begin by reviewing the variational structure of compressible fluid mechanics. We derive the barotropic compressible Euler equations from a variational principle in both material and spatial frames. Writing the resulting equations of motion requires certain Lie-algebraic calculations that we carry out in detail for expository purposes. Next, we extend the derivation of the Lagrangian averaged Euler (LAE-alpha) equations to the case of barotropic compressible flows. The derivation in this thesis involves averaging over a tube of trajectories etaepsilon centered around a given Lagrangian flow eta. With this tube framework, the LAE-alpha equations are derived by following a simple procedure: start with a given action, expand via Taylor series in terms of small-scale fluid fluctuations xi, truncate, average, and then model those terms that are nonlinear functions of xi. We then analyze a one-dimensional subcase of the general models derived above. We prove the existence of a large family of traveling wave solutions. Computing the dispersion relation for this model, we find it is nonlinear, implying that the equation is dispersive. We carry out numerical experiments that show that the model possesses smooth, bounded solutions that display interesting pattern formation. Finally, we examine a Hamiltonian partial differential equation (PDE) that regularizes the inviscid Burgers equation without the addition of standard viscosity. Here alpha is a small parameter that controls a nonlinear smoothing term that we have added to the inviscid Burgers equation. We show the existence of a large family of traveling front solutions. We analyze the initial-value problem and prove well-posedness for a certain class of initial data. We prove that in the zero-alpha limit, without any standard viscosity, solutions of the PDE converge strongly to weak solutions of the inviscid Burgers equation. We provide numerical evidence that this limit satisfies an entropy inequality for the inviscid Burgers equation. We demonstrate a Hamiltonian structure for the PDE.

Collective phase description of oscillatory convection

NASA Astrophysics Data System (ADS)

Kawamura, Yoji; Nakao, Hiroya

2013-12-01

We formulate a theory for the collective phase description of oscillatory convection in Hele-Shaw cells. It enables us to describe the dynamics of the oscillatory convection by a single degree of freedom which we call the collective phase. The theory can be considered as a phase reduction method for limit-cycle solutions in infinite-dimensional dynamical systems, namely, stable time-periodic solutions to partial differential equations, representing the oscillatory convection. We derive the phase sensitivity function, which quantifies the phase response of the oscillatory convection to weak perturbations applied at each spatial point, and analyze the phase synchronization between two weakly coupled Hele-Shaw cells exhibiting oscillatory convection on the basis of the derived phase equations.

On the origins and foundations of Laplacian determinism.

PubMed

van Strien, Marij

2014-03-01

In this paper I examine the foundations of Laplace's famous statement of determinism in 1814, and argue that rather than derived from his mechanics, this statement is based on general philosophical principles, namely the principle of sufficient reason and the law of continuity. It is usually supposed that Laplace's statement is based on the fact that each system in classical mechanics has an equation of motion which has a unique solution. But Laplace never proved this result, and in fact he could not have proven it, since it depends on a theorem about uniqueness of solutions to differential equations that was only developed later on. I show that the idea that is at the basis of Laplace's determinism was in fact widespread in enlightenment France, and is ultimately based on a re-interpretation of Leibnizian metaphysics, specifically the principle of sufficient reason and the law of continuity. Since the law of continuity also lies at the basis of the application of differential calculus in physics, one can say that Laplace's determinism and the idea that systems in physics can be described by differential equations with unique solutions have a common foundation.

Pharmacokinetics and "RC" Circuit Concepts

ERIC Educational Resources Information Center

De Cock, Mieke; Janssen, Paul

2013-01-01

Most introductory physics courses include a chapter on "RC" circuits in which the differential equations for the charging and discharging of a capacitor are derived. A number of papers in this journal describe lab experiments dealing with the measurement of different parameters in such "RC" circuits. In this contribution, we…

Erratum for "Symmetry of stochastic non-variational differential equations" [Phys. Rep. 686 (2017) 1-62

NASA Astrophysics Data System (ADS)

Gaeta, G.

2017-11-01

In my recent paper [1], due to a regrettable and rather trivial mistake, a mixed derivatives term is missing in the expression (5.3) for the Ito Laplacian - which is essentially a Taylor expansion. The correct formula is, of course

Reduced equations of motion for quantum systems driven by diffusive Markov processes.

PubMed

Sarovar, Mohan; Grace, Matthew D

2012-09-28

The expansion of a stochastic Liouville equation for the coupled evolution of a quantum system and an Ornstein-Uhlenbeck process into a hierarchy of coupled differential equations is a useful technique that simplifies the simulation of stochastically driven quantum systems. We expand the applicability of this technique by completely characterizing the class of diffusive Markov processes for which a useful hierarchy of equations can be derived. The expansion of this technique enables the examination of quantum systems driven by non-Gaussian stochastic processes with bounded range. We present an application of this extended technique by simulating Stark-tuned Förster resonance transfer in Rydberg atoms with nonperturbative position fluctuations.