Sample records for abstract integro-differential equations

  1. 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.

  2. ADM For Solving Linear Second-Order Fredholm Integro-Differential Equations

    NASA Astrophysics Data System (ADS)

    Karim, Mohd F.; Mohamad, Mahathir; Saifullah Rusiman, Mohd; Che-Him, Norziha; Roslan, Rozaini; Khalid, Kamil

    2018-04-01

    In this paper, we apply Adomian Decomposition Method (ADM) as numerically analyse linear second-order Fredholm Integro-differential Equations. The approximate solutions of the problems are calculated by Maple package. Some numerical examples have been considered to illustrate the ADM for solving this equation. The results are compared with the existing exact solution. Thus, the Adomian decomposition method can be the best alternative method for solving linear second-order Fredholm Integro-Differential equation. It converges to the exact solution quickly and in the same time reduces computational work for solving the equation. The result obtained by ADM shows the ability and efficiency for solving these equations.

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

    NASA Astrophysics Data System (ADS)

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

    2017-07-01

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

  4. New stability conditions for mixed linear Levin-Nohel integro-differential equations

    NASA Astrophysics Data System (ADS)

    Dung, Nguyen Tien

    2013-08-01

    For the mixed Levin-Nohel integro-differential equation, we obtain new necessary and sufficient conditions of asymptotic stability. These results improve those obtained by Becker and Burton ["Stability, fixed points and inverse of delays," Proc. - R. Soc. Edinburgh, Sect. A 136, 245-275 (2006)], 10.1017/S0308210500004546 and Jin and Luo ["Stability of an integro-differential equation," Comput. Math. Appl. 57(7), 1080-1088 (2009)], 10.1016/j.camwa.2009.01.006 when b(t) = 0 and supplement the 3/2-stability theorem when a(t, s) = 0. In addition, the case of the equations with several delays is discussed as well.

  5. Weak Solution Classes for Parabolic Integro-Differential Equations

    DTIC Science & Technology

    1982-09-01

    different existence argument for solutions of (I). It is partly based on a method that was used in (2) and (6] to treat a Hilbert - space version of (I) and...xx Differential Equations 35 (1980), 200-231. 121 V. Barbut Integro-Oifferential Squatton. in Hilbert Spaces. Ann. St. Univ. *Al. 1. Cuaxa 19 (1973... Greenberg : O,% the Existence, Uniqueness, and stability of the Equation 00 Xtt - 3(XX)X) AX *x . J Math. Anal. Appl. 25 (1969), S75-591. (131 7

  6. Relationship Between Integro-Differential Schrodinger Equation with a Symmetric Kernel and Position-Dependent Effective Mass

    NASA Astrophysics Data System (ADS)

    Khosropour, B.; Moayedi, S. K.; Sabzali, R.

    2018-07-01

    The solution of integro-differential Schrodinger equation (IDSE) which was introduced by physicists has a great role in the fields of science. The purpose of this paper comes in two parts. First, studying the relationship between integro-differential Schrodinger equation with a symmetric non-local potential and one-dimensional Schrodinger equation with a position-dependent effective mass. Second, we show that the quantum Hamiltonian for a particle with position-dependent mass after applying Liouville-Green transformations will be converted to a quantum Hamiltonian for a particle with constant mass.

  7. A pertinent approach to solve nonlinear fuzzy integro-differential equations.

    PubMed

    Narayanamoorthy, S; Sathiyapriya, S P

    2016-01-01

    Fuzzy integro-differential equations is one of the important parts of fuzzy analysis theory that holds theoretical as well as applicable values in analytical dynamics and so an appropriate computational algorithm to solve them is in essence. In this article, we use parametric forms of fuzzy numbers and suggest an applicable approach for solving nonlinear fuzzy integro-differential equations using homotopy perturbation method. A clear and detailed description of the proposed method is provided. Our main objective is to illustrate that the construction of appropriate convex homotopy in a proper way leads to highly accurate solutions with less computational work. The efficiency of the approximation technique is expressed via stability and convergence analysis so as to guarantee the efficiency and performance of the methodology. Numerical examples are demonstrated to verify the convergence and it reveals the validity of the presented numerical technique. Numerical results are tabulated and examined by comparing the obtained approximate solutions with the known exact solutions. Graphical representations of the exact and acquired approximate fuzzy solutions clarify the accuracy of the approach.

  8. Dielectric metasurfaces solve differential and integro-differential equations.

    PubMed

    Abdollahramezani, Sajjad; Chizari, Ata; Dorche, Ali Eshaghian; Jamali, Mohammad Vahid; Salehi, Jawad A

    2017-04-01

    Leveraging subwavelength resonant nanostructures, plasmonic metasurfaces have recently attracted much attention as a breakthrough concept for engineering optical waves both spatially and spectrally. However, inherent ohmic losses concomitant with low coupling efficiencies pose fundamental impediments over their practical applications. Not only can all-dielectric metasurfaces tackle such substantial drawbacks, but also their CMOS-compatible configurations support both Mie resonances that are invariant to the incident angle. Here, we report on a transmittive metasurface comprising arrayed silicon nanodisks embedded in a homogeneous dielectric medium to manipulate phase and amplitude of incident light locally and almost independently. By taking advantage of the interplay between the electric/magnetic resonances and employing general concepts of spatial Fourier transformation, a highly efficient metadevice is proposed to perform mathematical operations including solution of ordinary differential and integro-differential equations with constant coefficients. Our findings further substantiate dielectric metasurfaces as promising candidates for miniaturized, two-dimensional, and planar optical analog computing systems that are much thinner than their conventional lens-based counterparts.

  9. On exponential stability of linear Levin-Nohel integro-differential equations

    NASA Astrophysics Data System (ADS)

    Tien Dung, Nguyen

    2015-02-01

    The aim of this paper is to investigate the exponential stability for linear Levin-Nohel integro-differential equations with time-varying delays. To the best of our knowledge, the exponential stability for such equations has not yet been discussed. In addition, since we do not require that the kernel and delay are continuous, our results improve those obtained in Becker and Burton [Proc. R. Soc. Edinburgh, Sect. A: Math. 136, 245-275 (2006)]; Dung [J. Math. Phys. 54, 082705 (2013)]; and Jin and Luo [Comput. Math. Appl. 57(7), 1080-1088 (2009)].

  10. 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.

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

    NASA Astrophysics Data System (ADS)

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

    2017-04-01

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

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

    NASA Astrophysics Data System (ADS)

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

    2017-08-01

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

  13. On a numerical method for solving integro-differential equations with variable coefficients with applications in finance

    NASA Astrophysics Data System (ADS)

    Kudryavtsev, O.; Rodochenko, V.

    2018-03-01

    We propose a new general numerical method aimed to solve integro-differential equations with variable coefficients. The problem under consideration arises in finance where in the context of pricing barrier options in a wide class of stochastic volatility models with jumps. To handle the effect of the correlation between the price and the variance, we use a suitable substitution for processes. Then we construct a Markov-chain approximation for the variation process on small time intervals and apply a maturity randomization technique. The result is a system of boundary problems for integro-differential equations with constant coefficients on the line in each vertex of the chain. We solve the arising problems using a numerical Wiener-Hopf factorization method. The approximate formulae for the factors are efficiently implemented by means of the Fast Fourier Transform. Finally, we use a recurrent procedure that moves backwards in time on the variance tree. We demonstrate the convergence of the method using Monte-Carlo simulations and compare our results with the results obtained by the Wiener-Hopf method with closed-form expressions of the factors.

  14. Note on Solutions to a Class of Nonlinear Singular Integro-Differential Equations,

    DTIC Science & Technology

    1986-08-01

    KdV) ut + 2uu x +Uxx x a 0, (1) the sine-Gordon equation Uxt a sin u, (2) and the Kadomtsev - Petviashvili (KP) equation (Ut + 2uu x + UXXx)x -3a 2u yy...SOUIN OA LSFNN ! /" / M.. \\boiz A.S ::-:- and ,M.O.. .- :1/1 / NOTE ON SOLUTIONS TO A CLASS OF NON \\ / LINEAR SINGULAR INTEGRO-DIFFERENTIA[ EQUATIONS by...important nonlinear evolution equations which can be linearized. Many of these equations fall into the category of linearization via soliton theory and

  15. Lipschitz regularity for integro-differential equations with coercive Hamiltonians and application to large time behavior

    NASA Astrophysics Data System (ADS)

    Barles, Guy; Ley, Olivier; Topp, Erwin

    2017-02-01

    In this paper, we provide suitable adaptations of the ‘weak version of Bernstein method’ introduced by the first author in 1991, in order to obtain Lipschitz regularity results and Lipschitz estimates for nonlinear integro-differential elliptic and parabolic equations set in the whole space. Our interest is to obtain such Lipschitz results to possibly degenerate equations, or to equations which are indeed ‘uniformly elliptic’ (maybe in the nonlocal sense) but which do not satisfy the usual ‘growth condition’ on the gradient term allowing to use (for example) the Ishii-Lions’ method. We treat the case of a model equation with a superlinear coercivity on the gradient term which has a leading role in the equation. This regularity result together with comparison principle provided for the problem allow to obtain the ergodic large time behavior of the evolution problem in the periodic setting.

  16. Darboux transformation and solitons for an integrable nonautonomous nonlinear integro-differential Schrödinger equation

    NASA Astrophysics Data System (ADS)

    Yong, Xuelin; Fan, Yajing; Huang, Yehui; Ma, Wen-Xiu; Tian, Jing

    2017-10-01

    By modifying the scheme for an isospectral problem, the non-isospectral Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy is constructed via allowing the time varying spectrum. In this paper, we consider an integrable nonautonomous nonlinear integro-differential Schrödinger equation discussed before in “Multi-soliton management by the integrable nonautonomous nonlinear integro-differential Schrödinger equation” [Y. J. Zhang, D. Zhao and H. G. Luo, Ann. Phys. 350 (2014) 112]. We first analyze the integrability conditions and identify the model. Second, we modify the existing Darboux transformation (DT) for such a non-isospectral problem. Third, the nonautonomous soliton solutions are obtained via the resulting DT and basic properties of these solutions in the inhomogeneous media are discussed graphically to illustrate the influences of the variable coefficients. In the process, a technique by selecting appropriate spectral parameters instead of the variable inhomogeneities is employed to realize a different type of one-soliton management. Several novel optical solitons are constructed and their features are shown by some specific figures. In addition, four kinds of the special localized two-soliton solutions are obtained. The solitonic excitations localized both in space and time, which exhibit the feature of the so-called rogue waves but with a zero background, are discussed.

  17. A continuous time random walk (CTRW) integro-differential equation with chemical interaction

    NASA Astrophysics Data System (ADS)

    Ben-Zvi, Rami; Nissan, Alon; Scher, Harvey; Berkowitz, Brian

    2018-01-01

    A nonlocal-in-time integro-differential equation is introduced that accounts for close coupling between transport and chemical reaction terms. The structure of the equation contains these terms in a single convolution with a memory function M ( t), which includes the source of non-Fickian (anomalous) behavior, within the framework of a continuous time random walk (CTRW). The interaction is non-linear and second-order, relevant for a bimolecular reaction A + B → C. The interaction term ΓP A ( s, t) P B ( s, t) is symmetric in the concentrations of A and B (i.e. P A and P B ); thus the source terms in the equations for A, B and C are similar, but with a change in sign for that of C. Here, the chemical rate coefficient, Γ, is constant. The fully coupled equations are solved numerically using a finite element method (FEM) with a judicious representation of M ( t) that eschews the need for the entire time history, instead using only values at the former time step. To begin to validate the equations, the FEM solution is compared, in lieu of experimental data, to a particle tracking method (CTRW-PT); the results from the two approaches, particularly for the C profiles, are in agreement. The FEM solution, for a range of initial and boundary conditions, can provide a good model for reactive transport in disordered media.

  18. On an integro-differential equation model for the study of the response of an acoustically coupled panel

    NASA Technical Reports Server (NTRS)

    Yen, D. H. Y.; Maestrello, L.; Padula, S.

    1975-01-01

    The response of a clamped panel to supersonically convected turbulence is considered. A theoretical model in the form of an integro-differential equation is employed that takes into account the coupling between the panel motion and the surrounding acoustic medium. The kernels of the integrals, which represent induced pressures due to the panel motion, are Green's functions for sound radiations under various moving and stationary sources. An approximate analysis is made by following a finite-element Ritz-Galerkin procedure. Preliminary numerical results, in agreement with experimental findings, indicate that the acoustic damping is the controlling mechanism of the response.

  19. On implicit abstract neutral nonlinear differential equations

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

    Hernández, Eduardo, E-mail: lalohm@ffclrp.usp.br; O’Regan, Donal, E-mail: donal.oregan@nuigalway.ie

    2016-04-15

    In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.

  20. Propagation phenomena in monostable integro-differential equations: Acceleration or not?

    NASA Astrophysics Data System (ADS)

    Alfaro, Matthieu; Coville, Jérôme

    2017-11-01

    We consider the homogeneous integro-differential equation ∂t u = J * u - u + f (u) with a monostable nonlinearity f. Our interest is twofold: we investigate the existence/nonexistence of travelling waves, and the propagation properties of the Cauchy problem. When the dispersion kernel J is exponentially bounded, travelling waves are known to exist and solutions of the Cauchy problem typically propagate at a constant speed [7,10,11,22,26,27]. On the other hand, when the dispersion kernel J has heavy tails and the nonlinearity f is nondegenerate, i.e. f‧ (0) > 0, travelling waves do not exist and solutions of the Cauchy problem propagate by accelerating [14,20,27]. For a general monostable nonlinearity, a dichotomy between these two types of propagation behaviour is still not known. The originality of our work is to provide such dichotomy by studying the interplay between the tails of the dispersion kernel and the Allee effect induced by the degeneracy of f, i.e. f‧ (0) = 0. First, for algebraic decaying kernels, we prove the exact separation between existence and nonexistence of travelling waves. This in turn provides the exact separation between nonacceleration and acceleration in the Cauchy problem. In the latter case, we provide a first estimate of the position of the level sets of the solution.

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

    NASA Astrophysics Data System (ADS)

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

    2016-04-01

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

  2. An analytical theory of a scattering of radio waves on meteoric ionization - II. Solution of the integro-differential equation in case of backscatter

    NASA Astrophysics Data System (ADS)

    Pecina, P.

    2016-12-01

    The integro-differential equation for the polarization vector P inside the meteor trail, representing the analytical solution of the set of Maxwell equations, is solved for the case of backscattering of radio waves on meteoric ionization. The transversal and longitudinal dimensions of a typical meteor trail are small in comparison to the distances to both transmitter and receiver and so the phase factor appearing in the kernel of the integral equation is large and rapidly changing. This allows us to use the method of stationary phase to obtain an approximate solution of the integral equation for the scattered field and for the corresponding generalized radar equation. The final solution is obtained by expanding it into the complete set of Bessel functions, which results in solving a system of linear algebraic equations for the coefficients of the expansion. The time behaviour of the meteor echoes is then obtained using the generalized radar equation. Examples are given for values of the electron density spanning a range from underdense meteor echoes to overdense meteor echoes. We show that the time behaviour of overdense meteor echoes using this method is very different from the one obtained using purely numerical solutions of the Maxwell equations. Our results are in much better agreement with the observations performed e.g. by the Ondřejov radar.

  3. Some properties for integro-differential operator defined by a fractional formal.

    PubMed

    Abdulnaby, Zainab E; Ibrahim, Rabha W; Kılıçman, Adem

    2016-01-01

    Recently, the study of the fractional formal (operators, polynomials and classes of special functions) has been increased. This study not only in mathematics but extended to another topics. In this effort, we investigate a generalized integro-differential operator [Formula: see text] defined by a fractional formal (fractional differential operator) and study some its geometric properties by employing it in new subclasses of analytic univalent functions.

  4. SPREADING SPEEDS AND TRAVELING WAVES FOR NON-COOPERATIVE INTEGRO-DIFFERENCE SYSTEMS

    PubMed Central

    Wang, Haiyan; Castillo-Chavez, Carlos

    2014-01-01

    The study of spatially explicit integro-difference systems when the local population dynamics are given in terms of discrete-time generations models has gained considerable attention over the past two decades. These nonlinear systems arise naturally in the study of the spatial dispersal of organisms. The brunt of the mathematical research on these systems, particularly, when dealing with cooperative systems, has focused on the study of the existence of traveling wave solutions and the characterization of their spreading speed. Here, we characterize the minimum propagation (spreading) speed, via the convergence of initial data to wave solutions, for a large class of non cooperative nonlinear systems of integro-difference equations. The spreading speed turns out to be the slowest speed from a family of non-constant traveling wave solutions. The applicability of these theoretical results is illustrated through the explicit study of an integro-difference system with local population dynamics governed by Hassell and Comins’ non-cooperative competition model (1976). The corresponding integro-difference nonlinear systems that results from the redistribution of individuals via a dispersal kernel is shown to satisfy conditions that guarantee the existence of minimum speeds and traveling waves. This paper is dedicated to Avner Friedman as we celebrate his immense contributions to the fields of partial differential equations, integral equations, mathematical biology, industrial mathematics and applied mathematics in general. His leadership in the mathematical sciences and his mentorship of students and friends over several decades has made a huge difference in the personal and professional lives of many, including both of us. PMID:24899868

  5. Development and Retrospective Clinical Assessment of a Patient-Specific Closed-Form Integro-Differential Equation Model of Plasma Dilution.

    PubMed

    Atlas, Glen; Li, John K-J; Amin, Shawn; Hahn, Robert G

    2017-01-01

    A closed-form integro-differential equation (IDE) model of plasma dilution (PD) has been derived which represents both the intravenous (IV) infusion of crystalloid and the postinfusion period. Specifically, PD is mathematically represented using a combination of constant ratio, differential, and integral components. Furthermore, this model has successfully been applied to preexisting data, from a prior human study, in which crystalloid was infused for a period of 30 minutes at the beginning of thyroid surgery. Using Euler's formula and a Laplace transform solution to the IDE, patients could be divided into two distinct groups based on their response to PD during the infusion period. Explicitly, Group 1 patients had an infusion-based PD response which was modeled using an exponentially decaying hyperbolic sine function, whereas Group 2 patients had an infusion-based PD response which was modeled using an exponentially decaying trigonometric sine function. Both Group 1 and Group 2 patients had postinfusion PD responses which were modeled using the same combination of hyperbolic sine and hyperbolic cosine functions. Statistically significant differences, between Groups 1 and 2, were noted with respect to the area under their PD curves during both the infusion and postinfusion periods. Specifically, Group 2 patients exhibited a response to PD which was most likely consistent with a preoperative hypovolemia. Overall, this IDE model of PD appears to be highly "adaptable" and successfully fits clinically-obtained human data on a patient-specific basis, during both the infusion and postinfusion periods. In addition, patient-specific IDE modeling of PD may be a useful adjunct in perioperative fluid management and in assessing clinical volume kinetics, of crystalloid solutions, in real time.

  6. Development and Retrospective Clinical Assessment of a Patient-Specific Closed-Form Integro-Differential Equation Model of Plasma Dilution

    PubMed Central

    Atlas, Glen; Li, John K-J; Amin, Shawn; Hahn, Robert G

    2017-01-01

    A closed-form integro-differential equation (IDE) model of plasma dilution (PD) has been derived which represents both the intravenous (IV) infusion of crystalloid and the postinfusion period. Specifically, PD is mathematically represented using a combination of constant ratio, differential, and integral components. Furthermore, this model has successfully been applied to preexisting data, from a prior human study, in which crystalloid was infused for a period of 30 minutes at the beginning of thyroid surgery. Using Euler’s formula and a Laplace transform solution to the IDE, patients could be divided into two distinct groups based on their response to PD during the infusion period. Explicitly, Group 1 patients had an infusion-based PD response which was modeled using an exponentially decaying hyperbolic sine function, whereas Group 2 patients had an infusion-based PD response which was modeled using an exponentially decaying trigonometric sine function. Both Group 1 and Group 2 patients had postinfusion PD responses which were modeled using the same combination of hyperbolic sine and hyperbolic cosine functions. Statistically significant differences, between Groups 1 and 2, were noted with respect to the area under their PD curves during both the infusion and postinfusion periods. Specifically, Group 2 patients exhibited a response to PD which was most likely consistent with a preoperative hypovolemia. Overall, this IDE model of PD appears to be highly “adaptable” and successfully fits clinically-obtained human data on a patient-specific basis, during both the infusion and postinfusion periods. In addition, patient-specific IDE modeling of PD may be a useful adjunct in perioperative fluid management and in assessing clinical volume kinetics, of crystalloid solutions, in real time. PMID:29123436

  7. 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.

  8. Characterization of non-diffusive transport in plasma turbulence by means of flux-gradient integro-differential kernels

    NASA Astrophysics Data System (ADS)

    Alcuson, J. A.; Reynolds-Barredo, J. M.; Mier, J. A.; Sanchez, Raul; Del-Castillo-Negrete, Diego; Newman, David E.; Tribaldos, V.

    2015-11-01

    A method to determine fractional transport exponents in systems dominated by fluid or plasma turbulence is proposed. The method is based on the estimation of the integro-differential kernel that relates values of the fluxes and gradients of the transported field, and its comparison with the family of analytical kernels of the linear fractional transport equation. Although use of this type of kernels has been explored before in this context, the methodology proposed here is rather unique since the connection with specific fractional equations is exploited from the start. The procedure has been designed to be particularly well-suited for application in experimental setups, taking advantage of the fact that kernel determination only requires temporal data of the transported field measured on an Eulerian grid. The simplicity and robustness of the method is tested first by using fabricated data from continuous-time random walk models built with prescribed transport characteristics. Its strengths are then illustrated on numerical Eulerian data gathered from simulations of a magnetically confined turbulent plasma in a near-critical regime, that is known to exhibit superdiffusive radial transport

  9. 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

  10. 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

  11. Study of viscous flow about airfoils by the integro-differential method

    NASA Technical Reports Server (NTRS)

    Wu, J. C.; Sampath, S.

    1975-01-01

    An integro-differential method was used for numerically solving unsteady incompressible viscous flow problems. A computer program was prepared to solve the problem of an impulsively started 9% thick symmetric Joukowski airfoil at an angle of attack of 15 deg and a Reynolds number of 1000. Some of the results obtained for this problem were discussed and compared with related work completed previously. Two numerical procedures were used, an Alternating Direction Implicit (ADI) method and a Successive Line Relaxation (SLR) method. Generally, the ADI solution agrees well with the SLR solution and with previous results are stations away from the trailing edge. At the trailing edge station, the ADI solution differs substantially from previous results, while the vorticity profiles obtained from the SLR method there are in good qualitative agreement with previous results.

  12. A Lower Bound for the Norm of the Solution of a Nonlinear Volterra Equation in One-Dimensional Viscoelasticity.

    DTIC Science & Technology

    1980-12-09

    34, Symp. on Non-well-posed Problems and Logarithmic Convexity (Lecture Notes on Math. #316), pp. 31-5h, Springer, 1973. 3. Greenberg , J.M., MacCamy, R.C...34Continuous Data Dependence for an Abstract Volterra Integro- Differential Equation in Hilbert Space with Applications to Viscoelasticity", Annali Scuola... Hilbert Space", to appear in the J. Applicable Analysis. 8. Slemrod, M., "Instability of Steady Shearing Flows in a Nonlinear Viscoelastic Fluid", Arch

  13. Integro-differential modeling of ICRH wave propagation and damping at arbitrary cyclotron harmonics and wavelengths in tokamaks

    NASA Astrophysics Data System (ADS)

    Van Eester, D.; Lerche, E.

    2014-02-01

    Both at low and higher cyclotron harmonics, properly accounting for finite Larmor radius effects is crucial in many ion cyclotron resonance frequency heating scenarios creating high energy tails. The present paper discusses ongoing work to extend the 1D TOMCAT wave equation solver [D. Van Eester & R. Koch, Plasma Phys. Contr. Fusion 40 (1998) 1949] to arbitrary harmonics and arbitrary wavelengths. Rather than adopting the particle position, the guiding center position is used as the independent variable when writing down an expression for the dielectric response. Adopting a philosophy originally due to Kaufman [A.N. Kaufman, Phys. Fluids 15 (1972) 1063], the relevant dielectric response in the Galerkin formalism is written in a form where the electric field and the test function vector appear symmetrically, which yields a power balance equation that guarantees non-negative absorption for any wave type for Maxwellian plasmas. Moreover, this choice of independent variable yields intuitive expressions that can directly be linked to the corresponding expressions in the RF diffusion operator. It also guarantees that a positive definite power transfer from waves to particles is ensured for any of the wave modes in a plasma in which all populations have a Maxwellian distribution, as is expected from first principles. Rather than relying on a truncated Taylor series expansion of the dielectric response, an integro-differential approach that retains all finite Larmor radius effects [D. Van Eester & E. Lerche, Plasma Phys. Control. Fusion 55 (2013) 055008] is proposed.

  14. Relating coupled map lattices to integro-difference equations: dispersal-driven instabilities in coupled map lattices.

    PubMed

    White, Steven M; White, K A Jane

    2005-08-21

    Recently there has been a great deal of interest within the ecological community about the interactions of local populations that are coupled only by dispersal. Models have been developed to consider such scenarios but the theory needed to validate model outcomes has been somewhat lacking. In this paper, we present theory which can be used to understand these types of interaction when population exhibit discrete time dynamics. In particular, we consider a spatial extension to discrete-time models, known as coupled map lattices (CMLs) which are discrete in space. We introduce a general form of the CML and link this to integro-difference equations via a special redistribution kernel. General conditions are then derived for dispersal-driven instabilities. We then apply this theory to two discrete-time models; a predator-prey model and a host-pathogen model.

  15. Computation and visualization of geometric partial differential equations

    NASA Astrophysics Data System (ADS)

    Tiee, Christopher L.

    The chief goal of this work is to explore a modern framework for the study and approximation of partial differential equations, recast common partial differential equations into this framework, and prove theorems about such equations and their approximations. A central motivation is to recognize and respect the essential geometric nature of such problems, and take it into consideration when approximating. The hope is that this process will lead to the discovery of more refined algorithms and processes and apply them to new problems. In the first part, we introduce our quantities of interest and reformulate traditional boundary value problems in the modern framework. We see how Hilbert complexes capture and abstract the most important properties of such boundary value problems, leading to generalizations of important classical results such as the Hodge decomposition theorem. They also provide the proper setting for numerical approximations. We also provide an abstract framework for evolution problems in these spaces: Bochner spaces. We next turn to approximation. We build layers of abstraction, progressing from functions, to differential forms, and finally, to Hilbert complexes. We explore finite element exterior calculus (FEEC), which allows us to approximate solutions involving differential forms, and analyze the approximation error. In the second part, we prove our central results. We first prove an extension of current error estimates for the elliptic problem in Hilbert complexes. This extension handles solutions with nonzero harmonic part. Next, we consider evolution problems in Hilbert complexes and prove abstract error estimates. We apply these estimates to the problem for Riemannian hypersurfaces in R. {n+1},generalizing current results for open subsets of R. {n}. Finally, we applysome of the concepts to a nonlinear problem, the Ricci flow on surfaces, and use tools from nonlinear analysis to help develop and analyze the equations. In the appendices, we

  16. Interpreting Abstract Interpretations in Membership Equational Logic

    NASA Technical Reports Server (NTRS)

    Fischer, Bernd; Rosu, Grigore

    2001-01-01

    We present a logical framework in which abstract interpretations can be naturally specified and then verified. Our approach is based on membership equational logic which extends equational logics by membership axioms, asserting that a term has a certain sort. We represent an abstract interpretation as a membership equational logic specification, usually as an overloaded order-sorted signature with membership axioms. It turns out that, for any term, its least sort over this specification corresponds to its most concrete abstract value. Maude implements membership equational logic and provides mechanisms to calculate the least sort of a term efficiently. We first show how Maude can be used to get prototyping of abstract interpretations "for free." Building on the meta-logic facilities of Maude, we further develop a tool that automatically checks and abstract interpretation against a set of user-defined properties. This can be used to select an appropriate abstract interpretation, to characterize the specified loss of information during abstraction, and to compare different abstractions with each other.

  17. 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.

  18. A novel technique to solve nonlinear higher-index Hessenberg differential-algebraic equations by Adomian decomposition method.

    PubMed

    Benhammouda, Brahim

    2016-01-01

    Since 1980, the Adomian decomposition method (ADM) has been extensively used as a simple powerful tool that applies directly to solve different kinds of nonlinear equations including functional, differential, integro-differential and algebraic equations. However, for differential-algebraic equations (DAEs) the ADM is applied only in four earlier works. There, the DAEs are first pre-processed by some transformations like index reductions before applying the ADM. The drawback of such transformations is that they can involve complex algorithms, can be computationally expensive and may lead to non-physical solutions. The purpose of this paper is to propose a novel technique that applies the ADM directly to solve a class of nonlinear higher-index Hessenberg DAEs systems efficiently. The main advantage of this technique is that; firstly it avoids complex transformations like index reductions and leads to a simple general algorithm. Secondly, it reduces the computational work by solving only linear algebraic systems with a constant coefficient matrix at each iteration, except for the first iteration where the algebraic system is nonlinear (if the DAE is nonlinear with respect to the algebraic variable). To demonstrate the effectiveness of the proposed technique, we apply it to a nonlinear index-three Hessenberg DAEs system with nonlinear algebraic constraints. This technique is straightforward and can be programmed in Maple or Mathematica to simulate real application problems.

  19. Method of mechanical quadratures for solving singular integral equations of various types

    NASA Astrophysics Data System (ADS)

    Sahakyan, A. V.; Amirjanyan, H. A.

    2018-04-01

    The method of mechanical quadratures is proposed as a common approach intended for solving the integral equations defined on finite intervals and containing Cauchy-type singular integrals. This method can be used to solve singular integral equations of the first and second kind, equations with generalized kernel, weakly singular equations, and integro-differential equations. The quadrature rules for several different integrals represented through the same coefficients are presented. This allows one to reduce the integral equations containing integrals of different types to a system of linear algebraic equations.

  20. Soliton structure versus singularity analysis: Third-order completely intergrable nonlinear differential equations in 1 + 1-dimensions

    NASA Astrophysics Data System (ADS)

    Fuchssteiner, Benno; Carillo, Sandra

    1989-01-01

    Bäcklund transformations between all known completely integrable third-order differential equations in (1 + 1)-dimensions are established and the corresponding transformations formulas for their hereditary operators and Hamiltonian formulations are exhibited. Some of these Bäcklund transformations are not injective; therefore additional non-commutative symmetry groups are found for some equations. These non-commutative symmetry groups are classified as having a semisimple part isomorphic to the affine algebra A(1)1. New completely integrable third-order integro-differential equations, some depending explicitly on x, are given. These new equations give rise to nonin equation. Connections between the singularity equations (from the Painlevé analysis) and the nonlinear equations for interacting solitons are established. A common approach to singularity analysis and soliton structure is introduced. The Painlevé analysis is modified in such a sense that it carries over directly and without difficulty to the time evolution of singularity manifolds of equations like the sine-Gordon and nonlinear Schrödinger equation. A method to recover the Painlevé series from its constant level term is exhibit. The soliton-singularity transform is recognized to be connected to the Möbius group. This gives rise to a Darboux-like result for the spectral properties of the recursion operator. These connections are used in order to explain why poles of soliton equations move like trajectories of interacting solitons. Furthermore it is explicitly computed how solitons of singularity equations behave under the effect of this soliton-singularity transform. This then leads to the result that only for scaling degrees α = -1 and α = -2 the usual Painlevé analysis can be carried out. A new invariance principle, connected to kernels of differential operators is discovered. This new invariance, for example, connects the explicit solutions of the Liouville equation with the Miura transform

  1. Nonlinear differential equations

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

    Dresner, L.

    1988-01-01

    This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis ismore » on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.« less

  2. 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.

  3. LORENE: Spectral methods differential equations solver

    NASA Astrophysics Data System (ADS)

    Gourgoulhon, Eric; Grandclément, Philippe; Marck, Jean-Alain; Novak, Jérôme; Taniguchi, Keisuke

    2016-08-01

    LORENE (Langage Objet pour la RElativité NumériquE) solves various problems arising in numerical relativity, and more generally in computational astrophysics. It is a set of C++ classes and provides tools to solve partial differential equations by means of multi-domain spectral methods. LORENE classes implement basic structures such as arrays and matrices, but also abstract mathematical objects, such as tensors, and astrophysical objects, such as stars and black holes.

  4. pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

    DOE PAGES

    Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul; ...

    2017-12-20

    We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

  5. pyomo.dae: a modeling and automatic discretization framework for optimization with differential and algebraic equations

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

    Nicholson, Bethany; Siirola, John D.; Watson, Jean-Paul

    We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differentialmore » equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.« less

  6. The Form of the Solutions of the Linear Integro-Differential Equations of Subsonic Aeroelasticity.

    DTIC Science & Technology

    1979-09-01

    coefficients w (0) are given in Table 3; it V follows that, for T > 0 and (E - K v2) non-singular, the inverse transform of M- ) has the form, using (B-I) V...degree of freedom system by expanding )M- I in the form of equation (35), obtaining its inverse transform using the v -1results of Appendix A and hence...obtaining the inverse transform of M- l . The two-dimensional case, when the characteristic equation has a zero root, is not as simple. * Assuming all

  7. 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.

  8. SciCADE 95: International conference on scientific computation and differential equations

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

    NONE

    1995-12-31

    This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included.

  9. 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.

  10. Interface equation and viscosity contrast in Hele-Shaw flow

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

    Casademunt, J.; Jasnow, D.; Hernandez-Machado, A.

    1992-05-20

    In this paper, the authors derive an integro-differential equation for the evolution of the interface separating two immiscible viscous fluids in a Hele-Shaw cell with a channel geometry, for arbitrary viscosity contrast. The authors' equation differs from a previous one obtained by a vortex-sheet formulation of the problem, in that the normal component of the interface velocity is formally decoupled from the gauge-dependent tangential part. The result is thus a closed integral equation for the normal velocity. The authors briefly comment on the advantages of such a formulation and implement an alternative computational algorithm based on it. Preliminary numerical resultsmore » confirm a highly inefficient finger competition in the zero viscosity contrast limit.« less

  11. An analytical theory of radio-wave scattering from meteoric ionization - I. Basic equation

    NASA Astrophysics Data System (ADS)

    Pecina, P.

    2016-01-01

    We have developed an analytical theory of radio-wave scattering from ionization of meteoric origin. It is based on an integro-differential equation for the polarization vector, P, inside the meteor trail, representing an analytical solution of the set of Maxwell equations, in combination with a generalized radar equation involving an integral of the trail volume electron density, Ne, and P represented by an auxiliary vector, Q, taken over the whole trail volume. During the derivation of the final formulae, the following assumptions were applied: transversal as well as longitudinal dimensions of the meteor trail are small compared with the distances of the relevant trail point to both the transmitter and receiver and the ratio of these distances to the wavelength of the wave emitted by the radar is very large, so that the stationary-phase method can be employed for evaluation of the relevant integrals. Further, it is shown that in the case of sufficiently low electron density, Ne, corresponding to the case of underdense trails, the classical McKinley's radar equation results as a special case of the general theory. The same also applies regarding the Fresnel characteristics. Our approach is also capable of yielding solutions to the problems of the formation of Fresnel characteristics on trails having any electron density, forward scattering and scattering on trails immersed in the magnetic field. However, we have also shown that the geomagnetic field can be removed from consideration, due to its low strength. The full solution of the above integro-differential equation, valid for any electron volume densities, has been left to subsequent works dealing with this particular problem, due to its complexity.

  12. Non-Archimedean reaction-ultradiffusion equations and complex hierarchic systems

    NASA Astrophysics Data System (ADS)

    Zúñiga-Galindo, W. A.

    2018-06-01

    We initiate the study of non-Archimedean reaction-ultradiffusion equations and their connections with models of complex hierarchic systems. From a mathematical perspective, the equations studied here are the p-adic counterpart of the integro-differential models for phase separation introduced by Bates and Chmaj. Our equations are also generalizations of the ultradiffusion equations on trees studied in the 1980s by Ogielski, Stein, Bachas, Huberman, among others, and also generalizations of the master equations of the Avetisov et al models, which describe certain complex hierarchic systems. From a physical perspective, our equations are gradient flows of non-Archimedean free energy functionals and their solutions describe the macroscopic density profile of a bistable material whose space of states has an ultrametric structure. Some of our results are p-adic analogs of some well-known results in the Archimedean setting, however, the mechanism of diffusion is completely different due to the fact that it occurs in an ultrametric space.

  13. Dichotomies for generalized ordinary differential equations and applications

    NASA Astrophysics Data System (ADS)

    Bonotto, E. M.; Federson, M.; Santos, F. L.

    2018-03-01

    In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between generalized ordinary differential equations and other equations, we translate our results to measure differential equations and impulsive differential equations. The fact that we work in the framework of generalized ordinary differential equations allows us to manage functions with many discontinuities and of unbounded variation.

  14. Legendre-tau approximations for functional differential equations

    NASA Technical Reports Server (NTRS)

    Ito, K.; Teglas, R.

    1986-01-01

    The numerical approximation of solutions to linear retarded functional differential equations are considered using the so-called Legendre-tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time-differentiation. The approximate solution is then represented as a truncated Legendre series with time-varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximation is made.

  15. Legendre-Tau approximations for functional differential equations

    NASA Technical Reports Server (NTRS)

    Ito, K.; Teglas, R.

    1983-01-01

    The numerical approximation of solutions to linear functional differential equations are considered using the so called Legendre tau method. The functional differential equation is first reformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation. The approximate solution is then represented as a truncated Legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations. The method is very easy to code and yields very accurate approximations. Convergence is established, various numerical examples are presented, and comparison between the latter and cubic spline approximations is made.

  16. Stochastic differential equations

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

    Sobczyk, K.

    1990-01-01

    This book provides a unified treatment of both regular (or random) and Ito stochastic differential equations. It focuses on solution methods, including some developed only recently. Applications are discussed, in particular an insight is given into both the mathematical structure, and the most efficient solution methods (analytical as well as numerical). Starting from basic notions and results of the theory of stochastic processes and stochastic calculus (including Ito's stochastic integral), many principal mathematical problems and results related to stochastic differential equations are expounded here for the first time. Applications treated include those relating to road vehicles, earthquake excitations and offshoremore » structures.« less

  17. Derivation of kinetic equations from non-Wiener stochastic differential equations

    NASA Astrophysics Data System (ADS)

    Basharov, A. M.

    2013-12-01

    Kinetic differential-difference equations containing terms with fractional derivatives and describing α -stable Levy processes with 0 < α < 1 have been derived in a unified manner in terms of one-dimensional stochastic differential equations controlled merely by the Poisson processes.

  18. Application of the Sumudu Transform to Discrete Dynamic Systems

    ERIC Educational Resources Information Center

    Asiru, Muniru Aderemi

    2003-01-01

    The Sumudu transform is an integral transform introduced to solve differential equations and control engineering problems. The transform possesses many interesting properties that make visualization easier and application has been demonstrated in the solution of partial differential equations, integral equations, integro-differential equations and…

  19. 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.

  20. Radiation reaction on a classical charged particle: a modified form of the equation of motion.

    PubMed

    Alcaine, Guillermo García; Llanes-Estrada, Felipe J

    2013-09-01

    We present and numerically solve a modified form of the equation of motion for a charged particle under the influence of an external force, taking into account the radiation reaction. This covariant equation is integro-differential, as Dirac-Röhrlich's, but has several technical improvements. First, the equation has the form of Newton's second law, with acceleration isolated on the left hand side and the force depending only on positions and velocities: Thus, the equation is linear in the highest derivative. Second, the total four-force is by construction perpendicular to the four-velocity. Third, if the external force vanishes for all future times, the total force and the acceleration automatically vanish at the present time. We show the advantages of this equation by solving it numerically for several examples of external force.

  1. Radiation reaction on a classical charged particle: A modified form of the equation of motion

    NASA Astrophysics Data System (ADS)

    Alcaine, Guillermo García; Llanes-Estrada, Felipe J.

    2013-09-01

    We present and numerically solve a modified form of the equation of motion for a charged particle under the influence of an external force, taking into account the radiation reaction. This covariant equation is integro-differential, as Dirac-Röhrlich's, but has several technical improvements. First, the equation has the form of Newton's second law, with acceleration isolated on the left hand side and the force depending only on positions and velocities: Thus, the equation is linear in the highest derivative. Second, the total four-force is by construction perpendicular to the four-velocity. Third, if the external force vanishes for all future times, the total force and the acceleration automatically vanish at the present time. We show the advantages of this equation by solving it numerically for several examples of external force.

  2. 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

  3. 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.

  4. Ordinary differential equations.

    PubMed

    Lebl, Jiří

    2013-01-01

    In this chapter we provide an overview of the basic theory of ordinary differential equations (ODE). We give the basics of analytical methods for their solutions and also review numerical methods. The chapter should serve as a primer for the basic application of ODEs and systems of ODEs in practice. As an example, we work out the equations arising in Michaelis-Menten kinetics and give a short introduction to using Matlab for their numerical solution.

  5. Symbolic Solution of Linear Differential Equations

    NASA Technical Reports Server (NTRS)

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

    1981-01-01

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

  6. A tensor Banach algebra approach to abstract kinetic equations

    NASA Astrophysics Data System (ADS)

    Greenberg, W.; van der Mee, C. V. M.

    The study deals with a concrete algebraic construction providing the existence theory for abstract kinetic equation boundary-value problems, when the collision operator A is an accretive finite-rank perturbation of the identity operator in a Hilbert space H. An algebraic generalization of the Bochner-Phillips theorem is utilized to study solvability of the abstract boundary-value problem without any regulatory condition. A Banach algebra in which the convolution kernel acts is obtained explicitly, and this result is used to prove a perturbation theorem for bisemigroups, which then plays a vital role in solving the initial equations.

  7. Solving Differential Equations in R: Package deSolve

    EPA Science Inventory

    In this paper we present the R package deSolve to solve initial value problems (IVP) written as ordinary differential equations (ODE), differential algebraic equations (DAE) of index 0 or 1 and partial differential equations (PDE), the latter solved using the method of lines appr...

  8. Lie algebras and linear differential equations.

    NASA Technical Reports Server (NTRS)

    Brockett, R. W.; Rahimi, A.

    1972-01-01

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

  9. MACSYMA's symbolic ordinary differential equation solver

    NASA Technical Reports Server (NTRS)

    Golden, J. P.

    1977-01-01

    The MACSYMA's symbolic ordinary differential equation solver ODE2 is described. The code for this routine is delineated, which is of interest because it is written in top-level MACSYMA language, and may serve as a good example of programming in that language. Other symbolic ordinary differential equation solvers are mentioned.

  10. Solving Nonlinear Coupled Differential Equations

    NASA Technical Reports Server (NTRS)

    Mitchell, L.; David, J.

    1986-01-01

    Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.

  11. 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

  12. Schwarz maps of algebraic linear ordinary differential equations

    NASA Astrophysics Data System (ADS)

    Sanabria Malagón, Camilo

    2017-12-01

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

  13. A computer software system for the generation of global ocean tides including self-gravitation and crustal loading effects

    NASA Technical Reports Server (NTRS)

    Estes, R. H.

    1977-01-01

    A computer software system is described which computes global numerical solutions of the integro-differential Laplace tidal equations, including dissipation terms and ocean loading and self-gravitation effects, for arbitrary diurnal and semidiurnal tidal constituents. The integration algorithm features a successive approximation scheme for the integro-differential system, with time stepping forward differences in the time variable and central differences in spatial variables.

  14. Computational Algorithms or Identification of Distributed Parameter Systems

    DTIC Science & Technology

    1993-04-24

    delay-differential equations, Volterra integral equations, and partial differential equations with memory terms . In particular we investigated a...tested for estimating parameters in a Volterra integral equation arising from a viscoelastic model of a flexible structure with Boltzmann damping. In...particular, one of the parameters identified was the order of the derivative in Volterra integro-differential equations containing fractional

  15. Teaching Modeling with Partial Differential Equations: Several Successful Approaches

    ERIC Educational Resources Information Center

    Myers, Joseph; Trubatch, David; Winkel, Brian

    2008-01-01

    We discuss the introduction and teaching of partial differential equations (heat and wave equations) via modeling physical phenomena, using a new approach that encompasses constructing difference equations and implementing these in a spreadsheet, numerically solving the partial differential equations using the numerical differential equation…

  16. Some problems in fractal differential equations

    NASA Astrophysics Data System (ADS)

    Su, Weiyi

    2016-06-01

    Based upon the fractal calculus on local fields, or p-type calculus, or Gibbs-Butzer calculus ([1],[2]), we suggest a constructive idea for "fractal differential equations", beginning from some special examples to a general theory. However, this is just an original idea, it needs lots of later work to support. In [3], we show example "two dimension wave equations with fractal boundaries", and in this note, other examples, as well as an idea to construct fractal differential equations are shown.

  17. 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.

  18. Comparison of different notation for equations of motion of a body in a medium flow

    NASA Astrophysics Data System (ADS)

    Samsonov, V. A.; Selyutskii, Yu. D.

    2008-02-01

    In [1-6], a model of a nonstationary action of a medium flow on a body moving in this flow was constructed in the form of an associated dynamical system of second order. In the literature, the representation of the aerodynamic force in integral form with a Duhamel type integral is often used (e.g., see [7, 8]). In the present paper, we pay attention to the fact that a system of ODE is equivalent not to a single integro-differential equation but to a family of such equations. Therefore, it is necessary to discuss the problem of the correspondence between their solutions. The integro-differential representation of the aerodynamic force is reduced to a form convenient to realize the procedure of separation of motions. In this case, we single out the first two approximations with respect to a small parameter. It turns out that in the case of actual airfoils one can speak of "detached" rather than "attached" mass. In the problem on the forced drag of an airfoil in a flow, it is shown that for a sufficiently large acceleration the aerodynamic force can change its direction and turn from a drag force into an "accelerating" force for some time. At the same time, in the case of free drag of a sufficiently light plate, the "acceleration" effect is not observed, but in the course of deceleration the plate moves from it original position in the direction opposite to the initial direction of motion.

  19. 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.

  20. Ordinary differential equations with applications in molecular biology.

    PubMed

    Ilea, M; Turnea, M; Rotariu, M

    2012-01-01

    Differential equations are of basic importance in molecular biology mathematics because many biological laws and relations appear mathematically in the form of a differential equation. In this article we presented some applications of mathematical models represented by ordinary differential equations in molecular biology. The vast majority of quantitative models in cell and molecular biology are formulated in terms of ordinary differential equations for the time evolution of concentrations of molecular species. Assuming that the diffusion in the cell is high enough to make the spatial distribution of molecules homogenous, these equations describe systems with many participating molecules of each kind. We propose an original mathematical model with small parameter for biological phospholipid pathway. All the equations system includes small parameter epsilon. The smallness of epsilon is relative to the size of the solution domain. If we reduce the size of the solution region the same small epsilon will result in a different condition number. It is clear that the solution for a smaller region is less difficult. We introduce the mathematical technique known as boundary function method for singular perturbation system. In this system, the small parameter is an asymptotic variable, different from the independent variable. In general, the solutions of such equations exhibit multiscale phenomena. Singularly perturbed problems form a special class of problems containing a small parameter which may tend to zero. Many molecular biology processes can be quantitatively characterized by ordinary differential equations. Mathematical cell biology is a very active and fast growing interdisciplinary area in which mathematical concepts, techniques, and models are applied to a variety of problems in developmental medicine and bioengineering. Among the different modeling approaches, ordinary differential equations (ODE) are particularly important and have led to significant advances

  1. Solution of differential equations by application of transformation groups

    NASA Technical Reports Server (NTRS)

    Driskell, C. N., Jr.; Gallaher, L. J.; Martin, R. H., Jr.

    1968-01-01

    Report applies transformation groups to the solution of systems of ordinary differential equations and partial differential equations. Lies theorem finds an integrating factor for appropriate invariance group or groups can be found and can be extended to partial differential equations.

  2. 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.

  3. Modular Expression Language for Ordinary Differential Equation Editing

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

    Blake, Robert C.

    MELODEEis a system for describing systems of initial value problem ordinary differential equations, and a compiler for the language that produces optimized code to integrate the differential equations. Features include rational polynomial approximation for expensive functions and automatic differentiation for symbolic jacobians

  4. Laplace and the era of differential equations

    NASA Astrophysics Data System (ADS)

    Weinberger, Peter

    2012-11-01

    Between about 1790 and 1850 French mathematicians dominated not only mathematics, but also all other sciences. The belief that a particular physical phenomenon has to correspond to a single differential equation originates from the enormous influence Laplace and his contemporary compatriots had in all European learned circles. It will be shown that at the beginning of the nineteenth century Newton's "fluxionary calculus" finally gave way to a French-type notation of handling differential equations. A heated dispute in the Philosophical Magazine between Challis, Airy and Stokes, all three of them famous Cambridge professors of mathematics, then serves to illustrate the era of differential equations. A remark about Schrödinger and his equation for the hydrogen atom finally will lead back to present times.

  5. SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER

    DOEpatents

    Collier, D.M.; Meeks, L.A.; Palmer, J.P.

    1960-05-10

    A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.

  6. The existence of solutions of q-difference-differential equations.

    PubMed

    Wang, Xin-Li; Wang, Hua; Xu, Hong-Yan

    2016-01-01

    By using the Nevanlinna theory of value distribution, we investigate the existence of solutions of some types of non-linear q-difference differential equations. In particular, we generalize the Rellich-Wittich-type theorem and Malmquist-type theorem about differential equations to the case of q-difference differential equations (system).

  7. Solving Differential Equations Using Modified Picard Iteration

    ERIC Educational Resources Information Center

    Robin, W. A.

    2010-01-01

    Many classes of differential equations are shown to be open to solution through a method involving a combination of a direct integration approach with suitably modified Picard iterative procedures. The classes of differential equations considered include typical initial value, boundary value and eigenvalue problems arising in physics and…

  8. Variable-mesh method of solving differential equations

    NASA Technical Reports Server (NTRS)

    Van Wyk, R.

    1969-01-01

    Multistep predictor-corrector method for numerical solution of ordinary differential equations retains high local accuracy and convergence properties. In addition, the method was developed in a form conducive to the generation of effective criteria for the selection of subsequent step sizes in step-by-step solution of differential equations.

  9. Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

    NASA Astrophysics Data System (ADS)

    El, G. A.; Kamchatnov, A. M.; Pavlov, M. V.; Zykov, S. A.

    2011-04-01

    We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. These equations represent nonlinear integro-differential systems and have a novel structure, which we investigate by studying in detail the class of N-component `cold-gas' hydrodynamic reductions. We prove that these reductions represent integrable linearly degenerate hydrodynamic type systems for arbitrary N which is a strong evidence in favour of integrability of the full kinetic equation. We derive compact explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the `cold-gas' component densities and construct a number of exact solutions having special properties (quasiperiodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed light on the structure of a continuum limit for a large class of integrable systems of hydrodynamic type and are also relevant to the description of turbulent motion in conservative compressible flows.

  10. Parallels between control PDE's (Partial Differential Equations) and systems of ODE's (Ordinary Differential Equations)

    NASA Technical Reports Server (NTRS)

    Hunt, L. R.; Villarreal, Ramiro

    1987-01-01

    System theorists understand that the same mathematical objects which determine controllability for nonlinear control systems of ordinary differential equations (ODEs) also determine hypoellipticity for linear partial differentail equations (PDEs). Moreover, almost any study of ODE systems begins with linear systems. It is remarkable that Hormander's paper on hypoellipticity of second order linear p.d.e.'s starts with equations due to Kolmogorov, which are shown to be analogous to the linear PDEs. Eigenvalue placement by state feedback for a controllable linear system can be paralleled for a Kolmogorov equation if an appropriate type of feedback is introduced. Results concerning transformations of nonlinear systems to linear systems are similar to results for transforming a linear PDE to a Kolmogorov equation.

  11. 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.

  12. A complete analytical solution of the Fokker-Planck and balance equations for nucleation and growth of crystals

    NASA Astrophysics Data System (ADS)

    Makoveeva, Eugenya V.; Alexandrov, Dmitri V.

    2018-01-01

    This article is concerned with a new analytical description of nucleation and growth of crystals in a metastable mushy layer (supercooled liquid or supersaturated solution) at the intermediate stage of phase transition. The model under consideration consisting of the non-stationary integro-differential system of governing equations for the distribution function and metastability level is analytically solved by means of the saddle-point technique for the Laplace-type integral in the case of arbitrary nucleation kinetics and time-dependent heat or mass sources in the balance equation. We demonstrate that the time-dependent distribution function approaches the stationary profile in course of time. This article is part of the theme issue `From atomistic interfaces to dendritic patterns'.

  13. Mean-field approach to evolving spatial networks, with an application to osteocyte network formation

    NASA Astrophysics Data System (ADS)

    Taylor-King, Jake P.; Basanta, David; Chapman, S. Jonathan; Porter, Mason A.

    2017-07-01

    We consider evolving networks in which each node can have various associated properties (a state) in addition to those that arise from network structure. For example, each node can have a spatial location and a velocity, or it can have some more abstract internal property that describes something like a social trait. Edges between nodes are created and destroyed, and new nodes enter the system. We introduce a "local state degree distribution" (LSDD) as the degree distribution at a particular point in state space. We then make a mean-field assumption and thereby derive an integro-partial differential equation that is satisfied by the LSDD. We perform numerical experiments and find good agreement between solutions of the integro-differential equation and the LSDD from stochastic simulations of the full model. To illustrate our theory, we apply it to a simple model for osteocyte network formation within bones, with a view to understanding changes that may take place during cancer. Our results suggest that increased rates of differentiation lead to higher densities of osteocytes, but with a smaller number of dendrites. To help provide biological context, we also include an introduction to osteocytes, the formation of osteocyte networks, and the role of osteocytes in bone metastasis.

  14. Self-Consistent Sources Extensions of Modified Differential-Difference KP Equation

    NASA Astrophysics Data System (ADS)

    Gegenhasi; Li, Ya-Qian; Zhang, Duo-Duo

    2018-04-01

    In this paper, we investigate a modified differential-difference KP equation which is shown to have a continuum limit into the mKP equation. It is also shown that the solution of the modified differential-difference KP equation is related to the solution of the differential-difference KP equation through a Miura transformation. We first present the Grammian solution to the modified differential-difference KP equation, and then produce a coupled modified differential-difference KP system by applying the source generation procedure. The explicit N-soliton solution of the resulting coupled modified differential-difference system is expressed in compact forms by using the Grammian determinant and Casorati determinant. We also construct and solve another form of the self-consistent sources extension of the modified differential-difference KP equation, which constitutes a Bäcklund transformation for the differential-difference KP equation with self-consistent sources. Supported by the National Natural Science Foundation of China under Grant Nos. 11601247 and 11605096, the Natural Science Foundation of Inner Mongolia Autonomous Region under Grant Nos. 2016MS0115 and 2015MS0116 and the Innovation Fund Programme of Inner Mongolia University No. 20161115

  15. Algorithms For Integrating Nonlinear Differential Equations

    NASA Technical Reports Server (NTRS)

    Freed, A. D.; Walker, K. P.

    1994-01-01

    Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.

  16. A new treatment of nonlocality in scattering process

    NASA Astrophysics Data System (ADS)

    Upadhyay, N. J.; Bhagwat, A.; Jain, B. K.

    2018-01-01

    Nonlocality in the scattering potential leads to an integro-differential equation. In this equation nonlocality enters through an integral over the nonlocal potential kernel. The resulting Schrödinger equation is usually handled by approximating r,{r}{\\prime }-dependence of the nonlocal kernel. The present work proposes a novel method to solve the integro-differential equation. The method, using the mean value theorem of integral calculus, converts the nonhomogeneous term to a homogeneous term. The effective local potential in this equation turns out to be energy independent, but has relative angular momentum dependence. This method is accurate and valid for any form of nonlocality. As illustrative examples, the total and differential cross sections for neutron scattering off 12C, 56Fe and 100Mo nuclei are calculated with this method in the low energy region (up to 10 MeV) and are found to be in reasonable accord with the experiments.

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

    NASA Astrophysics Data System (ADS)

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

    2002-11-01

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

  18. The differential equation of an arbitrary reflecting surface

    NASA Astrophysics Data System (ADS)

    Melka, Richard F.; Berrettini, Vincent D.; Yousif, Hashim A.

    2018-05-01

    A differential equation describing the reflection of a light ray incident upon an arbitrary reflecting surface is obtained using the law of reflection. The derived equation is written in terms of a parameter and the value of this parameter determines the nature of the reflecting surface. Under various parametric constraints, the solution of the differential equation leads to the various conic surfaces but is not generally solvable. In addition, the dynamics of the light reflections from the conic surfaces are executed in the Mathematica software. Our derivation is the converse of the traditional approach and our analysis assumes a relation between the object distance and the image distance. This leads to the differential equation of the reflecting surface.

  19. Differential equations driven by rough paths with jumps

    NASA Astrophysics Data System (ADS)

    Friz, Peter K.; Zhang, Huilin

    2018-05-01

    We develop the rough path counterpart of Itô stochastic integration and differential equations driven by general semimartingales. This significantly enlarges the classes of (Itô/forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.

  20. Non-invertible transformations of differential-difference equations

    NASA Astrophysics Data System (ADS)

    Garifullin, R. N.; Yamilov, R. I.; Levi, D.

    2016-09-01

    We discuss aspects of the theory of non-invertible transformations of differential-difference equations and, in particular, the notion of Miura type transformation. We introduce the concept of non-Miura type linearizable transformation and we present techniques that allow one to construct simple linearizable transformations and might help one to solve classification problems. This theory is illustrated by the example of a new integrable differential-difference equation depending on five lattice points, interesting from the viewpoint of the non-invertible transformation, which relate it to an Itoh-Narita-Bogoyavlensky equation.

  1. A representation of solution of stochastic differential equations

    NASA Astrophysics Data System (ADS)

    Kim, Yoon Tae; Jeon, Jong Woo

    2006-03-01

    We prove that the logarithm of the formal power series, obtained from a stochastic differential equation, is an element in the closure of the Lie algebra generated by vector fields being coefficients of equations. By using this result, we obtain a representation of the solution of stochastic differential equations in terms of Lie brackets and iterated Stratonovich integrals in the algebra of formal power series.

  2. Long-Term Dynamics of Autonomous Fractional Differential Equations

    NASA Astrophysics Data System (ADS)

    Liu, Tao; Xu, Wei; Xu, Yong; Han, Qun

    This paper aims to investigate long-term dynamic behaviors of autonomous fractional differential equations with effective numerical method. The long-term dynamic behaviors predict where systems are heading after long-term evolution. We make some modification and transplant cell mapping methods to autonomous fractional differential equations. The mapping time duration of cell mapping is enlarged to deal with the long memory effect. Three illustrative examples, i.e. fractional Lotka-Volterra equation, fractional van der Pol oscillator and fractional Duffing equation, are studied with our revised generalized cell mapping method. We obtain long-term dynamics, such as attractors, basins of attraction, and saddles. Compared with some existing stability and numerical results, the validity of our method is verified. Furthermore, we find that the fractional order has its effect on the long-term dynamics of autonomous fractional differential equations.

  3. 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.

  4. Optimal moving grids for time-dependent partial differential equations

    NASA Technical Reports Server (NTRS)

    Wathen, A. J.

    1989-01-01

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

  5. 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

  6. On the singular perturbations for fractional differential equation.

    PubMed

    Atangana, Abdon

    2014-01-01

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

  7. Quasi-Newton methods for parameter estimation in functional differential equations

    NASA Technical Reports Server (NTRS)

    Brewer, Dennis W.

    1988-01-01

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

  8. 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.

  9. Solving Differential Equations Analytically. Elementary Differential Equations. Modules and Monographs in Undergraduate Mathematics and Its Applications Project. UMAP Unit 335.

    ERIC Educational Resources Information Center

    Goldston, J. W.

    This unit introduces analytic solutions of ordinary differential equations. The objective is to enable the student to decide whether a given function solves a given differential equation. Examples of problems from biology and chemistry are covered. Problem sets, quizzes, and a model exam are included, and answers to all items are provided. The…

  10. Liouvillian propagators, Riccati equation and differential Galois theory

    NASA Astrophysics Data System (ADS)

    Acosta-Humánez, Primitivo; Suazo, Erwin

    2013-11-01

    In this paper a Galoisian approach to building propagators through Riccati equations is presented. The main result corresponds to the relationship between the Galois integrability of the linear Schrödinger equation and the virtual solvability of the differential Galois group of its associated characteristic equation. As the main application of this approach we solve Ince’s differential equation through the Hamiltonian algebrization procedure and the Kovacic algorithm to find the propagator for a generalized harmonic oscillator. This propagator has applications which describe the process of degenerate parametric amplification in quantum optics and light propagation in a nonlinear anisotropic waveguide. Toy models of propagators inspired by integrable Riccati equations and integrable characteristic equations are also presented.

  11. Oscillation of a class of fractional differential equations with damping term.

    PubMed

    Qin, Huizeng; Zheng, Bin

    2013-01-01

    We investigate the oscillation of a class of fractional differential equations with damping term. Based on a certain variable transformation, the fractional differential equations are converted into another differential equations of integer order with respect to the new variable. Then, using Riccati transformation, inequality, and integration average technique, some new oscillatory criteria for the equations are established. As for applications, oscillation for two certain fractional differential equations with damping term is investigated by the use of the presented results.

  12. 1/f Noise from nonlinear stochastic differential equations.

    PubMed

    Ruseckas, J; Kaulakys, B

    2010-03-01

    We consider a class of nonlinear stochastic differential equations, giving the power-law behavior of the power spectral density in any desirably wide range of frequency. Such equations were obtained starting from the point process models of 1/fbeta noise. In this article the power-law behavior of spectrum is derived directly from the stochastic differential equations, without using the point process models. The analysis reveals that the power spectrum may be represented as a sum of the Lorentzian spectra. Such a derivation provides additional justification of equations, expands the class of equations generating 1/fbeta noise, and provides further insights into the origin of 1/fbeta noise.

  13. On the Singular Perturbations for Fractional Differential Equation

    PubMed Central

    Atangana, Abdon

    2014-01-01

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

  14. On the Inclusion of Difference Equation Problems and Z Transform Methods in Sophomore Differential Equation Classes

    ERIC Educational Resources Information Center

    Savoye, Philippe

    2009-01-01

    In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.

  15. Application of AWE Along with a Combined FEM/MoM Technique to Compute RCS of a Cavity-Backed Aperture in an Infinite Ground Plane Over a Frequency Range

    NASA Technical Reports Server (NTRS)

    Reddy, C.J.; Deshpande, M.D.

    1997-01-01

    A hybrid Finite Element Method (FEM)/Method of Moments (MoM) technique in conjunction with the Asymptotic Waveform Evaluation (AWE) technique is applied to obtain radar cross section (RCS) of a cavity-backed aperture in an infinite ground plane over a frequency range. The hybrid FEM/MoM technique when applied to the cavity-backed aperture results in an integro-differential equation with electric field as the unknown variable, the electric field obtained from the solution of the integro-differential equation is expanded in Taylor series. The coefficients of the Taylor series are obtained using the frequency derivatives of the integro-differential equation formed by the hybrid FEM/MoM technique. The series is then matched via the Pade approximation to a rational polynomial, which can be used to extrapolate the electric field over a frequency range. The RCS of the cavity-backed aperture is calculated using the electric field at different frequencies. Numerical results for a rectangular cavity, a circular cavity, and a material filled cavity are presented over a frequency range. Good agreement between AWE and the exact solution over the frequency range is obtained.

  16. 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.

  17. Program for solution of ordinary differential equations

    NASA Technical Reports Server (NTRS)

    Sloate, H.

    1973-01-01

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

  18. Transformation matrices between non-linear and linear differential equations

    NASA Technical Reports Server (NTRS)

    Sartain, R. L.

    1983-01-01

    In the linearization of systems of non-linear differential equations, those systems which can be exactly transformed into the second order linear differential equation Y"-AY'-BY=0 where Y, Y', and Y" are n x 1 vectors and A and B are constant n x n matrices of real numbers were considered. The 2n x 2n matrix was used to transform the above matrix equation into the first order matrix equation X' = MX. Specially the matrix M and the conditions which will diagonalize or triangularize M were studied. Transformation matrices P and P sub -1 were used to accomplish this diagonalization or triangularization to return to the solution of the second order matrix differential equation system from the first order system.

  19. Monograph - The Numerical Integration of Ordinary Differential Equations.

    ERIC Educational Resources Information Center

    Hull, T. E.

    The materials presented in this monograph are intended to be included in a course on ordinary differential equations at the upper division level in a college mathematics program. These materials provide an introduction to the numerical integration of ordinary differential equations, and they can be used to supplement a regular text on this…

  20. 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

  1. A procedure to construct exact solutions of nonlinear fractional differential equations.

    PubMed

    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.

  2. 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.

  3. Modelling by Differential Equations

    ERIC Educational Resources Information Center

    Chaachoua, Hamid; Saglam, Ayse

    2006-01-01

    This paper aims to show the close relation between physics and mathematics taking into account especially the theory of differential equations. By analysing the problems posed by scientists in the seventeenth century, we note that physics is very important for the emergence of this theory. Taking into account this analysis, we show the…

  4. The role of nonlinear critical layers in boundary layer transition

    NASA Technical Reports Server (NTRS)

    Goldstein, M.E.

    1995-01-01

    Asymptotic methods are used to describe the nonlinear self-interaction between pairs of oblique instability modes that eventually develops when initially linear spatially growing instability waves evolve downstream in nominally two-dimensional laminar boundary layers. The first nonlinear reaction takes place locally within a so-called 'critical layer', with the flow outside this layer consisting of a locally parallel mean flow plus a pair of oblique instability waves - which may or may not be accompanied by an associated plane wave. The amplitudes of these waves, which are completely determined by nonlinear effects within the critical layer, satisfy either a single integro-differential equation or a pair of integro-differential equations with quadratic to quartic-type nonlinearities. The physical implications of these equations are discussed.

  5. Nonstandard Topics for Student Presentations in Differential Equations

    ERIC Educational Resources Information Center

    LeMasurier, Michelle

    2006-01-01

    An interesting and effective way to showcase the wide variety of fields to which differential equations can be applied is to have students give short oral presentations on a specific application. These talks, which have been presented by 30-40 students per year in our differential equations classes, provide exposure to a diverse array of topics…

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

    NASA Technical Reports Server (NTRS)

    Geddes, K. O.

    1977-01-01

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

  7. Simplifying Differential Equations for Multiscale Feynman Integrals beyond Multiple Polylogarithms.

    PubMed

    Adams, Luise; Chaubey, Ekta; Weinzierl, Stefan

    2017-04-07

    In this Letter we exploit factorization properties of Picard-Fuchs operators to decouple differential equations for multiscale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator. As a side product, our method can be used to easily convert the differential equations for Feynman integrals which evaluate to multiple polylogarithms to an ϵ form.

  8. Wilsonian Renormalization Group and the Lippmann-Schwinger Equation with a Multitude of Cutoff Parameters

    NASA Astrophysics Data System (ADS)

    Epelbaum, E.; Gegelia, J.; Meißner, Ulf-G.

    2018-03-01

    The Wilsonian renormalization group approach to the Lippmann-Schwinger equation with a multitude of cutoff parameters is introduced. A system of integro-differential equations for the cutoff-dependent potential is obtained. As an illustration, a perturbative solution of these equations with two cutoff parameters for a simple case of an S-wave low-energy potential in the form of a Taylor series in momenta is obtained. The relevance of the obtained results for the effective field theory approach to nucleon-nucleon scattering is discussed. Supported in part by BMBF under Grant No. 05P2015 - NUSTAR R&D), DFG and NSFC through Funds Provided to the Sino- German CRC 110 “Symmetries and the Emergence of Structure in QCD”, National Natural Science Foundation of China under Grant No. 11621131001, DFG Grant No. TRR110, the Georgian Shota Rustaveli National Science Foundation (grant FR/417/6-100/14) and the CAS President’s International Fellowship Initiative (PIFI) under Grant No. 2017VMA0025

  9. Solving constant-coefficient differential equations with dielectric metamaterials

    NASA Astrophysics Data System (ADS)

    Zhang, Weixuan; Qu, Che; Zhang, Xiangdong

    2016-07-01

    Recently, the concept of metamaterial analog computing has been proposed (Silva et al 2014 Science 343 160-3). Some mathematical operations such as spatial differentiation, integration, and convolution, have been performed by using designed metamaterial blocks. Motivated by this work, we propose a practical approach based on dielectric metamaterial to solve differential equations. The ordinary differential equation can be solved accurately by the correctly designed metamaterial system. The numerical simulations using well-established numerical routines have been performed to successfully verify all theoretical analyses.

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

    PubMed

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

    2013-03-25

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

  11. Informed Conjecturing of Solutions for Differential Equations in a Modeling Context

    ERIC Educational Resources Information Center

    Winkel, Brian

    2015-01-01

    We examine two differential equations. (i) first-order exponential growth or decay; and (ii) second order, linear, constant coefficient differential equations, and show the advantage of learning differential equations in a modeling context for informed conjectures of their solution. We follow with a discussion of the complete analysis afforded by…

  12. Solving Differential Equations in R

    EPA Science Inventory

    Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here w...

  13. Lattice Boltzmann model for high-order nonlinear partial differential equations

    NASA Astrophysics Data System (ADS)

    Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

    2018-01-01

    In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂tϕ +∑k=1mαk∂xkΠk(ϕ ) =0 (1 ≤k ≤m ≤6 ), αk are constant coefficients, Πk(ϕ ) are some known differential functions of ϕ . As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K (n ,n ) -Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009), 10.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009), 10.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

  14. Lattice Boltzmann model for high-order nonlinear partial differential equations.

    PubMed

    Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

    2018-01-01

    In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

  15. Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation.

    PubMed

    Müller, Eike H; Scheichl, Rob; Shardlow, Tony

    2015-04-08

    This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.

  16. Differential geometry techniques for sets of nonlinear partial differential equations

    NASA Technical Reports Server (NTRS)

    Estabrook, Frank B.

    1990-01-01

    An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.

  17. On the hierarchy of partially invariant submodels of differential equations

    NASA Astrophysics Data System (ADS)

    Golovin, Sergey V.

    2008-07-01

    It is noted that the partially invariant solution (PIS) of differential equations in many cases can be represented as an invariant reduction of some PISs of the higher rank. This introduces a hierarchic structure in the set of all PISs of a given system of differential equations. An equivalence of the two-step and the direct ways of construction of PISs is proved. The hierarchy simplifies the process of enumeration and analysis of partially invariant submodels to the given system of differential equations. In this framework, the complete classification of regular partially invariant solutions of ideal MHD equations is given.

  18. Polynomial mixture method of solving ordinary differential equations

    NASA Astrophysics Data System (ADS)

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

    2017-11-01

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

  19. Real-time optical laboratory solution of parabolic differential equations

    NASA Technical Reports Server (NTRS)

    Casasent, David; Jackson, James

    1988-01-01

    An optical laboratory matrix-vector processor is used to solve parabolic differential equations (the transient diffusion equation with two space variables and time) by an explicit algorithm. This includes optical matrix-vector nonbase-2 encoded laboratory data, the combination of nonbase-2 and frequency-multiplexed data on such processors, a high-accuracy optical laboratory solution of a partial differential equation, new data partitioning techniques, and a discussion of a multiprocessor optical matrix-vector architecture.

  20. A neuro approach to solve fuzzy Riccati differential equations

    NASA Astrophysics Data System (ADS)

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

    2015-10-01

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

  1. A neuro approach to solve fuzzy Riccati differential equations

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

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

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

  2. Undergraduate Students' Mental Operations in Systems of Differential Equations

    ERIC Educational Resources Information Center

    Whitehead, Karen; Rasmussen, Chris

    2003-01-01

    This paper reports on research conducted to understand undergraduate students' ways of reasoning about systems of differential equations (SDEs). As part of a semester long classroom teaching experiment in a first course in differential equations, we conducted task-based interviews with six students after their study of first order differential…

  3. Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

    PubMed Central

    Müller, Eike H.; Scheichl, Rob; Shardlow, Tony

    2015-01-01

    This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy. PMID:27547075

  4. Periodicity and positivity of a class of fractional differential equations.

    PubMed

    Ibrahim, Rabha W; Ahmad, M Z; Mohammed, M Jasim

    2016-01-01

    Fractional differential equations have been discussed in this study. We utilize the Riemann-Liouville fractional calculus to implement it within the generalization of the well known class of differential equations. The Rayleigh differential equation has been generalized of fractional second order. The existence of periodic and positive outcome is established in a new method. The solution is described in a fractional periodic Sobolev space. Positivity of outcomes is considered under certain requirements. We develop and extend some recent works. An example is constructed.

  5. Modeling animal movements using stochastic differential equations

    Treesearch

    Haiganoush K. Preisler; Alan A. Ager; Bruce K. Johnson; John G. Kie

    2004-01-01

    We describe the use of bivariate stochastic differential equations (SDE) for modeling movements of 216 radiocollared female Rocky Mountain elk at the Starkey Experimental Forest and Range in northeastern Oregon. Spatially and temporally explicit vector fields were estimated using approximating difference equations and nonparametric regression techniques. Estimated...

  6. Synchronization with propagation - The functional differential equations

    NASA Astrophysics Data System (ADS)

    Rǎsvan, Vladimir

    2016-06-01

    The structure represented by one or several oscillators couple to a one-dimensional transmission environment (e.g. a vibrating string in the mechanical case or a lossless transmission line in the electrical case) turned to be attractive for the research in the field of complex structures and/or complex behavior. This is due to the fact that such a structure represents some generalization of various interconnection modes with lumped parameters for the oscillators. On the other hand the lossless and distortionless propagation along transmission lines has generated several research in electrical, thermal, hydro and control engineering leading to the association of some functional differential equations to the basic initial boundary value problems. The present research is performed at the crossroad of the aforementioned directions. We shall associate to the starting models some functional differential equations - in most cases of neutral type - and make use of the general theorems for existence and stability of forced oscillations for functional differential equations. The challenges introduced by the analyzed problems for the general theory are emphasized, together with the implication of the results for various applications.

  7. Approximation of Quantum Stochastic Differential Equations for Input-Output Model Reduction

    DTIC Science & Technology

    2016-02-25

    Approximation of Quantum Stochastic Differential Equations for Input-Output Model Reduction We have completed a short program of theoretical research...on dimensional reduction and approximation of models based on quantum stochastic differential equations. Our primary results lie in the area of...2211 quantum probability, quantum stochastic differential equations REPORT DOCUMENTATION PAGE 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 10. SPONSOR

  8. 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.

  9. 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.

  10. The nonlinear evolution of modes on unstable stratified shear layers

    NASA Technical Reports Server (NTRS)

    Blackaby, Nicholas; Dando, Andrew; Hall, Philip

    1993-01-01

    The nonlinear development of disturbances in stratified shear flows (having a local Richardson number of value less than one quarter) is considered. Such modes are initially fast growing but, like related studies, we assume that the viscous, non-parallel spreading of the shear layer results in them evolving in a linear fashion until they reach a position where their amplitudes are large enough and their growth rates have diminished sufficiently so that amplitude equations can be derived using weakly nonlinear and non-equilibrium critical-layer theories. Four different basic integro-differential amplitude equations are possible, including one due to a novel mechanism; the relevant choice of amplitude equation, at a particular instance, being dependent on the relative sizes of the disturbance amplitude, the growth rate of the disturbance, its wavenumber, and the viscosity of the fluid. This richness of choice of possible nonlinearities arises mathematically from the indicial Frobenius roots of the governing linear inviscid equation (the Taylor-Goldstein equation) not, in general, differing by an integer. The initial nonlinear evolution of a mode will be governed by an integro-differential amplitude equations with a cubic nonlinearity but the resulting significant increase in the size of the disturbance's amplitude leads on to the next stage of the evolution process where the evolution of the mode is governed by an integro-differential amplitude equations with a quintic nonlinearity. Continued growth of the disturbance amplitude is expected during this stage, resulting in the effects of nonlinearity spreading to outside the critical level, by which time the flow has become fully nonlinear.

  11. 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.

  12. 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.

  13. Algorithmic framework for group analysis of differential equations and its application to generalized Zakharov-Kuznetsov equations

    NASA Astrophysics Data System (ADS)

    Huang, Ding-jiang; Ivanova, Nataliya M.

    2016-02-01

    In this paper, we explain in more details the modern treatment of the problem of group classification of (systems of) partial differential equations (PDEs) from the algorithmic point of view. More precisely, we revise the classical Lie algorithm of construction of symmetries of differential equations, describe the group classification algorithm and discuss the process of reduction of (systems of) PDEs to (systems of) equations with smaller number of independent variables in order to construct invariant solutions. The group classification algorithm and reduction process are illustrated by the example of the generalized Zakharov-Kuznetsov (GZK) equations of form ut +(F (u)) xxx +(G (u)) xyy +(H (u)) x = 0. As a result, a complete group classification of the GZK equations is performed and a number of new interesting nonlinear invariant models which have non-trivial invariance algebras are obtained. Lie symmetry reductions and exact solutions for two important invariant models, i.e., the classical and modified Zakharov-Kuznetsov equations, are constructed. The algorithmic framework for group analysis of differential equations presented in this paper can also be applied to other nonlinear PDEs.

  14. On the integration of a class of nonlinear systems of ordinary differential equations

    NASA Astrophysics Data System (ADS)

    Talyshev, Aleksandr A.

    2017-11-01

    For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

  15. BOOK REVIEW: Partial Differential Equations in General Relativity

    NASA Astrophysics Data System (ADS)

    Halburd, Rodney G.

    2008-11-01

    Although many books on general relativity contain an overview of the relevant background material from differential geometry, very little attention is usually paid to background material from the theory of differential equations. This is understandable in a first course on relativity but it often limits the kinds of problems that can be studied rigorously. Einstein's field equations lie at the heart of general relativity. They are a system of partial differential equations (PDEs) relating the curvature of spacetime to properties of matter. A central part of most problems in general relativity is to extract information about solutions of these equations. Most standard texts achieve this by studying exact solutions or numerical and analytical approximations. In the book under review, Alan Rendall emphasises the role of rigorous qualitative methods in general relativity. There has long been a need for such a book, giving a broad overview of the relevant background from the theory of partial differential equations, and not just from differential geometry. It should be noted that the book also covers the basic theory of ordinary differential equations. Although there are many good books on the rigorous theory of PDEs, methods related to the Einstein equations deserve special attention, not only because of the complexity and importance of these equations, but because these equations do not fit into any of the standard classes of equations (elliptic, parabolic, hyperbolic) that one typically encounters in a course on PDEs. Even specifying exactly what ones means by a Cauchy problem in general relativity requires considerable care. The main problem here is that the manifold on which the solution is defined is determined by the solution itself. This means that one does not simply define data on a submanifold. Rendall's book gives a good overview of applications and results from the qualitative theory of PDEs to general relativity. It would be impossible to give detailed

  16. Parameter Estimates in Differential Equation Models for Chemical Kinetics

    ERIC Educational Resources Information Center

    Winkel, Brian

    2011-01-01

    We discuss the need for devoting time in differential equations courses to modelling and the completion of the modelling process with efforts to estimate the parameters in the models using data. We estimate the parameters present in several differential equation models of chemical reactions of order n, where n = 0, 1, 2, and apply more general…

  17. Intuitive Understanding of Solutions of Partially Differential Equations

    ERIC Educational Resources Information Center

    Kobayashi, Y.

    2008-01-01

    This article uses diagrams that help the observer see how solutions of the wave equation and heat conduction equation are obtained. The analytical approach cannot necessarily show the mechanisms of the key to the solution without transforming the differential equation into a more convenient form by separation of variables. The visual clues based…

  18. 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.

  19. A Unified Introduction to Ordinary Differential Equations

    ERIC Educational Resources Information Center

    Lutzer, Carl V.

    2006-01-01

    This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)

  20. Solution of some types of differential equations: operational calculus and inverse differential operators.

    PubMed

    Zhukovsky, K

    2014-01-01

    We present a general method of operational nature to analyze and obtain solutions for a variety of equations of mathematical physics and related mathematical problems. We construct inverse differential operators and produce operational identities, involving inverse derivatives and families of generalised orthogonal polynomials, such as Hermite and Laguerre polynomial families. We develop the methodology of inverse and exponential operators, employing them for the study of partial differential equations. Advantages of the operational technique, combined with the use of integral transforms, generating functions with exponentials and their integrals, for solving a wide class of partial derivative equations, related to heat, wave, and transport problems, are demonstrated.

  1. 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.

  2. Diffusion Influenced Adsorption Kinetics.

    PubMed

    Miura, Toshiaki; Seki, Kazuhiko

    2015-08-27

    When the kinetics of adsorption is influenced by the diffusive flow of solutes, the solute concentration at the surface is influenced by the surface coverage of solutes, which is given by the Langmuir-Hinshelwood adsorption equation. The diffusion equation with the boundary condition given by the Langmuir-Hinshelwood adsorption equation leads to the nonlinear integro-differential equation for the surface coverage. In this paper, we solved the nonlinear integro-differential equation using the Grünwald-Letnikov formula developed to solve fractional kinetics. Guided by the numerical results, analytical expressions for the upper and lower bounds of the exact numerical results were obtained. The upper and lower bounds were close to the exact numerical results in the diffusion- and reaction-controlled limits, respectively. We examined the validity of the two simple analytical expressions obtained in the diffusion-controlled limit. The results were generalized to include the effect of dispersive diffusion. We also investigated the effect of molecular rearrangement of anisotropic molecules on surface coverage.

  3. Compatible Spatial Discretizations for Partial Differential Equations

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

    Arnold, Douglas, N, ed.

    From May 11--15, 2004, the Institute for Mathematics and its Applications held a hot topics workshop on Compatible Spatial Discretizations for Partial Differential Equations. The numerical solution of partial differential equations (PDE) is a fundamental task in science and engineering. The goal of the workshop was to bring together a spectrum of scientists at the forefront of the research in the numerical solution of PDEs to discuss compatible spatial discretizations. We define compatible spatial discretizations as those that inherit or mimic fundamental properties of the PDE such as topology, conservation, symmetries, and positivity structures and maximum principles. A wide varietymore » of discretization methods applied across a wide range of scientific and engineering applications have been designed to or found to inherit or mimic intrinsic spatial structure and reproduce fundamental properties of the solution of the continuous PDE model at the finite dimensional level. A profusion of such methods and concepts relevant to understanding them have been developed and explored: mixed finite element methods, mimetic finite differences, support operator methods, control volume methods, discrete differential forms, Whitney forms, conservative differencing, discrete Hodge operators, discrete Helmholtz decomposition, finite integration techniques, staggered grid and dual grid methods, etc. This workshop seeks to foster communication among the diverse groups of researchers designing, applying, and studying such methods as well as researchers involved in practical solution of large scale problems that may benefit from advancements in such discretizations; to help elucidate the relations between the different methods and concepts; and to generally advance our understanding in the area of compatible spatial discretization methods for PDE. Particular points of emphasis included: + Identification of intrinsic properties of PDE models that are critical for the fidelity of

  4. Solution of Volterra and Fredholm Classes of Equations via Triangular Orthogonal Function (A Combination of Right Hand Triangular Function and Left Hand Triangular Function) and Hybrid Orthogonal Function (A Combination of Sample Hold Function and Right Hand Triangular Function)

    NASA Astrophysics Data System (ADS)

    Mukhopadhyay, Anirban; Ganguly, Anindita; Chatterjee, Saumya Deep

    2018-04-01

    In this paper the authors have dealt with seven kinds of non-linear Volterra and Fredholm classes of equations. The authors have formulated an algorithm for solving the aforementioned equation types via Hybrid Function (HF) and Triangular Function (TF) piecewise-linear orthogonal approach. In this approach the authors have reduced integral equation or integro-differential equation into equivalent system of simultaneous non-linear equation and have employed either Newton's method or Broyden's method to solve the simultaneous non-linear equations. The authors have calculated the L2-norm error and the max-norm error for both HF and TF method for each kind of equations. Through the illustrated examples, the authors have shown that the HF based algorithm produces stable result, on the contrary TF-computational method yields either stable, anomalous or unstable results.

  5. Chaotic attractors in tumor growth and decay: a differential equation model.

    PubMed

    Harney, Michael; Yim, Wen-sau

    2015-01-01

    Tumorigenesis can be modeled as a system of chaotic nonlinear differential equations. A simulation of the system is realized by converting the differential equations to difference equations. The results of the simulation show that an increase in glucose in the presence of low oxygen levels decreases tumor growth.

  6. Optimal moving grids for time-dependent partial differential equations

    NASA Technical Reports Server (NTRS)

    Wathen, A. J.

    1992-01-01

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

  7. Entropy and convexity for nonlinear partial differential equations

    PubMed Central

    Ball, John M.; Chen, Gui-Qiang G.

    2013-01-01

    Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue. PMID:24249768

  8. Entropy and convexity for nonlinear partial differential equations.

    PubMed

    Ball, John M; Chen, Gui-Qiang G

    2013-12-28

    Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue.

  9. [Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (1)].

    PubMed

    Murase, Kenya

    2014-01-01

    Utilization of differential equations and methods for solving them in medical physics are presented. First, the basic concept and the kinds of differential equations were overviewed. Second, separable differential equations and well-known first-order and second-order differential equations were introduced, and the methods for solving them were described together with several examples. In the next issue, the symbolic and series expansion methods for solving differential equations will be mainly introduced.

  10. Dual exponential polynomials and linear differential equations

    NASA Astrophysics Data System (ADS)

    Wen, Zhi-Tao; Gundersen, Gary G.; Heittokangas, Janne

    2018-01-01

    We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.

  11. Pendulum Motion and Differential Equations

    ERIC Educational Resources Information Center

    Reid, Thomas F.; King, Stephen C.

    2009-01-01

    A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…

  12. 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.

  13. Solving Fuzzy Fractional Differential Equations Using Zadeh's Extension Principle

    PubMed Central

    Ahmad, M. Z.; Hasan, M. K.; Abbasbandy, S.

    2013-01-01

    We study a fuzzy fractional differential equation (FFDE) and present its solution using Zadeh's extension principle. The proposed study extends the case of fuzzy differential equations of integer order. We also propose a numerical method to approximate the solution of FFDEs. To solve nonlinear problems, the proposed numerical method is then incorporated into an unconstrained optimisation technique. Several numerical examples are provided. PMID:24082853

  14. Connecting Related Rates and Differential Equations

    ERIC Educational Resources Information Center

    Brandt, Keith

    2012-01-01

    This article points out a simple connection between related rates and differential equations. The connection can be used for in-class examples or homework exercises, and it is accessible to students who are familiar with separation of variables.

  15. [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.

  16. Thermoacoustic tomography for an integro-differential wave equation modeling attenuation

    NASA Astrophysics Data System (ADS)

    Acosta, Sebastián; Palacios, Benjamín

    2018-02-01

    In this article we study the inverse problem of thermoacoustic tomography (TAT) on a medium with attenuation represented by a time-convolution (or memory) term, and whose consideration is motivated by the modeling of ultrasound waves in heterogeneous tissue via fractional derivatives with spatially dependent parameters. Under the assumption of being able to measure data on the whole boundary, we prove uniqueness and stability, and propose a convergent reconstruction method for a class of smooth variable sound speeds. By a suitable modification of the time reversal technique, we obtain a Neumann series reconstruction formula.

  17. A Bifurcation Problem for a Nonlinear Partial Differential Equation of Parabolic Type,

    DTIC Science & Technology

    NONLINEAR DIFFERENTIAL EQUATIONS, INTEGRATION), (*PARTIAL DIFFERENTIAL EQUATIONS, BOUNDARY VALUE PROBLEMS), BANACH SPACE , MAPPING (TRANSFORMATIONS), SET THEORY, TOPOLOGY, ITERATIONS, STABILITY, THEOREMS

  18. Entire solutions of nonlinear differential-difference equations.

    PubMed

    Li, Cuiping; Lü, Feng; Xu, Junfeng

    2016-01-01

    In this paper, we describe the properties of entire solutions of a nonlinear differential-difference equation and a Fermat type equation, and improve several previous theorems greatly. In addition, we also deduce a uniqueness result for an entire function f(z) that shares a set with its shift [Formula: see text], which is a generalization of a result of Liu.

  19. Sourcing for Parameter Estimation and Study of Logistic Differential Equation

    ERIC Educational Resources Information Center

    Winkel, Brian J.

    2012-01-01

    This article offers modelling opportunities in which the phenomena of the spread of disease, perception of changing mass, growth of technology, and dissemination of information can be described by one differential equation--the logistic differential equation. It presents two simulation activities for students to generate real data, as well as…

  20. The Hartman-Grobman theorem for semilinear hyperbolic evolution equations

    NASA Astrophysics Data System (ADS)

    Hein, Marie-Luise; Prüss, Jan

    2016-10-01

    The famous Hartman-Grobman theorem for ordinary differential equations is extended to abstract semilinear hyperbolic evolution equations in Banach spaces by means of simple direct proof. It is also shown that the linearising map is Hölder continuous. Several applications to abstract and specific damped wave equations are given, to demonstrate the strength of our results.

  1. Adaptation and Extension of the Framework of Reducing Abstraction in the Case of Differential Equations

    ERIC Educational Resources Information Center

    Raychaudhuri, Debasree

    2014-01-01

    Although there is no consensus in regard to a unique meaning for abstraction, there is a recognition of the existence of several theories of abstraction, and that the ability to abstract is imperative to learning and doing meaningful mathematics. The theory of "reducing abstraction" maps the abstract nature of mathematics to the nature…

  2. An Engineering-Oriented Approach to the Introductory Differential Equations Course

    ERIC Educational Resources Information Center

    Pennell, S.; Avitabile, P.; White, J.

    2009-01-01

    The introductory differential equations course can be made more relevant to engineering students by including more of the engineering viewpoint, in which differential equations are regarded as systems with inputs and outputs. This can be done without sacrificing any of the usual topical coverage. This point of view is conducive to student…

  3. Auto-Bäcklund transformations for a matrix partial differential equation

    NASA Astrophysics Data System (ADS)

    Gordoa, P. R.; Pickering, A.

    2018-07-01

    We derive auto-Bäcklund transformations, analogous to those of the matrix second Painlevé equation, for a matrix partial differential equation. We also then use these auto-Bäcklund transformations to derive matrix equations involving shifts in a discrete variable, a process analogous to the use of the auto-Bäcklund transformations of the matrix second Painlevé equation to derive a discrete matrix first Painlevé equation. The equations thus derived then include amongst other examples a semidiscrete matrix equation which can be considered to be an extension of this discrete matrix first Painlevé equation. The application of this technique to the auto-Bäcklund transformations of the scalar case of our partial differential equation has not been considered before, and so the results obtained here in this scalar case are also new. Other equations obtained here using this technique include a scalar semidiscrete equation which arises in the case of the second Painlevé equation, and which does not seem to have been thus derived previously.

  4. 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.

  5. Razumikhin-Type Stability Criteria for Differential Equations with Delayed Impulses.

    PubMed

    Wang, Qing; Zhu, Quanxin

    2013-01-01

    This paper studies stability problems of general impulsive differential equations where time delays occur in both differential and difference equations. Based on the method of Lyapunov functions, Razumikhin technique and mathematical induction, several stability criteria are obtained for differential equations with delayed impulses. Our results show that some systems with delayed impulses may be exponentially stabilized by impulses even if the system matrices are unstable. Some less restrictive sufficient conditions are also given to keep the good stability property of systems subject to certain type of impulsive perturbations. Examples with numerical simulations are discussed to illustrate the theorems. Our results may be applied to complex problems where impulses depend on both current and past states.

  6. Effect of Differential Item Functioning on Test Equating

    ERIC Educational Resources Information Center

    Kabasakal, Kübra Atalay; Kelecioglu, Hülya

    2015-01-01

    This study examines the effect of differential item functioning (DIF) items on test equating through multilevel item response models (MIRMs) and traditional IRMs. The performances of three different equating models were investigated under 24 different simulation conditions, and the variables whose effects were examined included sample size, test…

  7. 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.

  8. 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.

  9. Computational attributes of the integral form of the equation of transfer

    NASA Technical Reports Server (NTRS)

    Frankel, J. I.

    1991-01-01

    Difficulties can arise in radiative and neutron transport calculations when a highly anisotropic scattering phase function is present. In the presence of anisotropy, currently used numerical solutions are based on the integro-differential form of the linearized Boltzmann transport equation. This paper, departs from classical thought and presents an alternative numerical approach based on application of the integral form of the transport equation. Use of the integral formalism facilitates the following steps: a reduction in dimensionality of the system prior to discretization, the use of symbolic manipulation to augment the computational procedure, and the direct determination of key physical quantities which are derivable through the various Legendre moments of the intensity. The approach is developed in the context of radiative heat transfer in a plane-parallel geometry, and results are presented and compared with existing benchmark solutions. Encouraging results are presented to illustrate the potential of the integral formalism for computation. The integral formalism appears to possess several computational attributes which are well-suited to radiative and neutron transport calculations.

  10. Variations in the Solution of Linear First-Order Differential Equations. Classroom Notes

    ERIC Educational Resources Information Center

    Seaman, Brian; Osler, Thomas J.

    2004-01-01

    A special project which can be given to students of ordinary differential equations is described in detail. Students create new differential equations by changing the dependent variable in the familiar linear first-order equation (dv/dx)+p(x)v=q(x) by means of a substitution v=f(y). The student then creates a table of the new equations and…

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

    NASA Astrophysics Data System (ADS)

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

    2018-04-01

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

  12. Analysis of backward differentiation formula for nonlinear differential-algebraic equations with 2 delays.

    PubMed

    Sun, Leping

    2016-01-01

    This paper is concerned with the backward differential formula or BDF methods for a class of nonlinear 2-delay differential algebraic equations. We obtain two sufficient conditions under which the methods are stable and asymptotically stable. At last, examples show that our methods are true.

  13. Transforming parts of a differential equations system to difference equations as a method for run-time savings in NONMEM.

    PubMed

    Petersson, K J F; Friberg, L E; Karlsson, M O

    2010-10-01

    Computer models of biological systems grow more complex as computing power increase. Often these models are defined as differential equations and no analytical solutions exist. Numerical integration is used to approximate the solution; this can be computationally intensive, time consuming and be a large proportion of the total computer runtime. The performance of different integration methods depend on the mathematical properties of the differential equations system at hand. In this paper we investigate the possibility of runtime gains by calculating parts of or the whole differential equations system at given time intervals, outside of the differential equations solver. This approach was tested on nine models defined as differential equations with the goal to reduce runtime while maintaining model fit, based on the objective function value. The software used was NONMEM. In four models the computational runtime was successfully reduced (by 59-96%). The differences in parameter estimates, compared to using only the differential equations solver were less than 12% for all fixed effects parameters. For the variance parameters, estimates were within 10% for the majority of the parameters. Population and individual predictions were similar and the differences in OFV were between 1 and -14 units. When computational runtime seriously affects the usefulness of a model we suggest evaluating this approach for repetitive elements of model building and evaluation such as covariate inclusions or bootstraps.

  14. Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations

    NASA Astrophysics Data System (ADS)

    Demina, Maria V.; Kudryashov, Nikolay A.

    2011-03-01

    Meromorphic solutions of autonomous nonlinear ordinary differential equations are studied. An algorithm for constructing meromorphic solutions in explicit form is presented. General expressions for meromorphic solutions (including rational, periodic, elliptic) are found for a wide class of autonomous nonlinear ordinary differential equations.

  15. 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.

  16. Turbulence in the Ott-Antonsen equation for arrays of coupled phase oscillators

    NASA Astrophysics Data System (ADS)

    Wolfrum, M.; Gurevich, S. V.; Omel'chenko, O. E.

    2016-02-01

    In this paper we study the transition to synchrony in an one-dimensional array of oscillators with non-local coupling. For its description in the continuum limit of a large number of phase oscillators, we use a corresponding Ott-Antonsen equation, which is an integro-differential equation for the evolution of the macroscopic profiles of the local mean field. Recently, it was reported that in the spatially extended case at the synchronisation threshold there appear partially coherent plane waves with different wave numbers, which are organised in the well-known Eckhaus scenario. In this paper, we show that for Kuramoto-Sakaguchi phase oscillators the phase lag parameter in the interaction function can induce a Benjamin-Feir-type instability of the partially coherent plane waves. The emerging collective macroscopic chaos appears as an intermediate stage between complete incoherence and stable partially coherent plane waves. We give an analytic treatment of the Benjamin-Feir instability and its onset in a codimension-two bifurcation in the Ott-Antonsen equation as well as a numerical study of the transition from phase turbulence to amplitude turbulence inside the Benjamin-Feir unstable region.

  17. Use of artificial bee colonies algorithm as numerical approximation of differential equations solution

    NASA Astrophysics Data System (ADS)

    Fikri, Fariz Fahmi; Nuraini, Nuning

    2018-03-01

    The differential equation is one of the branches in mathematics which is closely related to human life problems. Some problems that occur in our life can be modeled into differential equations as well as systems of differential equations such as the Lotka-Volterra model and SIR model. Therefore, solving a problem of differential equations is very important. Some differential equations are difficult to solve, so numerical methods are needed to solve that problems. Some numerical methods for solving differential equations that have been widely used are Euler Method, Heun Method, Runge-Kutta and others. However, some of these methods still have some restrictions that cause the method cannot be used to solve more complex problems such as an evaluation interval that we cannot change freely. New methods are needed to improve that problems. One of the method that can be used is the artificial bees colony algorithm. This algorithm is one of metaheuristic algorithm method, which can come out from local search space and do exploration in solution search space so that will get better solution than other method.

  18. An Estimation Theory for Differential Equations and other Problems, with Applications.

    DTIC Science & Technology

    1981-11-01

    order differential -8- operators and M-operators, in particular, the Perron - Frobenius theory and generalizations. Convergence theory for iterative... THEORY FOR DIFFERENTIAL 0EQUATIONS AND OTHER FROBLEMS, WITH APPLICATIONS 0 ,Final Technical Report by Johann Schr6der November, 1981 EUROPEAN RESEARCH...COVERED An estimation theory for differential equations Final Report and other problrms, with app)lications A981 6. PERFORMING ORG. RN,-ORT NUMfFR 7

  19. 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.

  20. 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.

  1. A one-step method for modelling longitudinal data with differential equations.

    PubMed

    Hu, Yueqin; Treinen, Raymond

    2018-04-06

    Differential equation models are frequently used to describe non-linear trajectories of longitudinal data. This study proposes a new approach to estimate the parameters in differential equation models. Instead of estimating derivatives from the observed data first and then fitting a differential equation to the derivatives, our new approach directly fits the analytic solution of a differential equation to the observed data, and therefore simplifies the procedure and avoids bias from derivative estimations. A simulation study indicates that the analytic solutions of differential equations (ASDE) approach obtains unbiased estimates of parameters and their standard errors. Compared with other approaches that estimate derivatives first, ASDE has smaller standard error, larger statistical power and accurate Type I error. Although ASDE obtains biased estimation when the system has sudden phase change, the bias is not serious and a solution is also provided to solve the phase problem. The ASDE method is illustrated and applied to a two-week study on consumers' shopping behaviour after a sale promotion, and to a set of public data tracking participants' grammatical facial expression in sign language. R codes for ASDE, recommendations for sample size and starting values are provided. Limitations and several possible expansions of ASDE are also discussed. © 2018 The British Psychological Society.

  2. Existence of anti-periodic (differentiable) mild solutions to semilinear differential equations with nondense domain.

    PubMed

    Liu, Jinghuai; Zhang, Litao

    2016-01-01

    In this paper, we investigate the existence of anti-periodic (or anti-periodic differentiable) mild solutions to the semilinear differential equation [Formula: see text] with nondense domain. Furthermore, an example is given to illustrate our results.

  3. 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.

  4. Generalized Lie symmetry approach for fractional order systems of differential equations. III

    NASA Astrophysics Data System (ADS)

    Singla, Komal; Gupta, R. K.

    2017-06-01

    The generalized Lie symmetry technique is proposed for the derivation of point symmetries for systems of fractional differential equations with an arbitrary number of independent as well as dependent variables. The efficiency of the method is illustrated by its application to three higher dimensional nonlinear systems of fractional order partial differential equations consisting of the (2 + 1)-dimensional asymmetric Nizhnik-Novikov-Veselov system, (3 + 1)-dimensional Burgers system, and (3 + 1)-dimensional Navier-Stokes equations. With the help of derived Lie point symmetries, the corresponding invariant solutions transform each of the considered systems into a system of lower-dimensional fractional partial differential equations.

  5. Alternate solution to generalized Bernoulli equations via an integrating factor: an exact differential equation approach

    NASA Astrophysics Data System (ADS)

    Tisdell, C. C.

    2017-08-01

    Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem through a substitution. The purpose of this note is to present an alternative approach using 'exact methods', illustrating that a substitution and linearization of the problem is unnecessary. The ideas may be seen as forming a complimentary and arguably simpler approach to Azevedo and Valentino that have the potential to be assimilated and adapted to pedagogical needs of those learning and teaching exact differential equations in schools, colleges, universities and polytechnics. We illustrate how to apply the ideas through an analysis of the Gompertz equation, which is of interest in biomathematical models of tumour growth.

  6. Modeling Noisy Data with Differential Equations Using Observed and Expected Matrices

    ERIC Educational Resources Information Center

    Deboeck, Pascal R.; Boker, Steven M.

    2010-01-01

    Complex intraindividual variability observed in psychology may be well described using differential equations. It is difficult, however, to apply differential equation models in psychological contexts, as time series are frequently short, poorly sampled, and have large proportions of measurement and dynamic error. Furthermore, current methods for…

  7. Specular reflection treatment for the 3D radiative transfer equation solved with the discrete ordinates method

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

    Le Hardy, D.; Favennec, Y., E-mail: yann.favennec@univ-nantes.fr; Rousseau, B.

    The contribution of this paper relies in the development of numerical algorithms for the mathematical treatment of specular reflection on borders when dealing with the numerical solution of radiative transfer problems. The radiative transfer equation being integro-differential, the discrete ordinates method allows to write down a set of semi-discrete equations in which weights are to be calculated. The calculation of these weights is well known to be based on either a quadrature or on angular discretization, making the use of such method straightforward for the state equation. Also, the diffuse contribution of reflection on borders is usually well taken intomore » account. However, the calculation of accurate partition ratio coefficients is much more tricky for the specular condition applied on arbitrary geometrical borders. This paper presents algorithms that calculate analytically partition ratio coefficients needed in numerical treatments. The developed algorithms, combined with a decentered finite element scheme, are validated with the help of comparisons with analytical solutions before being applied on complex geometries.« less

  8. Application of the Green's function method for 2- and 3-dimensional steady transonic flows

    NASA Technical Reports Server (NTRS)

    Tseng, K.

    1984-01-01

    A Time-Domain Green's function method for the nonlinear time-dependent three-dimensional aerodynamic potential equation is presented. The Green's theorem is being used to transform the partial differential equation into an integro-differential-delay equation. Finite-element and finite-difference methods are employed for the spatial and time discretizations to approximate the integral equation by a system of differential-delay equations. Solution may be obtained by solving for this nonlinear simultaneous system of equations in time. This paper discusses the application of the method to the Transonic Small Disturbance Equation and numerical results for lifting and nonlifting airfoils and wings in steady flows are presented.

  9. Parallel Algorithm Solves Coupled Differential Equations

    NASA Technical Reports Server (NTRS)

    Hayashi, A.

    1987-01-01

    Numerical methods adapted to concurrent processing. Algorithm solves set of coupled partial differential equations by numerical integration. Adapted to run on hypercube computer, algorithm separates problem into smaller problems solved concurrently. Increase in computing speed with concurrent processing over that achievable with conventional sequential processing appreciable, especially for large problems.

  10. GHM method for obtaining rationalsolutions of nonlinear differential equations.

    PubMed

    Vazquez-Leal, Hector; Sarmiento-Reyes, Arturo

    2015-01-01

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

  11. 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

  12. 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.

  13. 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.

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

    NASA Astrophysics Data System (ADS)

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

    2017-05-01

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

  15. Differential emotional processing in concrete and abstract words.

    PubMed

    Yao, Bo; Keitel, Anne; Bruce, Gillian; Scott, Graham G; O'Donnell, Patrick J; Sereno, Sara C

    2018-02-12

    Emotion (positive and negative) words are typically recognized faster than neutral words. Recent research suggests that emotional valence, while often treated as a unitary semantic property, may be differentially represented in concrete and abstract words. Studies that have explicitly examined the interaction of emotion and concreteness, however, have demonstrated inconsistent patterns of results. Moreover, these findings may be limited as certain key lexical variables (e.g., familiarity, age of acquisition) were not taken into account. We investigated the emotion-concreteness interaction in a large-scale, highly controlled lexical decision experiment. A 3 (Emotion: negative, neutral, positive) × 2 (Concreteness: abstract, concrete) design was used, with 45 items per condition and 127 participants. We found a significant interaction between emotion and concreteness. Although positive and negative valenced words were recognized faster than neutral words, this emotion advantage was significantly larger in concrete than in abstract words. We explored potential contributions of participant alexithymia level and item imageability to this interactive pattern. We found that only word imageability significantly modulated the emotion-concreteness interaction. While both concrete and abstract emotion words are advantageously processed relative to comparable neutral words, the mechanisms of this facilitation are paradoxically more dependent on imageability in abstract words. (PsycINFO Database Record (c) 2018 APA, all rights reserved).

  16. Adaptive Grid Generation for Numerical Solution of Partial Differential Equations.

    DTIC Science & Technology

    1983-12-01

    numerical solution of fluid dynamics problems is presented. However, the method is applicable to the numer- ical evaluation of any partial differential...emphasis is being placed on numerical solution of the governing differential equations by finite difference methods . In the past two decades, considerable...original equations presented in that paper. The solution of the second problem is more difficult. 2 The method of Thompson et al. provides control for

  17. 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.

  18. 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.

  19. The Local Brewery: A Project for Use in Differential Equations Courses

    ERIC Educational Resources Information Center

    Starling, James K.; Povich, Timothy J.; Findlay, Michael

    2016-01-01

    We describe a modeling project designed for an ordinary differential equations (ODEs) course using first-order and systems of first-order differential equations to model the fermentation process in beer. The project aims to expose the students to the modeling process by creating and solving a mathematical model and effectively communicating their…

  20. Differential equations with applications in cancer diseases.

    PubMed

    Ilea, M; Turnea, M; Rotariu, M

    2013-01-01

    Mathematical modeling is a process by which a real world problem is described by a mathematical formulation. The cancer modeling is a highly challenging problem at the frontier of applied mathematics. A variety of modeling strategies have been developed, each focusing on one or more aspects of cancer. The vast majority of mathematical models in cancer diseases biology are formulated in terms of differential equations. We propose an original mathematical model with small parameter for the interactions between these two cancer cell sub-populations and the mathematical model of a vascular tumor. We work on the assumption that, the quiescent cells' nutrient consumption is long. One the equations system includes small parameter epsilon. The smallness of epsilon is relative to the size of the solution domain. MATLAB simulations obtained for transition rate from the quiescent cells' nutrient consumption is long, we show a similar asymptotic behavior for two solutions of the perturbed problem. In this system, the small parameter is an asymptotic variable, different from the independent variable. The graphical output for a mathematical model of a vascular tumor shows the differences in the evolution of the tumor populations of proliferating, quiescent and necrotic cells. The nutrient concentration decreases sharply through the viable rim and tends to a constant level in the core due to the nearly complete necrosis in this region. Many mathematical models can be quantitatively characterized by ordinary differential equations or partial differential equations. The use of MATLAB in this article illustrates the important role of informatics in research in mathematical modeling. The study of avascular tumor growth cells is an exciting and important topic in cancer research and will profit considerably from theoretical input. Interpret these results to be a permanent collaboration between math's and medical oncologists.

  1. Differential Equations and Computational Simulations

    DTIC Science & Technology

    1999-06-18

    divergence operator of a vector field, which can be defined in terms of the Levi - Civita connection. Let $(x, t) be the orbit passing through x g M...differential equations 31 Junping Chen and Dadi Yang The limit cycle of two species predator-prey model with general functional response > 34 S. S...analysis of two -species nonlinear competition system with periodic coefficients 286 X. H. Tang and J. S. Yu Oscillation of first order delay

  2. Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations.

    PubMed

    Baranwal, Vipul K; Pandey, Ram K; Singh, Om P

    2014-01-01

    We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.

  3. 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.

  4. 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.

  5. Stabilisation of time-varying linear systems via Lyapunov differential equations

    NASA Astrophysics Data System (ADS)

    Zhou, Bin; Cai, Guang-Bin; Duan, Guang-Ren

    2013-02-01

    This article studies stabilisation problem for time-varying linear systems via state feedback. Two types of controllers are designed by utilising solutions to Lyapunov differential equations. The first type of feedback controllers involves the unique positive-definite solution to a parametric Lyapunov differential equation, which can be solved when either the state transition matrix of the open-loop system is exactly known, or the future information of the system matrices are accessible in advance. Different from the first class of controllers which may be difficult to implement in practice, the second type of controllers can be easily implemented by solving a state-dependent Lyapunov differential equation with a given positive-definite initial condition. In both cases, explicit conditions are obtained to guarantee the exponentially asymptotic stability of the associated closed-loop systems. Numerical examples show the effectiveness of the proposed approaches.

  6. Conformal invariance of the Lungren-Monin-Novikov equations for vorticity fields in 2D turbulence

    NASA Astrophysics Data System (ADS)

    Grebenev, V. N.; Wacławczyk, M.; Oberlack, M.

    2017-10-01

    We study the statistical properties of the vorticity field in two-dimensional turbulence. The field is described in terms of the infinite Lundgren-Monin-Novikov (LMN) chain of equations for multi-point probability density functions (pdf’s) of vorticity. We perform a Lie group analysis of the first equation in this chain using the direct method based on the canonical Lie-Bäcklund transformations devised for integro-differential equations. We analytically show that the conformal group is broken for the first LMN equation i.e. for the 1-point pdf at least for the inviscid case but the equation is still conformally invariant on the associated characteristic with zero-vorticity. Then, we demonstrate that this characteristic is conformally transformed. We find this outcome coincides with the numerical results about the conformal invariance of the statistics of zero-vorticity isolines, see e.g. Falkovich (2007 Russian Math. Surv. 63 497-510). The conformal symmetry can be understood as a ‘local scaling’ and its traces in two-dimensional turbulence were already discussed in the literature, i.e. it was conjectured more than twenty years ago in Polyakov (1993 Nucl. Phys. B 396 367-85) and clearly validated experimentally in Bernard et al (2006 Nat. Phys. 2 124-8).

  7. Alternate Solution to Generalized Bernoulli Equations via an Integrating Factor: An Exact Differential Equation Approach

    ERIC Educational Resources Information Center

    Tisdell, C. C.

    2017-01-01

    Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem…

  8. Andrei Andreevich Bolibrukh's works on the analytic theory of differential equations

    NASA Astrophysics Data System (ADS)

    Anosov, Dmitry V.; Leksin, Vladimir P.

    2011-02-01

    This paper contains an account of A.A. Bolibrukh's results obtained in the new directions of research that arose in the analytic theory of differential equations as a consequence of his sensational counterexample to the Riemann-Hilbert problem. A survey of results of his students in developing topics first considered by Bolibrukh is also presented. The main focus is on the role of the reducibility/irreducibility of systems of linear differential equations and their monodromy representations. A brief synopsis of results on the multidimensional Riemann-Hilbert problem and on isomonodromic deformations of Fuchsian systems is presented, and the main methods in the modern analytic theory of differential equations are sketched. Bibliography: 69 titles.

  9. Noncommutative differential geometry related to the Young-Baxter equation

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

    Gurevich, D.; Radul, A.; Rubtsov, V.

    1995-11-10

    An analogue of the differential calculus associated with a unitary solution of the quantum Young-Baxter equation is constructed. An example of a ring sheaf Z`s considered in which local solutions of the Young-Baxter quantum equation are defined but there is no global section.

  10. Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials

    NASA Astrophysics Data System (ADS)

    Volkmer, Hans

    2008-04-01

    Sequences of polynomials, orthogonal with respect to signed measures, are associated with a class of differential equations including the Mathieu, Lame and Whittaker-Hill equation. It is shown that the zeros of pn form sequences which converge to the eigenvalues of the corresponding differential equations. Moreover, interlacing properties of the zeros of pn are found. Applications to the numerical treatment of eigenvalue problems are given.

  11. On the Solution of Elliptic Partial Differential Equations on Regions with Corners

    DTIC Science & Technology

    2015-07-09

    In this report we investigate the solution of boundary value problems on polygonal domains for elliptic partial differential equations . We observe...that when the problems are formulated as the boundary integral equations of classical potential theory, the solutions are representable by series of...efficient numerical algorithms. The results are illustrated by a number of numerical examples. On the solution of elliptic partial differential equations on

  12. Power-spectral-density relationship for retarded differential equations

    NASA Technical Reports Server (NTRS)

    Barker, L. K.

    1974-01-01

    The power spectral density (PSD) relationship between input and output of a set of linear differential-difference equations of the retarded type with real constant coefficients and delays is discussed. The form of the PSD relationship is identical with that applicable to unretarded equations. Since the PSD relationship is useful if and only if the system described by the equations is stable, the stability must be determined before applying the PSD relationship. Since it is sometimes difficult to determine the stability of retarded equations, such equations are often approximated by simpler forms. It is pointed out that some common approximations can lead to erroneous conclusions regarding the stability of a system and, therefore, to the possibility of obtaining PSD results which are not valid.

  13. Spatial complexity of solutions of higher order partial differential equations

    NASA Astrophysics Data System (ADS)

    Kukavica, Igor

    2004-03-01

    We address spatial oscillation properties of solutions of higher order parabolic partial differential equations. In the case of the Kuramoto-Sivashinsky equation ut + uxxxx + uxx + u ux = 0, we prove that for solutions u on the global attractor, the quantity card {x epsi [0, L]:u(x, t) = lgr}, where L > 0 is the spatial period, can be bounded by a polynomial function of L for all \\lambda\\in{\\Bbb R} . A similar property is proven for a general higher order partial differential equation u_t+(-1)^{s}\\partial_x^{2s}u+ \\sum_{k=0}^{2s-1}v_k(x,t)\\partial_x^k u =0 .

  14. A partial differential equation for pseudocontact shift.

    PubMed

    Charnock, G T P; Kuprov, Ilya

    2014-10-07

    It is demonstrated that pseudocontact shift (PCS), viewed as a scalar or a tensor field in three dimensions, obeys an elliptic partial differential equation with a source term that depends on the Hessian of the unpaired electron probability density. The equation enables straightforward PCS prediction and analysis in systems with delocalized unpaired electrons, particularly for the nuclei located in their immediate vicinity. It is also shown that the probability density of the unpaired electron may be extracted, using a regularization procedure, from PCS data.

  15. On the Number of Periodic Solutions of Delay Differential Equations

    NASA Astrophysics Data System (ADS)

    Han, Maoan; Xu, Bing; Tian, Huanhuan; Bai, Yuzhen

    In this paper, we consider the existence and number of periodic solutions for a class of delay differential equations of the form ẋ(t) = bx(t ‑ 1) + 𝜀f(x(t),x(t ‑ 1),𝜀), based on the Kaplan-Yorke method. Especially, we consider a kind of delay differential equations with f as a polynomial having parameters and find the number of periodic solutions with period 4 4k+1 or 4 4k+3.

  16. A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

    NASA Technical Reports Server (NTRS)

    Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Gray, Hugh R. (Technical Monitor)

    2002-01-01

    We discuss an Adams-type predictor-corrector method for the numerical solution of fractional differential equations. The method may be used both for linear and for nonlinear problems, and it may be extended to multi-term equations (involving more than one differential operator) too.

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

    NASA Astrophysics Data System (ADS)

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

    2015-07-01

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

  18. Green function of the double-fractional Fokker-Planck equation: path integral and stochastic differential equations.

    PubMed

    Kleinert, H; Zatloukal, V

    2013-11-01

    The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.

  19. 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…

  20. 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

  1. Solving Nonlinear Differential Equations in the Engineering Curriculum

    ERIC Educational Resources Information Center

    Auslander, David M.

    1977-01-01

    Described is the Dynamic System Simulation Language (SIM) mini-computer system utilized at the University of California, Los Angeles. It is used by engineering students for solving nonlinear differential equations. (SL)

  2. 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.

  3. Existence of solutions for a host-parasite model

    NASA Astrophysics Data System (ADS)

    Milner, Fabio Augusto; Patton, Curtis Allan

    2001-12-01

    The sea bass Dicentrarchus labrax has several gill ectoparasites. Diplectanum aequans (Plathelminth, Monogenea) is one of these species. Under certain demographic conditions, this flat worm can trigger pathological problems, in particular in fish farms. The life cycle of the parasite is described and a model for the dynamics of its interaction with the fish is described and analyzed. The model consists of a coupled system of ordinary differential equations and one integro-differential equation.

  4. The Use of Kruskal-Newton Diagrams for Differential Equations

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

    T. Fishaleck and R.B. White

    2008-02-19

    The method of Kruskal-Newton diagrams for the solution of differential equations with boundary layers is shown to provide rapid intuitive understanding of layer scaling and can result in the conceptual simplification of some problems. The method is illustrated using equations arising in the theory of pattern formation and in plasma physics.

  5. Operator Factorization and the Solution of Second-Order Linear Ordinary Differential Equations

    ERIC Educational Resources Information Center

    Robin, W.

    2007-01-01

    The theory and application of second-order linear ordinary differential equations is reviewed from the standpoint of the operator factorization approach to the solution of ordinary differential equations (ODE). Using the operator factorization approach, the general second-order linear ODE is solved, exactly, in quadratures and the resulting…

  6. A novel coupled system of non-local integro-differential equations modelling Young's modulus evolution, nutrients' supply and consumption during bone fracture healing

    NASA Astrophysics Data System (ADS)

    Lu, Yanfei; Lekszycki, Tomasz

    2016-10-01

    During fracture healing, a series of complex coupled biological and mechanical phenomena occurs. They include: (i) growth and remodelling of bone, whose Young's modulus varies in space and time; (ii) nutrients' diffusion and consumption by living cells. In this paper, we newly propose to model these evolution phenomena. The considered features include: (i) a new constitutive equation for growth simulation involving the number of sensor cells; (ii) an improved equation for nutrient concentration accounting for the switch between Michaelis-Menten kinetics and linear consumption regime; (iii) a new constitutive equation for Young's modulus evolution accounting for its dependence on nutrient concentration and variable number of active cells. The effectiveness of the model and its predictive capability are qualitatively verified by numerical simulations (using COMSOL) describing the healing of bone in the presence of damaged tissue between fractured parts.

  7. 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.

  8. Lie group classification of first-order delay ordinary differential equations

    NASA Astrophysics Data System (ADS)

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

    2018-05-01

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

  9. Concatenons as the solutions for non-linear partial differential equations

    NASA Astrophysics Data System (ADS)

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

    2017-07-01

    New class of solutions for nonlinear partial differential equations is introduced. We call them the concaten solutions. As an example we consider equations for the description of wave processes in the Fermi-Pasta-Ulam mass chain and construct the concatenon solutions for these equation. Stability of the concatenon-type solutions is investigated numerically. Interaction between the concatenon and solitons is discussed.

  10. The method of averages applied to the KS differential equations

    NASA Technical Reports Server (NTRS)

    Graf, O. F., Jr.; Mueller, A. C.; Starke, S. E.

    1977-01-01

    A new approach for the solution of artificial satellite trajectory problems is proposed. The basic idea is to apply an analytical solution method (the method of averages) to an appropriate formulation of the orbital mechanics equations of motion (the KS-element differential equations). The result is a set of transformed equations of motion that are more amenable to numerical solution.

  11. 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

  12. 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.

  13. 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.

  14. Revealing Numerical Solutions of a Differential Equation

    ERIC Educational Resources Information Center

    Glaister, P.

    2006-01-01

    In this article, the author considers a student exercise that involves determining the exact and numerical solutions of a particular differential equation. He shows how a typical student solution is at variance with a numerical solution, suggesting that the numerical solution is incorrect. However, further investigation shows that this numerical…

  15. A computer software system for the generation of global ocean tides including self-gravitation and crustal loading effects

    NASA Technical Reports Server (NTRS)

    Estes, R. H.

    1977-01-01

    A computer software system is described which computes global numerical solutions of the integro-differential Laplace tidal equations, including dissipation terms and ocean loading and self-gravitation effects, for arbitrary diurnal and semidiurnal tidal constituents. The integration algorithm features a successive approximation scheme for the integro-differential system, with time stepping forward differences in the time variable and central differences in spatial variables. Solutions for M2, S2, N2, K2, K1, O1, P1 tidal constituents neglecting the effects of ocean loading and self-gravitation and a converged M2, solution including ocean loading and self-gravitation effects are presented in the form of cotidal and corange maps.

  16. Stationary conditions for stochastic differential equations

    NASA Technical Reports Server (NTRS)

    Adomian, G.; Walker, W. W.

    1972-01-01

    This is a preliminary study of possible necessary and sufficient conditions to insure stationarity in the solution process for a stochastic differential equation. It indirectly sheds some light on ergodicity properties and shows that the spectral density is generally inadequate as a statistical measure of the solution. Further work is proceeding on a more general theory which gives necessary and sufficient conditions in a form useful for applications.

  17. Spurious Solutions Of Nonlinear Differential Equations

    NASA Technical Reports Server (NTRS)

    Yee, H. C.; Sweby, P. K.; Griffiths, D. F.

    1992-01-01

    Report utilizes nonlinear-dynamics approach to investigate possible sources of errors and slow convergence and non-convergence of steady-state numerical solutions when using time-dependent approach for problems containing nonlinear source terms. Emphasizes implications for development of algorithms in CFD and computational sciences in general. Main fundamental conclusion of study is that qualitative features of nonlinear differential equations cannot be adequately represented by finite-difference method and vice versa.

  18. Resonant vibrations of a submerged beam

    NASA Astrophysics Data System (ADS)

    Achenbach, J. D.; Qu, J.

    1986-03-01

    Forced vibration of a simply supported submerged beam of circular cross section is investigated by the use of two mathematical methods. In the first approach the problem formulation is reduced to a singular integro-differential equation for the transverse deflection. In the second approach the method of matched asymptotic expansions is employed. The integro-differential equation is solved numerically, to yield an exact solution for the frequency response. Subsequent use of a representation integral yields the radiated far field acoustic pressure. The exact results for the beam deflection are compared with approximate results that are available in the literature. Next, a matched asymptotic expansion is worked out by constructing "inner" and "outer" expansions for frequencies near and not near resonance frequencies, respectively. The two expansions are matched in an appropriate manner to yield a uniformly valid solution. The leading term of the matched asymptotic solution is compared with exact numerical results.

  19. Hidden physics models: Machine learning of nonlinear partial differential equations

    NASA Astrophysics Data System (ADS)

    Raissi, Maziar; Karniadakis, George Em

    2018-03-01

    While there is currently a lot of enthusiasm about "big data", useful data is usually "small" and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from small data. In particular, we introduce hidden physics models, which are essentially data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and nonlinear partial differential equations, to extract patterns from high-dimensional data generated from experiments. The proposed methodology may be applied to the problem of learning, system identification, or data-driven discovery of partial differential equations. Our framework relies on Gaussian processes, a powerful tool for probabilistic inference over functions, that enables us to strike a balance between model complexity and data fitting. The effectiveness of the proposed approach is demonstrated through a variety of canonical problems, spanning a number of scientific domains, including the Navier-Stokes, Schrödinger, Kuramoto-Sivashinsky, and time dependent linear fractional equations. The methodology provides a promising new direction for harnessing the long-standing developments of classical methods in applied mathematics and mathematical physics to design learning machines with the ability to operate in complex domains without requiring large quantities of data.

  20. 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.

  1. The convergence of the order sequence and the solution function sequence on fractional partial differential equation

    NASA Astrophysics Data System (ADS)

    Rusyaman, E.; Parmikanti, K.; Chaerani, D.; Asefan; Irianingsih, I.

    2018-03-01

    One of the application of fractional ordinary differential equation is related to the viscoelasticity, i.e., a correlation between the viscosity of fluids and the elasticity of solids. If the solution function develops into function with two or more variables, then its differential equation must be changed into fractional partial differential equation. As the preliminary study for two variables viscoelasticity problem, this paper discusses about convergence analysis of function sequence which is the solution of the homogenous fractional partial differential equation. The method used to solve the problem is Homotopy Analysis Method. The results show that if given two real number sequences (αn) and (βn) which converge to α and β respectively, then the solution function sequences of fractional partial differential equation with order (αn, βn) will also converge to the solution function of fractional partial differential equation with order (α, β).

  2. 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.

  3. 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.

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

    PubMed

    Lingala, N; Namachchivaya, N Sri

    2016-06-01

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

  5. 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.

  6. Differential equation based method for accurate approximations in optimization

    NASA Technical Reports Server (NTRS)

    Pritchard, Jocelyn I.; Adelman, Howard M.

    1990-01-01

    This paper describes a method to efficiently and accurately approximate the effect of design changes on structural response. The key to this new method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in msot cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacement are used to approximate bending stresses.

  7. Differential equation based method for accurate approximations in optimization

    NASA Technical Reports Server (NTRS)

    Pritchard, Jocelyn I.; Adelman, Howard M.

    1990-01-01

    A method to efficiently and accurately approximate the effect of design changes on structural response is described. The key to this method is to interpret sensitivity equations as differential equations that may be solved explicitly for closed form approximations, hence, the method is denoted the Differential Equation Based (DEB) method. Approximations were developed for vibration frequencies, mode shapes and static displacements. The DEB approximation method was applied to a cantilever beam and results compared with the commonly-used linear Taylor series approximations and exact solutions. The test calculations involved perturbing the height, width, cross-sectional area, tip mass, and bending inertia of the beam. The DEB method proved to be very accurate, and in most cases, was more accurate than the linear Taylor series approximation. The method is applicable to simultaneous perturbation of several design variables. Also, the approximations may be used to calculate other system response quantities. For example, the approximations for displacements are used to approximate bending stresses.

  8. A gradual update method for simulating the steady-state solution of stiff differential equations in metabolic circuits.

    PubMed

    Shiraishi, Emi; Maeda, Kazuhiro; Kurata, Hiroyuki

    2009-02-01

    Numerical simulation of differential equation systems plays a major role in the understanding of how metabolic network models generate particular cellular functions. On the other hand, the classical and technical problems for stiff differential equations still remain to be solved, while many elegant algorithms have been presented. To relax the stiffness problem, we propose new practical methods: the gradual update of differential-algebraic equations based on gradual application of the steady-state approximation to stiff differential equations, and the gradual update of the initial values in differential-algebraic equations. These empirical methods show a high efficiency for simulating the steady-state solutions for the stiff differential equations that existing solvers alone cannot solve. They are effective in extending the applicability of dynamic simulation to biochemical network models.

  9. Model equations for the Eiffel Tower profile: historical perspective and new results

    NASA Astrophysics Data System (ADS)

    Weidman, Patrick; Pinelis, Iosif

    2004-07-01

    Model equations for the shape of the Eiffel Tower are investigated. One model purported to be based on Eiffel's writing does not give a tower with the correct curvature. A second popular model not connected with Eiffel's writings provides a fair approximation to the tower's skyline profile of 29 contiguous panels. Reported here is a third model derived from Eiffel's concern about wind loads on the tower, as documented in his communication to the French Civil Engineering Society on 30 March 1885. The result is a nonlinear, integro-differential equation which is solved to yield an exponential tower profile. It is further verified that, as Eiffel wrote, "in reality the curve exterior of the tower reproduces, at a determined scale, the same curve of the moments produced by the wind". An analysis of the actual tower profile shows that it is composed of two piecewise continuous exponentials with different growth rates. This is explained by specific safety factors for wind loading that Eiffel & Company incorporated in the design of the free-standing tower. To cite this article: P. Weidman, I. Pinelis, C. R. Mecanique 332 (2004).

  10. 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.

  11. Advanced methods for the solution of differential equations

    NASA Technical Reports Server (NTRS)

    Goldstein, M. E.; Braun, W. H.

    1973-01-01

    This book is based on a course presented at the Lewis Research Center for engineers and scientists who were interested in increasing their knowledge of differential equations. Those results which can actually be used to solve equations are therefore emphasized; and detailed proofs of theorems are, for the most part, omitted. However, the conclusions of the theorems are stated in a precise manner, and enough references are given so that the interested reader can find the steps of the proofs.

  12. Sensitivity of rough differential equations: An approach through the Omega lemma

    NASA Astrophysics Data System (ADS)

    Coutin, Laure; Lejay, Antoine

    2018-03-01

    The Itô map gives the solution of a Rough Differential Equation, a generalization of an Ordinary Differential Equation driven by an irregular path, when existence and uniqueness hold. By studying how a path is transformed through the vector field which is integrated, we prove that the Itô map is Hölder or Lipschitz continuous with respect to all its parameters. This result unifies and weakens the hypotheses of the regularity results already established in the literature.

  13. Small-x asymptotics of the quark helicity distribution: Analytic results

    DOE PAGES

    Kovchegov, Yuri V.; Pitonyak, Daniel; Sievert, Matthew D.

    2017-06-15

    In this Letter, we analytically solve the evolution equations for the small-x asymptotic behavior of the (flavor singlet) quark helicity distribution in the large- N c limit. Here, these evolution equations form a set of coupled integro-differential equations, which previously could only be solved numerically. This approximate numerical solution, however, revealed simplifying properties of the small-x asymptotics, which we exploit here to obtain an analytic solution.

  14. 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.

  15. Observability of discretized partial differential equations

    NASA Technical Reports Server (NTRS)

    Cohn, Stephen E.; Dee, Dick P.

    1988-01-01

    It is shown that complete observability of the discrete model used to assimilate data from a linear partial differential equation (PDE) system is necessary and sufficient for asymptotic stability of the data assimilation process. The observability theory for discrete systems is reviewed and applied to obtain simple observability tests for discretized constant-coefficient PDEs. Examples are used to show how numerical dispersion can result in discrete dynamics with multiple eigenvalues, thereby detracting from observability.

  16. Spurious Numerical Solutions Of Differential Equations

    NASA Technical Reports Server (NTRS)

    Lafon, A.; Yee, H. C.

    1995-01-01

    Paper presents detailed study of spurious steady-state numerical solutions of differential equations that contain nonlinear source terms. Main objectives of this study are (1) to investigate how well numerical steady-state solutions of model nonlinear reaction/convection boundary-value problem mimic true steady-state solutions and (2) to relate findings of this investigation to implications for interpretation of numerical results from computational-fluid-dynamics algorithms and computer codes used to simulate reacting flows.

  17. Asymptotic problems for stochastic partial differential equations

    NASA Astrophysics Data System (ADS)

    Salins, Michael

    Stochastic partial differential equations (SPDEs) can be used to model systems in a wide variety of fields including physics, chemistry, and engineering. The main SPDEs of interest in this dissertation are the semilinear stochastic wave equations which model the movement of a material with constant mass density that is exposed to both determinstic and random forcing. Cerrai and Freidlin have shown that on fixed time intervals, as the mass density of the material approaches zero, the solutions of the stochastic wave equation converge uniformly to the solutions of a stochastic heat equation, in probability. This is called the Smoluchowski-Kramers approximation. In Chapter 2, we investigate some of the multi-scale behaviors that these wave equations exhibit. In particular, we show that the Freidlin-Wentzell exit place and exit time asymptotics for the stochastic wave equation in the small noise regime can be approximated by the exit place and exit time asymptotics for the stochastic heat equation. We prove that the exit time and exit place asymptotics are characterized by quantities called quasipotentials and we prove that the quasipotentials converge. We then investigate the special case where the equation has a gradient structure and show that we can explicitly solve for the quasipotentials, and that the quasipotentials for the heat equation and wave equation are equal. In Chapter 3, we study the Smoluchowski-Kramers approximation in the case where the material is electrically charged and exposed to a magnetic field. Interestingly, if the system is frictionless, then the Smoluchowski-Kramers approximation does not hold. We prove that the Smoluchowski-Kramers approximation is valid for systems exposed to both a magnetic field and friction. Notably, we prove that the solutions to the second-order equations converge to the solutions of the first-order equation in an Lp sense. This strengthens previous results where convergence was proved in probability.

  18. 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

  19. 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.

  20. Representing Sudden Shifts in Intensive Dyadic Interaction Data Using Differential Equation Models with Regime Switching.

    PubMed

    Chow, Sy-Miin; Ou, Lu; Ciptadi, Arridhana; Prince, Emily B; You, Dongjun; Hunter, Michael D; Rehg, James M; Rozga, Agata; Messinger, Daniel S

    2018-06-01

    A growing number of social scientists have turned to differential equations as a tool for capturing the dynamic interdependence among a system of variables. Current tools for fitting differential equation models do not provide a straightforward mechanism for diagnosing evidence for qualitative shifts in dynamics, nor do they provide ways of identifying the timing and possible determinants of such shifts. In this paper, we discuss regime-switching differential equation models, a novel modeling framework for representing abrupt changes in a system of differential equation models. Estimation was performed by combining the Kim filter (Kim and Nelson State-space models with regime switching: classical and Gibbs-sampling approaches with applications, MIT Press, Cambridge, 1999) and a numerical differential equation solver that can handle both ordinary and stochastic differential equations. The proposed approach was motivated by the need to represent discrete shifts in the movement dynamics of [Formula: see text] mother-infant dyads during the Strange Situation Procedure (SSP), a behavioral assessment where the infant is separated from and reunited with the mother twice. We illustrate the utility of a novel regime-switching differential equation model in representing children's tendency to exhibit shifts between the goal of staying close to their mothers and intermittent interest in moving away from their mothers to explore the room during the SSP. Results from empirical model fitting were supplemented with a Monte Carlo simulation study to evaluate the use of information criterion measures to diagnose sudden shifts in dynamics.

  1. Numerical integration of ordinary differential equations of various orders

    NASA Technical Reports Server (NTRS)

    Gear, C. W.

    1969-01-01

    Report describes techniques for the numerical integration of differential equations of various orders. Modified multistep predictor-corrector methods for general initial-value problems are discussed and new methods are introduced.

  2. Numerical modelling in biosciences using delay differential equations

    NASA Astrophysics Data System (ADS)

    Bocharov, Gennadii A.; Rihan, Fathalla A.

    2000-12-01

    Our principal purposes here are (i) to consider, from the perspective of applied mathematics, models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics, (ii) to review the application of numerical techniques to investigate these models. We show that there are prima facie reasons for using such models: (i) they have a richer mathematical framework (compared with ordinary differential equations) for the analysis of biosystem dynamics, (ii) they display better consistency with the nature of certain biological processes and predictive results. We analyze both the qualitative and quantitative role that delays play in basic time-lag models proposed in population dynamics, epidemiology, physiology, immunology, neural networks and cell kinetics. We then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences, comparing them with those implemented by the bio-modellers.

  3. Differential equation of exospheric lateral transport and its application to terrestrial hydrogen

    NASA Technical Reports Server (NTRS)

    Hodges, R. R., Jr.

    1973-01-01

    The differential equation description of exospheric lateral transport of Hodges and Johnson is reformulated to extend its utility to light gases. Accuracy of the revised equation is established by applying it to terrestrial hydrogen. The resulting global distributions for several static exobase models are shown to be essentially the same as those that have been computed by Quessette using an integral equation approach. The present theory is subsequently used to elucidate the effects of nonzero lateral flow, exobase rotation, and diurnal tidal winds on the hydrogen distribution. Finally it is shown that the differential equation of exospheric transport is analogous to a diffusion equation. Hence it is practical to consider exospheric transport as a continuation of thermospheric diffusion, a concept that alleviates the need for an artificial exobase dividing thermosphere and exosphere.

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

    PubMed

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

    2013-09-01

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

  5. 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.

  6. Given a one-step numerical scheme, on which ordinary differential equations is it exact?

    NASA Astrophysics Data System (ADS)

    Villatoro, Francisco R.

    2009-01-01

    A necessary condition for a (non-autonomous) ordinary differential equation to be exactly solved by a one-step, finite difference method is that the principal term of its local truncation error be null. A procedure to determine some ordinary differential equations exactly solved by a given numerical scheme is developed. Examples of differential equations exactly solved by the explicit Euler, implicit Euler, trapezoidal rule, second-order Taylor, third-order Taylor, van Niekerk's second-order rational, and van Niekerk's third-order rational methods are presented.

  7. Multigrid methods for differential equations with highly oscillatory coefficients

    NASA Technical Reports Server (NTRS)

    Engquist, Bjorn; Luo, Erding

    1993-01-01

    New coarse grid multigrid operators for problems with highly oscillatory coefficients are developed. These types of operators are necessary when the characters of the differential equations on coarser grids or longer wavelengths are different from that on the fine grid. Elliptic problems for composite materials and different classes of hyperbolic problems are practical examples. The new coarse grid operators can be constructed directly based on the homogenized differential operators or hierarchically computed from the finest grid. Convergence analysis based on the homogenization theory is given for elliptic problems with periodic coefficients and some hyperbolic problems. These are classes of equations for which there exists a fairly complete theory for the interaction between shorter and longer wavelengths in the problems. Numerical examples are presented.

  8. Lump solutions to nonlinear partial differential equations via Hirota bilinear forms

    NASA Astrophysics Data System (ADS)

    Ma, Wen-Xiu; Zhou, Yuan

    2018-02-01

    Lump solutions are analytical rational function solutions localized in all directions in space. We analyze a class of lump solutions, generated from quadratic functions, to nonlinear partial differential equations. The basis of success is the Hirota bilinear formulation and the primary object is the class of positive multivariate quadratic functions. A complete determination of quadratic functions positive in space and time is given, and positive quadratic functions are characterized as sums of squares of linear functions. Necessary and sufficient conditions for positive quadratic functions to solve Hirota bilinear equations are presented, and such polynomial solutions yield lump solutions to nonlinear partial differential equations under the dependent variable transformations u = 2(ln ⁡ f) x and u = 2(ln ⁡ f) xx, where x is one spatial variable. Applications are made for a few generalized KP and BKP equations.

  9. Nonexistence of global solutions of abstract wave equations with high energies.

    PubMed

    Esquivel-Avila, Jorge A

    2017-01-01

    We consider an undamped second order in time evolution equation. For any positive value of the initial energy, we give sufficient conditions to conclude nonexistence of global solutions. The analysis is based on a differential inequality. The success of our result is based in a detailed analysis which is different from the ones commonly used to prove blow-up. Several examples are given improving known results in the literature.

  10. On the systematic approach to the classification of differential equations by group theoretical methods

    NASA Astrophysics Data System (ADS)

    Andriopoulos, K.; Dimas, S.; Leach, P. G. L.; Tsoubelis, D.

    2009-08-01

    Complete symmetry groups enable one to characterise fully a given differential equation. By considering the reversal of an approach based upon complete symmetry groups we construct new classes of differential equations which have the equations of Bateman, Monge-Ampère and Born-Infeld as special cases. We develop a symbolic algorithm to decrease the complexity of the calculations involved.

  11. Oscillation theorems for second order nonlinear forced differential equations.

    PubMed

    Salhin, Ambarka A; Din, Ummul Khair Salma; Ahmad, Rokiah Rozita; Noorani, Mohd Salmi Md

    2014-01-01

    In this paper, a class of second order forced nonlinear differential equation is considered and several new oscillation theorems are obtained. Our results generalize and improve those known ones in the literature.

  12. A stochastic differential equation analysis of cerebrospinal fluid dynamics.

    PubMed

    Raman, Kalyan

    2011-01-18

    Clinical measurements of intracranial pressure (ICP) over time show fluctuations around the deterministic time path predicted by a classic mathematical model in hydrocephalus research. Thus an important issue in mathematical research on hydrocephalus remains unaddressed--modeling the effect of noise on CSF dynamics. Our objective is to mathematically model the noise in the data. The classic model relating the temporal evolution of ICP in pressure-volume studies to infusions is a nonlinear differential equation based on natural physical analogies between CSF dynamics and an electrical circuit. Brownian motion was incorporated into the differential equation describing CSF dynamics to obtain a nonlinear stochastic differential equation (SDE) that accommodates the fluctuations in ICP. The SDE is explicitly solved and the dynamic probabilities of exceeding critical levels of ICP under different clinical conditions are computed. A key finding is that the probabilities display strong threshold effects with respect to noise. Above the noise threshold, the probabilities are significantly influenced by the resistance to CSF outflow and the intensity of the noise. Fluctuations in the CSF formation rate increase fluctuations in the ICP and they should be minimized to lower the patient's risk. The nonlinear SDE provides a scientific methodology for dynamic risk management of patients. The dynamic output of the SDE matches the noisy ICP data generated by the actual intracranial dynamics of patients better than the classic model used in prior research.

  13. 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.

  14. Two-dimensional integrating matrices on rectangular grids. [solving differential equations associated with rotating structures

    NASA Technical Reports Server (NTRS)

    Lakin, W. D.

    1981-01-01

    The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.

  15. Power series solutions of ordinary differential equations in MACSYMA

    NASA Technical Reports Server (NTRS)

    Lafferty, E. L.

    1977-01-01

    A program is described which extends the differential equation solving capability of MACSYMA to power series solutions and is available via the SHARE library. The program is directed toward those classes of equations with variable coefficients (in particular, those with singularities) and uses the method of Frobenius. Probably the most important distinction between this package and others currently available or being developed is that, wherever possible, this program will attempt to provide a complete solution to the equation rather than an approximation, i.e., a finite number of terms. This solution will take the form of a sum of infinite series.

  16. Runge-Kutta Methods for Linear Ordinary Differential Equations

    NASA Technical Reports Server (NTRS)

    Zingg, David W.; Chisholm, Todd T.

    1997-01-01

    Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODES) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of the Runge-Kutta method must satisfy. This freedom is used to develop methods which are more efficient than conventional Runge-Kutta methods. A fourth-order method is presented which uses only two memory locations per dependent variable, while the classical fourth-order Runge-Kutta method uses three. This method is an excellent choice for simulations of linear wave phenomena if memory is a primary concern. In addition, fifth- and sixth-order methods are presented which require five and six stages, respectively, one fewer than their conventional counterparts, and are therefore more efficient. These methods are an excellent option for use with high-order spatial discretizations.

  17. 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.

  18. Additive schemes for certain operator-differential equations

    NASA Astrophysics Data System (ADS)

    Vabishchevich, P. N.

    2010-12-01

    Unconditionally stable finite difference schemes for the time approximation of first-order operator-differential systems with self-adjoint operators are constructed. Such systems arise in many applied problems, for example, in connection with nonstationary problems for the system of Stokes (Navier-Stokes) equations. Stability conditions in the corresponding Hilbert spaces for two-level weighted operator-difference schemes are obtained. Additive (splitting) schemes are proposed that involve the solution of simple problems at each time step. The results are used to construct splitting schemes with respect to spatial variables for nonstationary Navier-Stokes equations for incompressible fluid. The capabilities of additive schemes are illustrated using a two-dimensional model problem as an example.

  19. Saturation behavior: a general relationship described by a simple second-order differential equation.

    PubMed

    Kepner, Gordon R

    2010-04-13

    The numerous natural phenomena that exhibit saturation behavior, e.g., ligand binding and enzyme kinetics, have been approached, to date, via empirical and particular analyses. This paper presents a mechanism-free, and assumption-free, second-order differential equation, designed only to describe a typical relationship between the variables governing these phenomena. It develops a mathematical model for this relation, based solely on the analysis of the typical experimental data plot and its saturation characteristics. Its utility complements the traditional empirical approaches. For the general saturation curve, described in terms of its independent (x) and dependent (y) variables, a second-order differential equation is obtained that applies to any saturation phenomena. It shows that the driving factor for the basic saturation behavior is the probability of the interactive site being free, which is described quantitatively. Solving the equation relates the variables in terms of the two empirical constants common to all these phenomena, the initial slope of the data plot and the limiting value at saturation. A first-order differential equation for the slope emerged that led to the concept of the effective binding rate at the active site and its dependence on the calculable probability the interactive site is free. These results are illustrated using specific cases, including ligand binding and enzyme kinetics. This leads to a revised understanding of how to interpret the empirical constants, in terms of the variables pertinent to the phenomenon under study. The second-order differential equation revealed the basic underlying relations that describe these saturation phenomena, and the basic mathematical properties of the standard experimental data plot. It was shown how to integrate this differential equation, and define the common basic properties of these phenomena. The results regarding the importance of the slope and the new perspectives on the empirical

  20. 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.

  1. Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations

    NASA Astrophysics Data System (ADS)

    Eden, Burkhard; Smirnov, Vladimir A.

    2016-10-01

    We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.

  2. A Simple Method to Find out when an Ordinary Differential Equation Is Separable

    ERIC Educational Resources Information Center

    Cid, Jose Angel

    2009-01-01

    We present an alternative method to that of Scott (D. Scott, "When is an ordinary differential equation separable?", "Amer. Math. Monthly" 92 (1985), pp. 422-423) to teach the students how to discover whether a differential equation y[prime] = f(x,y) is separable or not when the nonlinearity f(x, y) is not explicitly factorized. Our approach is…

  3. Asymptotic (h tending to infinity) absolute stability for BDFs applied to stiff differential equations. [Backward Differentiation Formulas

    NASA Technical Reports Server (NTRS)

    Krogh, F. T.; Stewart, K.

    1984-01-01

    Methods based on backward differentiation formulas (BDFs) for solving stiff differential equations require iterating to approximate the solution of the corrector equation on each step. One hope for reducing the cost of this is to make do with iteration matrices that are known to have errors and to do no more iterations than are necessary to maintain the stability of the method. This paper, following work by Klopfenstein, examines the effect of errors in the iteration matrix on the stability of the method. Application of the results to an algorithm is discussed briefly.

  4. Periodicity computation of generalized mathematical biology problems involving delay differential equations.

    PubMed

    Jasim Mohammed, M; Ibrahim, Rabha W; Ahmad, M Z

    2017-03-01

    In this paper, we consider a low initial population model. Our aim is to study the periodicity computation of this model by using neutral differential equations, which are recognized in various studies including biology. We generalize the neutral Rayleigh equation for the third-order by exploiting the model of fractional calculus, in particular the Riemann-Liouville differential operator. We establish the existence and uniqueness of a periodic computational outcome. The technique depends on the continuation theorem of the coincidence degree theory. Besides, an example is presented to demonstrate the finding.

  5. Using Computer Symbolic Algebra to Solve Differential Equations.

    ERIC Educational Resources Information Center

    Mathews, John H.

    1989-01-01

    This article illustrates that mathematical theory can be incorporated into the process to solve differential equations by a computer algebra system, muMATH. After an introduction to functions of muMATH, several short programs for enhancing the capabilities of the system are discussed. Listed are six references. (YP)

  6. Design of TIR collimating lens for ordinary differential equation of extended light source

    NASA Astrophysics Data System (ADS)

    Zhan, Qianjing; Liu, Xiaoqin; Hou, Zaihong; Wu, Yi

    2017-10-01

    The source of LED has been widely used in our daily life. The intensity angle distribution of single LED is lambert distribution, which does not satisfy the requirement of people. Therefore, we need to distribute light and change the LED's intensity angle distribution. The most commonly method to change its intensity angle distribution is the free surface. Generally, using ordinary differential equations to calculate free surface can only be applied in a point source, but it will lead to a big error for the expand light. This paper proposes a LED collimating lens based on the ordinary differential equation, combined with the LED's light distribution curve, and adopt the method of calculating the center gravity of the extended light to get the normal vector. According to the law of Snell, the ordinary differential equations are constructed. Using the runge-kutta method for solution of ordinary differential equation solution, the curve point coordinates are gotten. Meanwhile, the edge point data of lens are imported into the optical simulation software TracePro. Based on 1mm×1mm single lambert body for light conditions, The degrees of collimating light can be close to +/-3. Furthermore, the energy utilization rate is higher than 85%. In this paper, the point light source is used to calculate partial differential equation method and compared with the simulation of the lens, which improve the effect of 1 degree of collimation.

  7. The reservoir model: a differential equation model of psychological regulation.

    PubMed

    Deboeck, Pascal R; Bergeman, C S

    2013-06-01

    Differential equation models can be used to describe the relationships between the current state of a system of constructs (e.g., stress) and how those constructs are changing (e.g., based on variable-like experiences). The following article describes a differential equation model based on the concept of a reservoir. With a physical reservoir, such as one for water, the level of the liquid in the reservoir at any time depends on the contributions to the reservoir (inputs) and the amount of liquid removed from the reservoir (outputs). This reservoir model might be useful for constructs such as stress, where events might "add up" over time (e.g., life stressors, inputs), but individuals simultaneously take action to "blow off steam" (e.g., engage coping resources, outputs). The reservoir model can provide descriptive statistics of the inputs that contribute to the "height" (level) of a construct and a parameter that describes a person's ability to dissipate the construct. After discussing the model, we describe a method of fitting the model as a structural equation model using latent differential equation modeling and latent distribution modeling. A simulation study is presented to examine recovery of the input distribution and output parameter. The model is then applied to the daily self-reports of negative affect and stress from a sample of older adults from the Notre Dame Longitudinal Study on Aging. (PsycINFO Database Record (c) 2013 APA, all rights reserved).

  8. The Reservoir Model: A Differential Equation Model of Psychological Regulation

    PubMed Central

    Deboeck, Pascal R.; Bergeman, C. S.

    2017-01-01

    Differential equation models can be used to describe the relationships between the current state of a system of constructs (e.g., stress) and how those constructs are changing (e.g., based on variable-like experiences). The following article describes a differential equation model based on the concept of a reservoir. With a physical reservoir, such as one for water, the level of the liquid in the reservoir at any time depends on the contributions to the reservoir (inputs) and the amount of liquid removed from the reservoir (outputs). This reservoir model might be useful for constructs such as stress, where events might “add up” over time (e.g., life stressors, inputs), but individuals simultaneously take action to “blow off steam” (e.g., engage coping resources, outputs). The reservoir model can provide descriptive statistics of the inputs that contribute to the “height” (level) of a construct and a parameter that describes a person's ability to dissipate the construct. After discussing the model, we describe a method of fitting the model as a structural equation model using latent differential equation modeling and latent distribution modeling. A simulation study is presented to examine recovery of the input distribution and output parameter. The model is then applied to the daily self-reports of negative affect and stress from a sample of older adults from the Notre Dame Longitudinal Study on Aging. PMID:23527605

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

    NASA Astrophysics Data System (ADS)

    Camporesi, Roberto

    2011-06-01

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

  10. The Parker-Sochacki Method--A Powerful New Method for Solving Systems of Differential Equations

    NASA Astrophysics Data System (ADS)

    Rudmin, Joseph W.

    2001-04-01

    The Parker-Sochacki Method--A Powerful New Method for Solving Systems of Differential Equations Joseph W. Rudmin (Physics Dept, James Madison University) A new system of solving systems of differential equations will be presented, which has been developed by J. Edgar Parker and James Sochacki, of the James Madison University Mathematics Department. The method produces MacClaurin Series solutions to systems of differential equations, with the coefficients in either algebraic or numerical form. The method yields high-degree solutions: 20th degree is easily obtainable. It is conceptually simple, fast, and extremely general. It has been applied to over a hundred systems of differential equations, some of which were previously unsolved, and has yet to fail to solve any system for which the MacClaurin series converges. The method is non-recursive: each coefficient in the series is calculated just once, in closed form, and its accuracy is limited only by the digital accuracy of the computer. Although the original differential equations may include any mathematical functions, the computational method includes ONLY the operations of addition, subtraction, and multiplication. Furthermore, it is perfectly suited to parallel -processing computer languages. Those who learn this system will never use Runge-Kutta or predictor-corrector methods again. Examples will be presented, including the classical many-body problem.

  11. Generalized Ordinary Differential Equation Models.

    PubMed

    Miao, Hongyu; Wu, Hulin; Xue, Hongqi

    2014-10-01

    Existing estimation methods for ordinary differential equation (ODE) models are not applicable to discrete data. The generalized ODE (GODE) model is therefore proposed and investigated for the first time. We develop the likelihood-based parameter estimation and inference methods for GODE models. We propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving ODEs. The simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ODE model estimation approach and the extended smoothing-based (ESB) method.

  12. Existence of a coupled system of fractional differential equations

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

    Ibrahim, Rabha W.; Siri, Zailan

    2015-10-22

    We manage the existence and uniqueness of a fractional coupled system containing Schrödinger equations. Such a system appears in quantum mechanics. We confirm that the fractional system under consideration admits a global solution in appropriate functional spaces. The solution is shown to be unique. The method is based on analytic technique of the fixed point theory. The fractional differential operator is considered from the virtue of the Riemann-Liouville differential operator.

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

    ERIC Educational Resources Information Center

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

    2009-01-01

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

  14. Implications of a quadratic stream definition in radiative transfer theory.

    NASA Technical Reports Server (NTRS)

    Whitney, C.

    1972-01-01

    An explicit definition of the radiation-stream concept is stated and applied to approximate the integro-differential equation of radiative transfer with a set of twelve coupled differential equations. Computational efficiency is enhanced by distributing the corresponding streams in three-dimensional space in a totally symmetric way. Polarization is then incorporated in this model. A computer program based on the model is briefly compared with a Monte Carlo program for simulation of horizon scans of the earth's atmosphere. It is found to be considerably faster.

  15. 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

  16. Systems of Differential Equations with Skew-Symmetric, Orthogonal Matrices

    ERIC Educational Resources Information Center

    Glaister, P.

    2008-01-01

    The solution of a system of linear, inhomogeneous differential equations is discussed. The particular class considered is where the coefficient matrix is skew-symmetric and orthogonal, and where the forcing terms are sinusoidal. More general matrices are also considered.

  17. Nonlinear grid error effects on numerical solution of partial differential equations

    NASA Technical Reports Server (NTRS)

    Dey, S. K.

    1980-01-01

    Finite difference solutions of nonlinear partial differential equations require discretizations and consequently grid errors are generated. These errors strongly affect stability and convergence properties of difference models. Previously such errors were analyzed by linearizing the difference equations for solutions. Properties of mappings of decadence were used to analyze nonlinear instabilities. Such an analysis is directly affected by initial/boundary conditions. An algorithm was developed, applied to nonlinear Burgers equations, and verified computationally. A preliminary test shows that Navier-Stokes equations may be treated similarly.

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

    NASA Technical Reports Server (NTRS)

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

    2002-01-01

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

  19. 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.

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

    PubMed

    Chen, Minfeng; Gao, Zongsheng; Du, Yunfei

    2017-01-01

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

  1. Fitting ordinary differential equations to short time course data.

    PubMed

    Brewer, Daniel; Barenco, Martino; Callard, Robin; Hubank, Michael; Stark, Jaroslav

    2008-02-28

    Ordinary differential equations (ODEs) are widely used to model many systems in physics, chemistry, engineering and biology. Often one wants to compare such equations with observed time course data, and use this to estimate parameters. Surprisingly, practical algorithms for doing this are relatively poorly developed, particularly in comparison with the sophistication of numerical methods for solving both initial and boundary value problems for differential equations, and for locating and analysing bifurcations. A lack of good numerical fitting methods is particularly problematic in the context of systems biology where only a handful of time points may be available. In this paper, we present a survey of existing algorithms and describe the main approaches. We also introduce and evaluate a new efficient technique for estimating ODEs linear in parameters particularly suited to situations where noise levels are high and the number of data points is low. It employs a spline-based collocation scheme and alternates linear least squares minimization steps with repeated estimates of the noise-free values of the variables. This is reminiscent of expectation-maximization methods widely used for problems with nuisance parameters or missing data.

  2. Minimal parameter solution of the orthogonal matrix differential equation

    NASA Technical Reports Server (NTRS)

    Bar-Itzhack, Itzhack Y.; Markley, F. Landis

    1990-01-01

    As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed emplying the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.

  3. Minimal parameter solution of the orthogonal matrix differential equation

    NASA Technical Reports Server (NTRS)

    Baritzhack, Itzhack Y.; Markley, F. Landis

    1988-01-01

    As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed employing the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.

  4. Solving ordinary differential equations by electrical analogy: a multidisciplinary teaching tool

    NASA Astrophysics Data System (ADS)

    Sanchez Perez, J. F.; Conesa, M.; Alhama, I.

    2016-11-01

    Ordinary differential equations are the mathematical formulation for a great variety of problems in science and engineering, and frequently, two different problems are equivalent from a mathematical point of view when they are formulated by the same equations. Students acquire the knowledge of how to solve these equations (at least some types of them) using protocols and strict algorithms of mathematical calculation without thinking about the meaning of the equation. The aim of this work is that students learn to design network models or circuits in this way; with simple knowledge of them, students can establish the association of electric circuits and differential equations and their equivalences, from a formal point of view, that allows them to associate knowledge of two disciplines and promote the use of this interdisciplinary approach to address complex problems. Therefore, they learn to use a multidisciplinary tool that allows them to solve these kinds of equations, even students of first course of engineering, whatever the order, grade or type of non-linearity. This methodology has been implemented in numerous final degree projects in engineering and science, e.g., chemical engineering, building engineering, industrial engineering, mechanical engineering, architecture, etc. Applications are presented to illustrate the subject of this manuscript.

  5. Estimation of periodic solutions number of first-order differential equations

    NASA Astrophysics Data System (ADS)

    Ivanov, Gennady; Alferov, Gennady; Gorovenko, Polina; Sharlay, Artem

    2018-05-01

    The paper deals with first-order differential equations under the assumption that the right-hand side is a periodic function of time and continuous in the set of arguments. Pliss V.A. obtained the first results for a particular class of equations and showed that a number of theorems can not be continued. In this paper, it was possible to reduce the restrictions on the degree of smoothness of the right-hand side of the equation and obtain upper and lower bounds on the number of possible periodic solutions.

  6. 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

  7. Application of the Finite Element Method in Atomic and Molecular Physics

    NASA Technical Reports Server (NTRS)

    Shertzer, Janine

    2007-01-01

    The finite element method (FEM) is a numerical algorithm for solving second order differential equations. It has been successfully used to solve many problems in atomic and molecular physics, including bound state and scattering calculations. To illustrate the diversity of the method, we present here details of two applications. First, we calculate the non-adiabatic dipole polarizability of Hi by directly solving the first and second order equations of perturbation theory with FEM. In the second application, we calculate the scattering amplitude for e-H scattering (without partial wave analysis) by reducing the Schrodinger equation to set of integro-differential equations, which are then solved with FEM.

  8. Partial differential equation models in macroeconomics.

    PubMed

    Achdou, Yves; Buera, Francisco J; Lasry, Jean-Michel; Lions, Pierre-Louis; Moll, Benjamin

    2014-11-13

    The purpose of this article is to get mathematicians interested in studying a number of partial differential equations (PDEs) that naturally arise in macroeconomics. These PDEs come from models designed to study some of the most important questions in economics. At the same time, they are highly interesting for mathematicians because their structure is often quite difficult. We present a number of examples of such PDEs, discuss what is known about their properties, and list some open questions for future research. © 2014 The Author(s) Published by the Royal Society. All rights reserved.

  9. FAST TRACK COMMUNICATION: On the Liouvillian solution of second-order linear differential equations and algebraic invariant curves

    NASA Astrophysics Data System (ADS)

    Man, Yiu-Kwong

    2010-10-01

    In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle-Singer method for computing first integrals of differential equations. Some examples on using this approach are provided.

  10. Interval oscillation criteria for second-order forced impulsive delay differential equations with damping term.

    PubMed

    Thandapani, Ethiraju; Kannan, Manju; Pinelas, Sandra

    2016-01-01

    In this paper, we present some sufficient conditions for the oscillation of all solutions of a second order forced impulsive delay differential equation with damping term. Three factors-impulse, delay and damping that affect the interval qualitative properties of solutions of equations are taken into account together. The results obtained in this paper extend and generalize some of the the known results for forced impulsive differential equations. An example is provided to illustrate the main result.

  11. 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

  12. Computations of Wall Distances Based on Differential Equations

    NASA Technical Reports Server (NTRS)

    Tucker, Paul G.; Rumsey, Chris L.; Spalart, Philippe R.; Bartels, Robert E.; Biedron, Robert T.

    2004-01-01

    The use of differential equations such as Eikonal, Hamilton-Jacobi and Poisson for the economical calculation of the nearest wall distance d, which is needed by some turbulence models, is explored. Modifications that could palliate some turbulence-modeling anomalies are also discussed. Economy is of especial value for deforming/adaptive grid problems. For these, ideally, d is repeatedly computed. It is shown that the Eikonal and Hamilton-Jacobi equations can be easy to implement when written in implicit (or iterated) advection and advection-diffusion equation analogous forms, respectively. These, like the Poisson Laplacian term, are commonly occurring in CFD solvers, allowing the re-use of efficient algorithms and code components. The use of the NASA CFL3D CFD program to solve the implicit Eikonal and Hamilton-Jacobi equations is explored. The re-formulated d equations are easy to implement, and are found to have robust convergence. For accurate Eikonal solutions, upwind metric differences are required. The Poisson approach is also found effective, and easiest to implement. Modified distances are not found to affect global outputs such as lift and drag significantly, at least in common situations such as airfoil flows.

  13. A new perturbative approach to nonlinear partial differential equations

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

    Bender, C.M.; Boettcher, S.; Milton, K.A.

    1991-11-01

    This paper shows how to solve some nonlinear wave equations as perturbation expansions in powers of a parameter that expresses the degree of nonlinearity. For the case of the Burgers equation {ital u}{sub {ital t}}+{ital uu}{sub {ital x}}={ital u}{sub {ital xx}}, the general nonlinear equation {ital u}{sub {ital t}}+{ital u}{sup {delta}}{ital u}{sub {ital x}}={ital u}{sub {ital xx}} is considered and expanded in powers of {delta}. The coefficients of the {delta} series to sixth order in powers of {delta} is determined and Pade summation is used to evaluate the perturbation series for large values of {delta}. The numerical results are accuratemore » and the method is very general; it applies to other well-studied partial differential equations such as the Korteweg--de Vries equation, {ital u}{sub {ital t}}+{ital uu}{sub {ital x}} ={ital u}{sub {ital xxx}}.« less

  14. The study of nonlinear almost periodic differential equations without recourse to the H-classes of these equations

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

    Slyusarchuk, V. E., E-mail: V.E.Slyusarchuk@gmail.com, E-mail: V.Ye.Slyusarchuk@NUWM.rv.ua

    2014-06-01

    The well-known theorems of Favard and Amerio on the existence of almost periodic solutions to linear and nonlinear almost periodic differential equations depend to a large extent on the H-classes and the requirement that the bounded solutions of these equations be separated. The present paper provides different conditions for the existence of almost periodic solutions. These conditions, which do not depend on the H-classes of the equations, are formulated in terms of a special functional on the set of bounded solutions of the equations under consideration. This functional is used, in particular, to test whether solutions are separated. Bibliography: 24more » titles. (paper)« less

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

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

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

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

  16. Varieties of operator manipulation. [for solving differential equations and calculating finite differences

    NASA Technical Reports Server (NTRS)

    Doohovskoy, A.

    1977-01-01

    A change in MACSYMA syntax is proposed to accommodate the operator manipulators necessary to implement direct and indirect methods for the solution of differential equations, calculus of finite differences, and the fractional calculus, as well as their modern counterparts. To illustrate the benefits and convenience of this syntax extension, an example is given to show how MACSYMA's pattern-matching capability can be used to implement a particular set of operator identities which can then be used to obtain exact solutions to nonlinear differential equations.

  17. 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.

  18. Asymptotic integration algorithms for nonhomogeneous, nonlinear, first order, ordinary differential equations

    NASA Technical Reports Server (NTRS)

    Walker, K. P.; Freed, A. D.

    1991-01-01

    New methods for integrating systems of stiff, nonlinear, first order, ordinary differential equations are developed by casting the differential equations into integral form. Nonlinear recursive relations are obtained that allow the solution to a system of equations at time t plus delta t to be obtained in terms of the solution at time t in explicit and implicit forms. Examples of accuracy obtained with the new technique are given by considering systems of nonlinear, first order equations which arise in the study of unified models of viscoplastic behaviors, the spread of the AIDS virus, and predator-prey populations. In general, the new implicit algorithm is unconditionally stable, and has a Jacobian of smaller dimension than that which is acquired by current implicit methods, such as the Euler backward difference algorithm; yet, it gives superior accuracy. The asymptotic explicit and implicit algorithms are suitable for solutions that are of the growing and decaying exponential kinds, respectively, whilst the implicit Euler-Maclaurin algorithm is superior when the solution oscillates, i.e., when there are regions in which both growing and decaying exponential solutions exist.

  19. Backscattering and absorption coefficients for electrons: Solutions of invariant embedding transport equations using a method of convergence

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

    Figueroa, C.; Brizuela, H.; Heluani, S. P.

    2014-05-21

    The backscattering coefficient is a magnitude whose measurement is fundamental for the characterization of materials with techniques that make use of particle beams and particularly when performing microanalysis. In this work, we report the results of an analytic method to calculate the backscattering and absorption coefficients of electrons in similar conditions to those of electron probe microanalysis. Starting on a five level states ladder model in 3D, we deduced a set of integro-differential coupled equations of the coefficients with a method know as invariant embedding. By means of a procedure proposed by authors, called method of convergence, two types ofmore » approximate solutions for the set of equations, namely complete and simple solutions, can be obtained. Although the simple solutions were initially proposed as auxiliary forms to solve higher rank equations, they turned out to be also useful for the estimation of the aforementioned coefficients. In previous reports, we have presented results obtained with the complete solutions. In this paper, we present results obtained with the simple solutions of the coefficients, which exhibit a good degree of fit with the experimental data. Both the model and the calculation method presented here can be generalized to other techniques that make use of different sorts of particle beams.« less

  20. A homotopy analysis method for the nonlinear partial differential equations arising in engineering

    NASA Astrophysics Data System (ADS)

    Hariharan, G.

    2017-05-01

    In this article, we have established the homotopy analysis method (HAM) for solving a few partial differential equations arising in engineering. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of nonlinear terms appearing in the governing differential equations. The convergence analysis of the proposed method is also discussed. Finally, we have given some illustrative examples to demonstrate the validity and applicability of the proposed method.

  1. Multiple positive solutions to a coupled systems of nonlinear fractional differential equations.

    PubMed

    Shah, Kamal; Khan, Rahmat Ali

    2016-01-01

    In this article, we study existence, uniqueness and nonexistence of positive solution to a highly nonlinear coupled system of fractional order differential equations. Necessary and sufficient conditions for the existence and uniqueness of positive solution are developed by using Perov's fixed point theorem for the considered problem. Further, we also established sufficient conditions for existence of multiplicity results for positive solutions. Also, we developed some conditions under which the considered coupled system of fractional order differential equations has no positive solution. Appropriate examples are also provided which demonstrate our results.

  2. Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (4).

    PubMed

    Murase, Kenya

    2016-01-01

    Partial differential equations are often used in the field of medical physics. In this (final) issue, the methods for solving the partial differential equations were introduced, which include separation of variables, integral transform (Fourier and Fourier-sine transforms), Green's function, and series expansion methods. Some examples were also introduced, in which the integral transform and Green's function methods were applied to solving Pennes' bioheat transfer equation and the Fourier series expansion method was applied to Navier-Stokes equation for analyzing the wall shear stress in blood vessels.Finally, the author hopes that this series will be helpful for people who engage in medical physics.

  3. A Second-Year Undergraduate Course in Applied Differential Equations.

    ERIC Educational Resources Information Center

    Fahidy, Thomas Z.

    1991-01-01

    Presents the framework for a chemical engineering course using ordinary differential equations to solve problems with the underlying strategy of concisely discussing the theory behind each solution technique without extensions to formal proofs. Includes typical class illustrations, student responses to this strategy, and reaction of the…

  4. Bounding solutions of geometrically nonlinear viscoelastic problems

    NASA Technical Reports Server (NTRS)

    Stubstad, J. M.; Simitses, G. J.

    1985-01-01

    Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

  5. Bounding solutions of geometrically nonlinear viscoelastic problems

    NASA Technical Reports Server (NTRS)

    Stubstad, J. M.; Simitses, G. J.

    1986-01-01

    Integral transform techniques, such as the Laplace transform, provide simple and direct methods for solving viscoelastic problems formulated within a context of linear material response and using linear measures for deformation. Application of the transform operator reduces the governing linear integro-differential equations to a set of algebraic relations between the transforms of the unknown functions, the viscoelastic operators, and the initial and boundary conditions. Inversion either directly or through the use of the appropriate convolution theorem, provides the time domain response once the unknown functions have been expressed in terms of sums, products or ratios of known transforms. When exact inversion is not possible approximate techniques may provide accurate results. The overall problem becomes substantially more complex when nonlinear effects must be included. Situations where a linear material constitutive law can still be productively employed but where the magnitude of the resulting time dependent deformations warrants the use of a nonlinear kinematic analysis are considered. The governing equations will be nonlinear integro-differential equations for this class of problems. Thus traditional as well as approximate techniques, such as cited above, cannot be employed since the transform of a nonlinear function is not explicitly expressible.

  6. Local bifurcations in differential equations with state-dependent delay.

    PubMed

    Sieber, Jan

    2017-11-01

    A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one encounters in a numerical bifurcation study guides follow-up computations. This paper builds on normal form algorithms for equilibria of delay differential equations with constant delay that were developed and implemented in DDE-Biftool recently. We show how one can extend these methods to delay-differential equations with state-dependent delay (sd-DDEs). Since higher degrees of regularity of local center manifolds are still open for sd-DDEs, we give an independent (still only partial) argument which phenomena from the truncated normal must persist in the full sd-DDE. In particular, we show that all invariant manifolds with a sufficient degree of normal hyperbolicity predicted by the normal form exist also in the full sd-DDE.

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

    NASA Technical Reports Server (NTRS)

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

    1966-01-01

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

  8. 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.

  9. 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.

  10. 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

  11. Solving Partial Differential Equations on Overlapping Grids

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

    Henshaw, W D

    2008-09-22

    We discuss the solution of partial differential equations (PDEs) on overlapping grids. This is a powerful technique for efficiently solving problems in complex, possibly moving, geometry. An overlapping grid consists of a set of structured grids that overlap and cover the computational domain. By allowing the grids to overlap, grids for complex geometries can be more easily constructed. The overlapping grid approach can also be used to remove coordinate singularities by, for example, covering a sphere with two or more patches. We describe the application of the overlapping grid approach to a variety of different problems. These include the solutionmore » of incompressible fluid flows with moving and deforming geometry, the solution of high-speed compressible reactive flow with rigid bodies using adaptive mesh refinement (AMR), and the solution of the time-domain Maxwell's equations of electromagnetism.« less

  12. Formulation and application of optimal homotopty asymptotic method to coupled differential-difference equations.

    PubMed

    Ullah, Hakeem; Islam, Saeed; Khan, Ilyas; Shafie, Sharidan; Fiza, Mehreen

    2015-01-01

    In this paper we applied a new analytic approximate technique Optimal Homotopy Asymptotic Method (OHAM) for treatment of coupled differential-difference equations (DDEs). To see the efficiency and reliability of the method, we consider Relativistic Toda coupled nonlinear differential-difference equation. It provides us a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature. The obtained solutions show that OHAM is effective, simpler, easier and explicit.

  13. Taguchi method for partial differential equations with application in tumor growth.

    PubMed

    Ilea, M; Turnea, M; Rotariu, M; Arotăriţei, D; Popescu, Marilena

    2014-01-01

    The growth of tumors is a highly complex process. To describe this process, mathematical models are needed. A variety of partial differential mathematical models for tumor growth have been developed and studied. Most of those models are based on the reaction-diffusion equations and mass conservation law. A variety of modeling strategies have been developed, each focusing on tumor growth. Systems of time-dependent partial differential equations occur in many branches of applied mathematics. The vast majority of mathematical models in tumor growth are formulated in terms of partial differential equations. We propose a mathematical model for the interactions between these three cancer cell populations. The Taguchi methods are widely used by quality engineering scientists to compare the effects of multiple variables, together with their interactions, with a simple and manageable experimental design. In Taguchi's design of experiments, variation is more interesting to study than the average. First, Taguchi methods are utilized to search for the significant factors and the optimal level combination of parameters. Except the three parameters levels, other factors levels other factors levels would not be considered. Second, cutting parameters namely, cutting speed, depth of cut, and feed rate are designed using the Taguchi method. Finally, the adequacy of the developed mathematical model is proved by ANOVA. According to the results of ANOVA, since the percentage contribution of the combined error is as small. Many mathematical models can be quantitatively characterized by partial differential equations. The use of MATLAB and Taguchi method in this article illustrates the important role of informatics in research in mathematical modeling. The study of tumor growth cells is an exciting and important topic in cancer research and will profit considerably from theoretical input. Interpret these results to be a permanent collaboration between math's and medical oncologists.

  14. Population stochastic modelling (PSM)--an R package for mixed-effects models based on stochastic differential equations.

    PubMed

    Klim, Søren; Mortensen, Stig Bousgaard; Kristensen, Niels Rode; Overgaard, Rune Viig; Madsen, Henrik

    2009-06-01

    The extension from ordinary to stochastic differential equations (SDEs) in pharmacokinetic and pharmacodynamic (PK/PD) modelling is an emerging field and has been motivated in a number of articles [N.R. Kristensen, H. Madsen, S.H. Ingwersen, Using stochastic differential equations for PK/PD model development, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 109-141; C.W. Tornøe, R.V. Overgaard, H. Agersø, H.A. Nielsen, H. Madsen, E.N. Jonsson, Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations, Pharm. Res. 22 (August(8)) (2005) 1247-1258; R.V. Overgaard, N. Jonsson, C.W. Tornøe, H. Madsen, Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 85-107; U. Picchini, S. Ditlevsen, A. De Gaetano, Maximum likelihood estimation of a time-inhomogeneous stochastic differential model of glucose dynamics, Math. Med. Biol. 25 (June(2)) (2008) 141-155]. PK/PD models are traditionally based ordinary differential equations (ODEs) with an observation link that incorporates noise. This state-space formulation only allows for observation noise and not for system noise. Extending to SDEs allows for a Wiener noise component in the system equations. This additional noise component enables handling of autocorrelated residuals originating from natural variation or systematic model error. Autocorrelated residuals are often partly ignored in PK/PD modelling although violating the hypothesis for many standard statistical tests. This article presents a package for the statistical program R that is able to handle SDEs in a mixed-effects setting. The estimation method implemented is the FOCE(1) approximation to the population likelihood which is generated from the individual likelihoods that are approximated using the Extended Kalman Filter's one-step predictions.

  15. Attempts at a numerical realisation of stochastic differential equations containing Preisach operator

    NASA Astrophysics Data System (ADS)

    McCarthy, S.; Rachinskii, D.

    2011-01-01

    We describe two Euler type numerical schemes obtained by discretisation of a stochastic differential equation which contains the Preisach memory operator. Equations of this type are of interest in areas such as macroeconomics and terrestrial hydrology where deterministic models containing the Preisach operator have been developed but do not fully encapsulate stochastic aspects of the area. A simple price dynamics model is presented as one motivating example for our studies. Some numerical evidence is given that the two numerical schemes converge to the same limit as the time step decreases. We show that the Preisach term introduces a damping effect which increases on the parts of the trajectory demonstrating a stronger upwards or downwards trend. The results are preliminary to a broader programme of research of stochastic differential equations with the Preisach hysteresis operator.

  16. Study of coupled nonlinear partial differential equations for finding exact analytical solutions.

    PubMed

    Khan, Kamruzzaman; Akbar, M Ali; Koppelaar, H

    2015-07-01

    Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G'/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.

  17. State-dependent differential Riccati equation to track control of time-varying systems with state and control nonlinearities.

    PubMed

    Korayem, M H; Nekoo, S R

    2015-07-01

    This work studies an optimal control problem using the state-dependent Riccati equation (SDRE) in differential form to track for time-varying systems with state and control nonlinearities. The trajectory tracking structure provides two nonlinear differential equations: the state-dependent differential Riccati equation (SDDRE) and the feed-forward differential equation. The independence of the governing equations and stability of the controller are proven along the trajectory using the Lyapunov approach. Backward integration (BI) is capable of solving the equations as a numerical solution; however, the forward solution methods require the closed-form solution to fulfill the task. A closed-form solution is introduced for SDDRE, but the feed-forward differential equation has not yet been obtained. Different ways of solving the problem are expressed and analyzed. These include BI, closed-form solution with corrective assumption, approximate solution, and forward integration. Application of the tracking problem is investigated to control robotic manipulators possessing rigid or flexible joints. The intention is to release a general program for automatic implementation of an SDDRE controller for any manipulator that obeys the Denavit-Hartenberg (D-H) principle when only D-H parameters are received as input data. Copyright © 2015 ISA. Published by Elsevier Ltd. All rights reserved.

  18. 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.

  19. Parameter Estimates in Differential Equation Models for Population Growth

    ERIC Educational Resources Information Center

    Winkel, Brian J.

    2011-01-01

    We estimate the parameters present in several differential equation models of population growth, specifically logistic growth models and two-species competition models. We discuss student-evolved strategies and offer "Mathematica" code for a gradient search approach. We use historical (1930s) data from microbial studies of the Russian biologist,…

  20. Control of functional differential equations with function space boundary conditions

    NASA Technical Reports Server (NTRS)

    Banks, H. T.

    1972-01-01

    Problems involving functional differential equations with terminal conditions in function space are considered. Their application to mechanical and electrical systems is discussed. Investigations of controllability, existence of optimal controls, and necessary and sufficient conditions for optimality are reported.

  1. exponential finite difference technique for solving partial differential equations

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

    Handschuh, R.F.

    1987-01-01

    An exponential finite difference algorithm, as first presented by Bhattacharya for one-dimensianal steady-state, heat conduction in Cartesian coordinates, has been extended. The finite difference algorithm developed was used to solve the diffusion equation in one-dimensional cylindrical coordinates and applied to two- and three-dimensional problems in Cartesian coordinates. The method was also used to solve nonlinear partial differential equations in one (Burger's equation) and two (Boundary Layer equations) dimensional Cartesian coordinates. Predicted results were compared to exact solutions where available, or to results obtained by other numerical methods. It was found that the exponential finite difference method produced results that weremore » more accurate than those obtained by other numerical methods, especially during the initial transient portion of the solution. Other applications made using the exponential finite difference technique included unsteady one-dimensional heat transfer with temperature varying thermal conductivity and the development of the temperature field in a laminar Couette flow.« less

  2. On new classes of solutions of nonlinear partial differential equations in the form of convergent special series

    NASA Astrophysics Data System (ADS)

    Filimonov, M. Yu.

    2017-12-01

    The method of special series with recursively calculated coefficients is used to solve nonlinear partial differential equations. The recurrence of finding the coefficients of the series is achieved due to a special choice of functions, in powers of which the solution is expanded in a series. We obtain a sequence of linear partial differential equations to find the coefficients of the series constructed. In many cases, one can deal with a sequence of linear ordinary differential equations. We construct classes of solutions in the form of convergent series for a certain class of nonlinear evolution equations. A new class of solutions of generalized Boussinesque equation with an arbitrary function in the form of a convergent series is constructed.

  3. Formulation and Application of Optimal Homotopty Asymptotic Method to Coupled Differential - Difference Equations

    PubMed Central

    Ullah, Hakeem; Islam, Saeed; Khan, Ilyas; Shafie, Sharidan; Fiza, Mehreen

    2015-01-01

    In this paper we applied a new analytic approximate technique Optimal Homotopy Asymptotic Method (OHAM) for treatment of coupled differential- difference equations (DDEs). To see the efficiency and reliability of the method, we consider Relativistic Toda coupled nonlinear differential-difference equation. It provides us a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature. The obtained solutions show that OHAM is effective, simpler, easier and explicit. PMID:25874457

  4. Homogeneous partial differential equations for superpositions of indeterminate functions of several variables

    NASA Astrophysics Data System (ADS)

    Asai, Kazuto

    2009-02-01

    We determine essentially all partial differential equations satisfied by superpositions of tree type and of a further special type. These equations represent necessary and sufficient conditions for an analytic function to be locally expressible as an analytic superposition of the type indicated. The representability of a real analytic function by a superposition of this type is independent of whether that superposition involves real-analytic functions or C^{\\rho}-functions, where the constant \\rho is determined by the structure of the superposition. We also prove that the function u defined by u^n=xu^a+yu^b+zu^c+1 is generally non-representable in any real (resp. complex) domain as f\\bigl(g(x,y),h(y,z)\\bigr) with twice differentiable f and differentiable g, h (resp. analytic f, g, h).

  5. Modelling biochemical reaction systems by stochastic differential equations with reflection.

    PubMed

    Niu, Yuanling; Burrage, Kevin; Chen, Luonan

    2016-05-07

    In this paper, we gave a new framework for modelling and simulating biochemical reaction systems by stochastic differential equations with reflection not in a heuristic way but in a mathematical way. The model is computationally efficient compared with the discrete-state Markov chain approach, and it ensures that both analytic and numerical solutions remain in a biologically plausible region. Specifically, our model mathematically ensures that species numbers lie in the domain D, which is a physical constraint for biochemical reactions, in contrast to the previous models. The domain D is actually obtained according to the structure of the corresponding chemical Langevin equations, i.e., the boundary is inherent in the biochemical reaction system. A variant of projection method was employed to solve the reflected stochastic differential equation model, and it includes three simple steps, i.e., Euler-Maruyama method was applied to the equations first, and then check whether or not the point lies within the domain D, and if not perform an orthogonal projection. It is found that the projection onto the closure D¯ is the solution to a convex quadratic programming problem. Thus, existing methods for the convex quadratic programming problem can be employed for the orthogonal projection map. Numerical tests on several important problems in biological systems confirmed the efficiency and accuracy of this approach. Copyright © 2016 Elsevier Ltd. All rights reserved.

  6. On method of solving third-order ordinary differential equations directly using Bernstein polynomials

    NASA Astrophysics Data System (ADS)

    Khataybeh, S. N.; Hashim, I.

    2018-04-01

    In this paper, we propose for the first time a method based on Bernstein polynomials for solving directly a class of third-order ordinary differential equations (ODEs). This method gives a numerical solution by converting the equation into a system of algebraic equations which is solved directly. Some numerical examples are given to show the applicability of the method.

  7. Description and use of LSODE, the Livermore Solver for Ordinary Differential Equations

    NASA Technical Reports Server (NTRS)

    Radhakrishnan, Krishnan; Hindmarsh, Alan C.

    1993-01-01

    LSODE, the Livermore Solver for Ordinary Differential Equations, is a package of FORTRAN subroutines designed for the numerical solution of the initial value problem for a system of ordinary differential equations. It is particularly well suited for 'stiff' differential systems, for which the backward differentiation formula method of orders 1 to 5 is provided. The code includes the Adams-Moulton method of orders 1 to 12, so it can be used for nonstiff problems as well. In addition, the user can easily switch methods to increase computational efficiency for problems that change character. For both methods a variety of corrector iteration techniques is included in the code. Also, to minimize computational work, both the step size and method order are varied dynamically. This report presents complete descriptions of the code and integration methods, including their implementation. It also provides a detailed guide to the use of the code, as well as an illustrative example problem.

  8. Application of Model Based Parameter Estimation for Fast Frequency Response Calculations of Input Characteristics of Cavity-Backed Aperture Antennas Using Hybrid FEM/MoM Technique

    NASA Technical Reports Server (NTRS)

    Reddy C. J.

    1998-01-01

    Model Based Parameter Estimation (MBPE) is presented in conjunction with the hybrid Finite Element Method (FEM)/Method of Moments (MoM) technique for fast computation of the input characteristics of cavity-backed aperture antennas over a frequency range. The hybrid FENI/MoM technique is used to form an integro-partial- differential equation to compute the electric field distribution of a cavity-backed aperture antenna. In MBPE, the electric field is expanded in a rational function of two polynomials. The coefficients of the rational function are obtained using the frequency derivatives of the integro-partial-differential equation formed by the hybrid FEM/ MoM technique. Using the rational function approximation, the electric field is obtained over a frequency range. Using the electric field at different frequencies, the input characteristics of the antenna are obtained over a wide frequency range. Numerical results for an open coaxial line, probe-fed coaxial cavity and cavity-backed microstrip patch antennas are presented. Good agreement between MBPE and the solutions over individual frequencies is observed.

  9. Frequency Response Calculations of Input Characteristics of Cavity-Backed Aperture Antennas Using AWE with Hybrid FEM/MoM Technique

    NASA Technical Reports Server (NTRS)

    Reddy, C. J.; Deshpande, M. D.

    1997-01-01

    Application of Asymptotic Waveform Evaluation (AWE) is presented in conjunction with a hybrid Finite Element Method (FEM)/Method of Moments (MoM) technique to calculate the input characteristics of cavity-backed aperture antennas over a frequency range. The hybrid FEM/MoM technique is used to form an integro-partial-differential equation to compute the electric field distribution of the cavity-backed aperture antenna. The electric field, thus obtained, is expanded in a Taylor series around the frequency of interest. The coefficients of 'Taylor series (called 'moments') are obtained using the frequency derivatives of the integro-partial-differential Equation formed by the hybrid FEM/MoM technique. Using the moments, the electric field in the cavity is obtained over a frequency range. Using the electric field at different frequencies, the input characteristics of the antenna are obtained over a wide frequency band. Numerical results for an open coaxial line, probe fed cavity, and cavity-backed microstrip patch antennas are presented. Good agreement between AWE and the exact solution over the frequency range is observed.

  10. Flap-lag-torsional dynamics of extensional and inextensional rotor blades in hover and in forward flight

    NASA Technical Reports Server (NTRS)

    Dasilva, C.

    1982-01-01

    The reduction of the O(cu epsilon) integro differential equations to ordinary differential equations using a set of orthogonal functions is described. Attention was focused on the hover flight condition. The set of Galerkin integrals that appear in the reduced equations was evaluated by making use of nonrotating beam modes. Although a large amount of computer time was needed to accomplish this task, the Galerkin integrals so evaluated were stored on tape on a permanent basis. Several of the coefficients were also obtained in closed form in order to check the accuracy of the numerical computations. The equilibrium solution to the set of 3n equations obtained was determined as the solution to a minimization problem.

  11. 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.

  12. Isostable reduction with applications to time-dependent partial differential equations.

    PubMed

    Wilson, Dan; Moehlis, Jeff

    2016-07-01

    Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system.

  13. A nonlinear ordinary differential equation associated with the quantum sojourn time

    NASA Astrophysics Data System (ADS)

    Benguria, Rafael D.; Duclos, Pierre; Fernández, Claudio; Sing-Long, Carlos

    2010-11-01

    We study a nonlinear ordinary differential equation on the half-line, with the Dirichlet boundary condition at the origin. This equation arises when studying the local maxima of the sojourn time for a free quantum particle whose states belong to an adequate subspace of the unit sphere of the corresponding Hilbert space. We establish several results concerning the existence and asymptotic behavior of the solutions.

  14. 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.

  15. 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.

  16. Study of coupled nonlinear partial differential equations for finding exact analytical solutions

    PubMed Central

    Khan, Kamruzzaman; Akbar, M. Ali; Koppelaar, H.

    2015-01-01

    Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G′/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd–Sokolov–Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics. PMID:26587256

  17. Numerical solution of stiff systems of ordinary differential equations with applications to electronic circuits

    NASA Technical Reports Server (NTRS)

    Rosenbaum, J. S.

    1971-01-01

    Systems of ordinary differential equations in which the magnitudes of the eigenvalues (or time constants) vary greatly are commonly called stiff. Such systems of equations arise in nuclear reactor kinetics, the flow of chemically reacting gas, dynamics, control theory, circuit analysis and other fields. The research reported develops an A-stable numerical integration technique for solving stiff systems of ordinary differential equations. The method, which is called the generalized trapezoidal rule, is a modification of the trapezoidal rule. However, the method is computationally more efficient than the trapezoidal rule when the solution of the almost-discontinuous segments is being calculated.

  18. Existence and stability of periodic solutions of an impulsive differential equation and application to CD8 T-cell differentiation.

    PubMed

    Girel, Simon; Crauste, Fabien

    2018-06-01

    Unequal partitioning of the molecular content at cell division has been shown to be a source of heterogeneity in a cell population. We propose to model this phenomenon with the help of a scalar, nonlinear impulsive differential equation (IDE). To study the effect of molecular partitioning at cell division on the effector/memory cell-fate decision in a CD8 T-cell lineage, we study an IDE describing the concentration of the protein Tbet in a CD8 T-cell, where impulses are associated to cell division. We discuss how the degree of asymmetry of molecular partitioning can affect the process of cell differentiation and the phenotypical heterogeneity of a cell population. We show that a moderate degree of asymmetry is necessary and sufficient to observe irreversible differentiation. We consider, in a second part, a general autonomous IDE with fixed times of impulse and a specific form of impulse function. We establish properties of the solutions of that equation, most of them obtained under the hypothesis that impulses occur periodically. In particular, we show how to investigate the existence of periodic solutions and their stability by studying the flow of an autonomous differential equation. Then we apply those properties to prove the results presented in the first part.

  19. Generalized Ordinary Differential Equation Models 1

    PubMed Central

    Miao, Hongyu; Wu, Hulin; Xue, Hongqi

    2014-01-01

    Existing estimation methods for ordinary differential equation (ODE) models are not applicable to discrete data. The generalized ODE (GODE) model is therefore proposed and investigated for the first time. We develop the likelihood-based parameter estimation and inference methods for GODE models. We propose robust computing algorithms and rigorously investigate the asymptotic properties of the proposed estimator by considering both measurement errors and numerical errors in solving ODEs. The simulation study and application of our methods to an influenza viral dynamics study suggest that the proposed methods have a superior performance in terms of accuracy over the existing ODE model estimation approach and the extended smoothing-based (ESB) method. PMID:25544787

  20. Oscillation of certain higher-order neutral partial functional differential equations.

    PubMed

    Li, Wei Nian; Sheng, Weihong

    2016-01-01

    In this paper, we study the oscillation of certain higher-order neutral partial functional differential equations with the Robin boundary conditions. Some oscillation criteria are established. Two examples are given to illustrate the main results in the end of this paper.

  1. On Polynomial Solutions of Linear Differential Equations with Polynomial Coefficients

    ERIC Educational Resources Information Center

    Si, Do Tan

    1977-01-01

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

  2. Einstein-Weyl spaces and third-order differential equations

    NASA Astrophysics Data System (ADS)

    Tod, K. P.

    2000-08-01

    The three-dimensional null-surface formalism of Tanimoto [M. Tanimoto, "On the null surface formalism," Report No. gr-qc/9703003 (1997)] and Forni et al. [Forni et al., "Null surfaces formation in 3D," J. Math Phys. (submitted)] are extended to describe Einstein-Weyl spaces, following Cartan [E. Cartan, "Les espaces généralisées et l'integration de certaines classes d'equations différentielles," C. R. Acad. Sci. 206, 1425-1429 (1938); "La geometria de las ecuaciones diferenciales de tercer order," Rev. Mat. Hispano-Am. 4, 1-31 (1941)]. In the resulting formalism, Einstein-Weyl spaces are obtained from a particular class of third-order differential equations. Some examples of the construction which include some new Einstein-Weyl spaces are given.

  3. In silico ordinary differential equation/partial differential equation hemodialysis model estimates methadone removal during dialysis

    PubMed Central

    Linares, Oscar A; Schiesser, William E; Fudin, Jeffrey; Pham, Thien C; Bettinger, Jeffrey J; Mathew, Roy O; Daly, Annemarie L

    2015-01-01

    Background There is a need to have a model to study methadone’s losses during hemodialysis to provide informed methadone dose recommendations for the practitioner. Aim To build a one-dimensional (1-D), hollow-fiber geometry, ordinary differential equation (ODE) and partial differential equation (PDE) countercurrent hemodialyzer model (ODE/PDE model). Methodology We conducted a cross-sectional study in silico that evaluated eleven hemodialysis patients. Patients received a ceiling dose of methadone hydrochloride 30 mg/day. Outcome measures included: the total amount of methadone removed during dialysis; methadone’s overall intradialytic mass transfer rate coefficient, km; and, methadone’s removal rate, jME. Each metric was measured at dialysate flow rates of 250 mL/min and 800 mL/min. Results The ODE/PDE model revealed a significant increase in the change of methadone’s mass transfer with increased dialysate flow rate, %Δkm=18.56, P=0.02, N=11. The total amount of methadone mass transferred across the dialyzer membrane with high dialysate flow rate significantly increased (0.042±0.016 versus 0.052±0.019 mg/kg, P=0.02, N=11). This was accompanied by a small significant increase in methadone’s mass transfer rate (0.113±0.002 versus 0.014±0.002 mg/kg/h, P=0.02, N=11). The ODE/PDE model accurately predicted methadone’s removal during dialysis. The absolute value of the prediction errors for methadone’s extraction and throughput were less than 2%. Conclusion ODE/PDE modeling of methadone’s hemodialysis is a new approach to study methadone’s removal, in particular, and opioid removal, in general, in patients with end-stage renal disease on hemodialysis. ODE/PDE modeling accurately quantified the fundamental phenomena of methadone’s mass transfer during hemodialysis. This methodology may lead to development of optimally designed intradialytic opioid treatment protocols, and allow dynamic monitoring of outflow plasma opioid concentrations for model

  4. In silico ordinary differential equation/partial differential equation hemodialysis model estimates methadone removal during dialysis.

    PubMed

    Linares, Oscar A; Schiesser, William E; Fudin, Jeffrey; Pham, Thien C; Bettinger, Jeffrey J; Mathew, Roy O; Daly, Annemarie L

    2015-01-01

    There is a need to have a model to study methadone's losses during hemodialysis to provide informed methadone dose recommendations for the practitioner. To build a one-dimensional (1-D), hollow-fiber geometry, ordinary differential equation (ODE) and partial differential equation (PDE) countercurrent hemodialyzer model (ODE/PDE model). We conducted a cross-sectional study in silico that evaluated eleven hemodialysis patients. Patients received a ceiling dose of methadone hydrochloride 30 mg/day. Outcome measures included: the total amount of methadone removed during dialysis; methadone's overall intradialytic mass transfer rate coefficient, km ; and, methadone's removal rate, j ME. Each metric was measured at dialysate flow rates of 250 mL/min and 800 mL/min. The ODE/PDE model revealed a significant increase in the change of methadone's mass transfer with increased dialysate flow rate, %Δkm =18.56, P=0.02, N=11. The total amount of methadone mass transferred across the dialyzer membrane with high dialysate flow rate significantly increased (0.042±0.016 versus 0.052±0.019 mg/kg, P=0.02, N=11). This was accompanied by a small significant increase in methadone's mass transfer rate (0.113±0.002 versus 0.014±0.002 mg/kg/h, P=0.02, N=11). The ODE/PDE model accurately predicted methadone's removal during dialysis. The absolute value of the prediction errors for methadone's extraction and throughput were less than 2%. ODE/PDE modeling of methadone's hemodialysis is a new approach to study methadone's removal, in particular, and opioid removal, in general, in patients with end-stage renal disease on hemodialysis. ODE/PDE modeling accurately quantified the fundamental phenomena of methadone's mass transfer during hemodialysis. This methodology may lead to development of optimally designed intradialytic opioid treatment protocols, and allow dynamic monitoring of outflow plasma opioid concentrations for model predictive control during dialysis in humans.

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

    PubMed

    Schlemm, Eckhard

    2015-09-01

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

  6. Numerical treatment for solving two-dimensional space-fractional advection-dispersion equation using meshless method

    NASA Astrophysics Data System (ADS)

    Cheng, Rongjun; Sun, Fengxin; Wei, Qi; Wang, Jufeng

    2018-02-01

    Space-fractional advection-dispersion equation (SFADE) can describe particle transport in a variety of fields more accurately than the classical models of integer-order derivative. Because of nonlocal property of integro-differential operator of space-fractional derivative, it is very challenging to deal with fractional model, and few have been reported in the literature. In this paper, a numerical analysis of the two-dimensional SFADE is carried out by the element-free Galerkin (EFG) method. The trial functions for the SFADE are constructed by the moving least-square (MLS) approximation. By the Galerkin weak form, the energy functional is formulated. Employing the energy functional minimization procedure, the final algebraic equations system is obtained. The Riemann-Liouville operator is discretized by the Grünwald formula. With center difference method, EFG method and Grünwald formula, the fully discrete approximation schemes for SFADE are established. Comparing with exact results and available results by other well-known methods, the computed approximate solutions are presented in the format of tables and graphs. The presented results demonstrate the validity, efficiency and accuracy of the proposed techniques. Furthermore, the error is computed and the proposed method has reasonable convergence rates in spatial and temporal discretizations.

  7. Multiscale functions, scale dynamics, and applications to partial differential equations

    NASA Astrophysics Data System (ADS)

    Cresson, Jacky; Pierret, Frédéric

    2016-05-01

    Modeling phenomena from experimental data always begins with a choice of hypothesis on the observed dynamics such as determinism, randomness, and differentiability. Depending on these choices, different behaviors can be observed. The natural question associated to the modeling problem is the following: "With a finite set of data concerning a phenomenon, can we recover its underlying nature? From this problem, we introduce in this paper the definition of multi-scale functions, scale calculus, and scale dynamics based on the time scale calculus [see Bohner, M. and Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (Springer Science & Business Media, 2001)] which is used to introduce the notion of scale equations. These definitions will be illustrated on the multi-scale Okamoto's functions. Scale equations are analysed using scale regimes and the notion of asymptotic model for a scale equation under a particular scale regime. The introduced formalism explains why a single scale equation can produce distinct continuous models even if the equation is scale invariant. Typical examples of such equations are given by the scale Euler-Lagrange equation. We illustrate our results using the scale Newton's equation which gives rise to a non-linear diffusion equation or a non-linear Schrödinger equation as asymptotic continuous models depending on the particular fractional scale regime which is considered.

  8. Solving differential equations for Feynman integrals by expansions near singular points

    NASA Astrophysics Data System (ADS)

    Lee, Roman N.; Smirnov, Alexander V.; Smirnov, Vladimir A.

    2018-03-01

    We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with two scales, i.e. non-trivially depending on one variable. The corresponding algorithm is oriented at situations where canonical form of the differential equations is impossible. We provide a computer code constructed with the help of our algorithm for a simple example of four-loop generalized sunset integrals with three equal non-zero masses and two zero masses. Our code gives values of the master integrals at any given point on the real axis with a required accuracy and a given order of expansion in the regularization parameter ɛ.

  9. Climate Modeling in the Calculus and Differential Equations Classroom

    ERIC Educational Resources Information Center

    Kose, Emek; Kunze, Jennifer

    2013-01-01

    Students in college-level mathematics classes can build the differential equations of an energy balance model of the Earth's climate themselves, from a basic understanding of the background science. Here we use variable albedo and qualitative analysis to find stable and unstable equilibria of such a model, providing a problem or perhaps a…

  10. ASP: Automated symbolic computation of approximate symmetries of differential equations

    NASA Astrophysics Data System (ADS)

    Jefferson, G. F.; Carminati, J.

    2013-03-01

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

  11. Calculation of prevalence estimates through differential equations: application to stroke-related disability.

    PubMed

    Mar, Javier; Sainz-Ezkerra, María; Moler-Cuiral, Jose Antonio

    2008-01-01

    Neurological diseases now make up 6.3% of the global burden of disease mainly because they cause disability. To assess disability, prevalence estimates are needed. The objective of this study is to apply a method based on differential equations to calculate the prevalence of stroke-related disability. On the basis of a flow diagram, a set of differential equations for each age group was constructed. The linear system was solved analytically and numerically. The parameters of the system were obtained from the literature. The model was validated and calibrated by comparison with previous results. The stroke prevalence rate per 100,000 men was 828, and the rate for stroke-related disability was 331. The rates steadily rose with age, but the group between the ages of 65 and 74 years had the highest total number of individuals. Differential equations are useful to represent the natural history of neurological diseases and to make possible the calculation of the prevalence for the various states of disability. In our experience, when compared with the results obtained by Markov models, the benefit of the continuous use of time outweighs the mathematical requirements of our model. (c) 2008 S. Karger AG, Basel.

  12. Computer transformation of partial differential equations into any coordinate system

    NASA Technical Reports Server (NTRS)

    Sullivan, R. D.

    1977-01-01

    The use of tensors to provide a compact way of writing partial differential equations in a form valid in all coordinate systems is discussed. In order to find solutions to the equations with their boundary conditions they must be expressed in terms of the coordinate system under consideration. The process of arriving at these expressions from the tensor formulation was automated by a software system, TENSR. An allied system that analyzes the resulting expressions term by term and drops those that are negligible is also described.

  13. A new multi-step technique with differential transform method for analytical solution of some nonlinear variable delay differential equations.

    PubMed

    Benhammouda, Brahim; Vazquez-Leal, Hector

    2016-01-01

    This work presents an analytical solution of some nonlinear delay differential equations (DDEs) with variable delays. Such DDEs are difficult to treat numerically and cannot be solved by existing general purpose codes. A new method of steps combined with the differential transform method (DTM) is proposed as a powerful tool to solve these DDEs. This method reduces the DDEs to ordinary differential equations that are then solved by the DTM. Furthermore, we show that the solutions can be improved by Laplace-Padé resummation method. Two examples are presented to show the efficiency of the proposed technique. The main advantage of this technique is that it possesses a simple procedure based on a few straight forward steps and can be combined with any analytical method, other than the DTM, like the homotopy perturbation method.

  14. A hybrid Pade-Galerkin technique for differential equations

    NASA Technical Reports Server (NTRS)

    Geer, James F.; Andersen, Carl M.

    1993-01-01

    A three-step hybrid analysis technique, which successively uses the regular perturbation expansion method, the Pade expansion method, and then a Galerkin approximation, is presented and applied to some model boundary value problems. In the first step of the method, the regular perturbation method is used to construct an approximation to the solution in the form of a finite power series in a small parameter epsilon associated with the problem. In the second step of the method, the series approximation obtained in step one is used to construct a Pade approximation in the form of a rational function in the parameter epsilon. In the third step, the various powers of epsilon which appear in the Pade approximation are replaced by new (unknown) parameters (delta(sub j)). These new parameters are determined by requiring that the residual formed by substituting the new approximation into the governing differential equation is orthogonal to each of the perturbation coordinate functions used in step one. The technique is applied to model problems involving ordinary or partial differential equations. In general, the technique appears to provide good approximations to the solution even when the perturbation and Pade approximations fail to do so. The method is discussed and topics for future investigations are indicated.

  15. Factors Affecting Differential Equation Problem Solving Ability of Students at Pre-University Level: A Conceptual Model

    ERIC Educational Resources Information Center

    Aisha, Bibi; Zamri, Sharifa NorulAkmar Syed; Abdallah, Nabeel; Abedalaziz, Mohammad; Ahmad, Mushtaq; Satti, Umbreen

    2017-01-01

    In this study, different factors affecting students' differential equations (DEs) solving abilities were explored at pre university level. To explore main factors affecting students' differential equations problem solving ability, articles for a 19-year period, from 1996 to 2015, were critically reviewed and analyzed. It was revealed that…

  16. Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method

    NASA Astrophysics Data System (ADS)

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

    2005-09-01

    We present a double ultraspherical spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomogeneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. Numerical applications of how to use these methods are described. Numerical results obtained compare favorably with those of the analytical solutions. Accurate double ultraspherical spectral approximations for Poisson's and Helmholtz's equations are also noted. Numerical experiments show that spectral approximation based on Chebyshev polynomials of the first kind is not always better than others based on ultraspherical polynomials.

  17. Geometry of Conservation Laws for a Class of Parabolic Partial Differential Equations

    NASA Astrophysics Data System (ADS)

    Clelland, Jeanne Nielsen

    1996-08-01

    I consider the problem of computing the space of conservation laws for a second-order, parabolic partial differential equation for one function of three independent variables. The PDE is formulated as an exterior differential system {cal I} on a 12 -manifold M, and its conservation laws are identified with the vector space of closed 3-forms in the infinite prolongation of {cal I} modulo the so -called "trivial" conservation laws. I use the tools of exterior differential systems and Cartan's method of equivalence to study the structure of the space of conservation laws. My main result is:. Theorem. Any conservation law for a second-order, parabolic PDE for one function of three independent variables can be represented by a closed 3-form in the differential ideal {cal I} on the original 12-manifold M. I show that if a nontrivial conservation law exists, then {cal I} has a deprolongation to an equivalent system {cal J} on a 7-manifold N, and any conservation law for {cal I} can be expressed as a closed 3-form on N which lies in {cal J}. Furthermore, any such system in the real analytic category is locally equivalent to a system generated by a (parabolic) equation of the formA(u _{xx}u_{yy}-u_sp {xy}{2}) + B_1u_{xx }+2B_2u_{xy} +B_3u_ {yy}+C=0crwhere A, B_{i}, C are functions of x, y, t, u, u_{x}, u _{y}, u_{t}. I compute the space of conservation laws for several examples, and I begin the process of analyzing the general case using Cartan's method of equivalence. I show that the non-linearizable equation u_{t} = {1over2}e ^{-u}(u_{xx}+u_ {yy})has an infinite-dimensional space of conservation laws. This stands in contrast to the two-variable case, for which Bryant and Griffiths showed that any equation whose space of conservation laws has dimension 4 or more is locally equivalent to a linear equation, i.e., is linearizable.

  18. U(1)-invariant membranes: The geometric formulation, Abel, and pendulum differential equations

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

    Zheltukhin, A. A.; Fysikum, AlbaNova, Stockholm University, 106 91 Stockholm; NORDITA, Roslagstullsbacken 23, 106 91 Stockholm

    The geometric approach to study the dynamics of U(1)-invariant membranes is developed. The approach reveals an important role of the Abel nonlinear differential equation of the first type with variable coefficients depending on time and one of the membrane extendedness parameters. The general solution of the Abel equation is constructed. Exact solutions of the whole system of membrane equations in the D=5 Minkowski space-time are found and classified. It is shown that if the radial component of the membrane world vector is only time dependent, then the dynamics is described by the pendulum equation.

  19. Quantization of wave equations and hermitian structures in partial differential varieties

    PubMed Central

    Paneitz, S. M.; Segal, I. E.

    1980-01-01

    Sufficiently close to 0, the solution variety of a nonlinear relativistic wave equation—e.g., of the form □ϕ + m2ϕ + gϕp = 0—admits a canonical Lorentz-invariant hermitian structure, uniquely determined by the consideration that the action of the differential scattering transformation in each tangent space be unitary. Similar results apply to linear time-dependent equations or to equations in a curved asymptotically flat space-time. A close relation of the Riemannian structure to the determination of vacuum expectation values is developed and illustrated by an explicit determination of a perturbative 2-point function for the case of interaction arising from curvature. The theory underlying these developments is in part a generalization of that of M. G. Krein and collaborators concerning stability of differential equations in Hilbert space and in part a precise relation between the unitarization of given symplectic linear actions and their full probabilistic quantization. The unique causal structure in the infinite symplectic group is instrumental in these developments. PMID:16592923

  20. Spectral methods for time dependent partial differential equations

    NASA Technical Reports Server (NTRS)

    Gottlieb, D.; Turkel, E.

    1983-01-01

    The theory of spectral methods for time dependent partial differential equations is reviewed. When the domain is periodic Fourier methods are presented while for nonperiodic problems both Chebyshev and Legendre methods are discussed. The theory is presented for both hyperbolic and parabolic systems using both Galerkin and collocation procedures. While most of the review considers problems with constant coefficients the extension to nonlinear problems is also discussed. Some results for problems with shocks are presented.

  1. A partial differential equation model and its reduction to an ordinary differential equation model for prostate tumor growth under intermittent hormone therapy.

    PubMed

    Tao, Youshan; Guo, Qian; Aihara, Kazuyuki

    2014-10-01

    Hormonal therapy with androgen suppression is a common treatment for advanced prostate tumors. The emergence of androgen-independent cells, however, leads to a tumor relapse under a condition of long-term androgen deprivation. Clinical trials suggest that intermittent androgen suppression (IAS) with alternating on- and off-treatment periods can delay the relapse when compared with continuous androgen suppression (CAS). In this paper, we propose a mathematical model for prostate tumor growth under IAS therapy. The model elucidates initial hormone sensitivity, an eventual relapse of a tumor under CAS therapy, and a delay of a relapse under IAS therapy, which are due to the coexistence of androgen-dependent cells, androgen-independent cells resulting from reversible changes by adaptation, and androgen-independent cells resulting from irreversible changes by genetic mutations. The model is formulated as a free boundary problem of partial differential equations that describe the evolution of populations of the abovementioned three types of cells during on-treatment periods and off-treatment periods. Moreover, the model can be transformed into a piecewise linear ordinary differential equation model by introducing three new volume variables, and the study of the resulting model may help to devise optimal IAS schedules.

  2. Radiation Transport Around Axisymmetric Blunt Body Vehicles Using a Modified Differential Approximation

    NASA Technical Reports Server (NTRS)

    Hartung, Lin C.; Hassan, H. A.

    1992-01-01

    A moment method for computing 3-D radiative transport is applied to axisymmetric flows in thermochemical nonequilibrium. Such flows are representative of proposed aerobrake missions. The method uses the P-1 approximation to reduce the governing system of integro-di erential equations to a coupled set of partial di erential equations. A numerical solution method for these equations given actual variations of the radiation properties in thermochemical nonequilibrium blunt body flows is developed. Initial results from the method are shown and compared to tangent slab calculations. The agreement between the transport methods is found to be about 10 percent in the stagnation region, with the difference increasing along the flank of the vehicle.

  3. Approximating a retarded-advanced differential equation that models human phonation

    NASA Astrophysics Data System (ADS)

    Teodoro, M. Filomena

    2017-11-01

    In [1, 2, 3] we have got the numerical solution of a linear mixed type functional differential equation (MTFDE) introduced initially in [4], considering the autonomous and non-autonomous case by collocation, least squares and finite element methods considering B-splines basis set. The present work introduces a numerical scheme using least squares method (LSM) and Gaussian basis functions to solve numerically a nonlinear mixed type equation with symmetric delay and advance which models human phonation. The preliminary results are promising. We obtain an accuracy comparable with the previous results.

  4. Estimation of time- and state-dependent delays and other parameters in functional differential equations

    NASA Technical Reports Server (NTRS)

    Murphy, K. A.

    1988-01-01

    A parameter estimation algorithm is developed which can be used to estimate unknown time- or state-dependent delays and other parameters (e.g., initial condition) appearing within a nonlinear nonautonomous functional differential equation. The original infinite dimensional differential equation is approximated using linear splines, which are allowed to move with the variable delay. The variable delays are approximated using linear splines as well. The approximation scheme produces a system of ordinary differential equations with nice computational properties. The unknown parameters are estimated within the approximating systems by minimizing a least-squares fit-to-data criterion. Convergence theorems are proved for time-dependent delays and state-dependent delays within two classes, which say essentially that fitting the data by using approximations will, in the limit, provide a fit to the data using the original system. Numerical test examples are presented which illustrate the method for all types of delay.

  5. Estimation of time- and state-dependent delays and other parameters in functional differential equations

    NASA Technical Reports Server (NTRS)

    Murphy, K. A.

    1990-01-01

    A parameter estimation algorithm is developed which can be used to estimate unknown time- or state-dependent delays and other parameters (e.g., initial condition) appearing within a nonlinear nonautonomous functional differential equation. The original infinite dimensional differential equation is approximated using linear splines, which are allowed to move with the variable delay. The variable delays are approximated using linear splines as well. The approximation scheme produces a system of ordinary differential equations with nice computational properties. The unknown parameters are estimated within the approximating systems by minimizing a least-squares fit-to-data criterion. Convergence theorems are proved for time-dependent delays and state-dependent delays within two classes, which say essentially that fitting the data by using approximations will, in the limit, provide a fit to the data using the original system. Numerical test examples are presented which illustrate the method for all types of delay.

  6. Comparing Three Methods of Geometrical Approach in Visualizing Differential Equations

    ERIC Educational Resources Information Center

    KarimiFardinpour, Younes; Gooya, Zahra

    2018-01-01

    This paper concerns "planes-coordination" and "long-term-prediction" difficulties. These are specifically the case when students attempt to visualize solution curves of autonomous differential equations for predicting the long-term behavior of various initial conditions. To address these issues, a study was conducted in which…

  7. Alternans promotion in cardiac electrophysiology models by delay differential equations.

    PubMed

    Gomes, Johnny M; Dos Santos, Rodrigo Weber; Cherry, Elizabeth M

    2017-09-01

    Cardiac electrical alternans is a state of alternation between long and short action potentials and is frequently associated with harmful cardiac conditions. Different dynamic mechanisms can give rise to alternans; however, many cardiac models based on ordinary differential equations are not able to reproduce this phenomenon. A previous study showed that alternans can be induced by the introduction of delay differential equations (DDEs) in the formulations of the ion channel gating variables of a canine myocyte model. The present work demonstrates that this technique is not model-specific by successfully promoting alternans using DDEs for five cardiac electrophysiology models that describe different types of myocytes, with varying degrees of complexity. By analyzing results across the different models, we observe two potential requirements for alternans promotion via DDEs for ionic gates: (i) the gate must have a significant influence on the action potential duration and (ii) a delay must significantly impair the gate's recovery between consecutive action potentials.

  8. Polynomial Solutions of Nth Order Non-Homogeneous Differential Equations

    ERIC Educational Resources Information Center

    Levine, Lawrence E.; Maleh, Ray

    2002-01-01

    It was shown by Costa and Levine that the homogeneous differential equation (1-x[superscript N])y([superscript N]) + A[subscript N-1]x[superscript N-1)y([superscript N-1]) + A[subscript N-2]x[superscript N-2])y([superscript N-2]) + ... + A[subscript 1]xy[prime] + A[subscript 0]y = 0 has a finite polynomial solution if and only if [for…

  9. Absolute Value Boundedness, Operator Decomposition, and Stochastic Media and Equations

    NASA Technical Reports Server (NTRS)

    Adomian, G.; Miao, C. C.

    1973-01-01

    The research accomplished during this period is reported. Published abstracts and technical reports are listed. Articles presented include: boundedness of absolute values of generalized Fourier coefficients, propagation in stochastic media, and stationary conditions for stochastic differential equations.

  10. 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.

  11. XMDS2: Fast, scalable simulation of coupled stochastic partial differential equations

    NASA Astrophysics Data System (ADS)

    Dennis, Graham R.; Hope, Joseph J.; Johnsson, Mattias T.

    2013-01-01

    XMDS2 is a cross-platform, GPL-licensed, open source package for numerically integrating initial value problems that range from a single ordinary differential equation up to systems of coupled stochastic partial differential equations. The equations are described in a high-level XML-based script, and the package generates low-level optionally parallelised C++ code for the efficient solution of those equations. It combines the advantages of high-level simulations, namely fast and low-error development, with the speed, portability and scalability of hand-written code. XMDS2 is a complete redesign of the XMDS package, and features support for a much wider problem space while also producing faster code. Program summaryProgram title: XMDS2 Catalogue identifier: AENK_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENK_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 2 No. of lines in distributed program, including test data, etc.: 872490 No. of bytes in distributed program, including test data, etc.: 45522370 Distribution format: tar.gz Programming language: Python and C++. Computer: Any computer with a Unix-like system, a C++ compiler and Python. Operating system: Any Unix-like system; developed under Mac OS X and GNU/Linux. RAM: Problem dependent (roughly 50 bytes per grid point) Classification: 4.3, 6.5. External routines: The external libraries required are problem-dependent. Uses FFTW3 Fourier transforms (used only for FFT-based spectral methods), dSFMT random number generation (used only for stochastic problems), MPI message-passing interface (used only for distributed problems), HDF5, GNU Scientific Library (used only for Bessel-based spectral methods) and a BLAS implementation (used only for non-FFT-based spectral methods). Nature of problem: General coupled initial-value stochastic partial differential equations. Solution method: Spectral method

  12. The ATOMFT integrator - Using Taylor series to solve ordinary differential equations

    NASA Technical Reports Server (NTRS)

    Berryman, Kenneth W.; Stanford, Richard H.; Breckheimer, Peter J.

    1988-01-01

    This paper discusses the application of ATOMFT, an integration package based on Taylor series solution with a sophisticated user interface. ATOMFT has the capabilities to allow the implementation of user defined functions and the solution of stiff and algebraic equations. Detailed examples, including the solutions to several astrodynamics problems, are presented. Comparisons with its predecessor ATOMCC and other modern integrators indicate that ATOMFT is a fast, accurate, and easy method to use to solve many differential equation problems.

  13. 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.

  14. 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.

  15. Ordinary Differential Equation Models for Adoptive Immunotherapy.

    PubMed

    Talkington, Anne; Dantoin, Claudia; Durrett, Rick

    2018-05-01

    Modified T cells that have been engineered to recognize the CD19 surface marker have recently been shown to be very successful at treating acute lymphocytic leukemias. Here, we explore four previous approaches that have used ordinary differential equations to model this type of therapy, compare their properties, and modify the models to address their deficiencies. Although the four models treat the workings of the immune system in slightly different ways, they all predict that adoptive immunotherapy can be successful to move a patient from the large tumor fixed point to an equilibrium with little or no tumor.

  16. Numerical solution of modified differential equations based on symmetry preservation.

    PubMed

    Ozbenli, Ersin; Vedula, Prakash

    2017-12-01

    In this paper, we propose a method to construct invariant finite-difference schemes for solution of partial differential equations (PDEs) via consideration of modified forms of the underlying PDEs. The invariant schemes, which preserve Lie symmetries, are obtained based on the method of equivariant moving frames. While it is often difficult to construct invariant numerical schemes for PDEs due to complicated symmetry groups associated with cumbersome discrete variable transformations, we note that symmetries associated with more convenient transformations can often be obtained by appropriately modifying the original PDEs. In some cases, modifications to the original PDEs are also found to be useful in order to avoid trivial solutions that might arise from particular selections of moving frames. In our proposed method, modified forms of PDEs can be obtained either by addition of perturbation terms to the original PDEs or through defect correction procedures. These additional terms, whose primary purpose is to enable symmetries with more convenient transformations, are then removed from the system by considering moving frames for which these specific terms go to zero. Further, we explore selection of appropriate moving frames that result in improvement in accuracy of invariant numerical schemes based on modified PDEs. The proposed method is tested using the linear advection equation (in one- and two-dimensions) and the inviscid Burgers' equation. Results obtained for these tests cases indicate that numerical schemes derived from the proposed method perform significantly better than existing schemes not only by virtue of improvement in numerical accuracy but also due to preservation of qualitative properties or symmetries of the underlying differential equations.

  17. Dynamics and Control of Constrained Multibody Systems modeled with Maggi's equation: Application to Differential Mobile Robots Part I

    NASA Astrophysics Data System (ADS)

    Amengonu, Yawo H.; Kakad, Yogendra P.

    2014-07-01

    Quasivelocity techniques such as Maggi's and Boltzmann-Hamel's equations eliminate Lagrange multipliers from the beginning as opposed to the Euler-Lagrange method where one has to solve for the n configuration variables and the multipliers as functions of time when there are m nonholonomic constraints. Maggi's equation produces n second-order differential equations of which (n-m) are derived using (n-m) independent quasivelocities and the time derivative of the m kinematic constraints which add the remaining m second order differential equations. This technique is applied to derive the dynamics of a differential mobile robot and a controller which takes into account these dynamics is developed.

  18. Solving differential equations with unknown constitutive relations as recurrent neural networks

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

    Hagge, Tobias J.; Stinis, Panagiotis; Yeung, Enoch H.

    We solve a system of ordinary differential equations with an unknown functional form of a sink (reaction rate) term. We assume that the measurements (time series) of state variables are partially available, and use a recurrent neural network to “learn” the reaction rate from this data. This is achieved by including discretized ordinary differential equations as part of a recurrent neural network training problem. We extend TensorFlow’s recurrent neural network architecture to create a simple but scalable and effective solver for the unknown functions, and apply it to a fedbatch bioreactor simulation problem. Use of techniques from recent deep learningmore » literature enables training of functions with behavior manifesting over thousands of time steps. Our networks are structurally similar to recurrent neural networks, but differ in purpose, and require modified training strategies.« less

  19. An electric-analog simulation of elliptic partial differential equations using finite element theory

    USGS Publications Warehouse

    Franke, O.L.; Pinder, G.F.; Patten, E.P.

    1982-01-01

    Elliptic partial differential equations can be solved using the Galerkin-finite element method to generate the approximating algebraic equations, and an electrical network to solve the resulting matrices. Some element configurations require the use of networks containing negative resistances which, while physically realizable, are more expensive and time-consuming to construct. ?? 1982.

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

    NASA Astrophysics Data System (ADS)

    Al-Islam, Najja Shakir

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

  1. Searching fundamental information in ordinary differential equations. Nondimensionalization technique.

    PubMed

    Sánchez Pérez, J F; Conesa, M; Alhama, I; Alhama, F; Cánovas, M

    2017-01-01

    Classical dimensional analysis and nondimensionalization are assumed to be two similar approaches in the search for dimensionless groups. Both techniques, simplify the study of many problems. The first approach does not need to know the mathematical model, being sufficient a deep understanding of the physical phenomenon involved, while the second one begins with the governing equations and reduces them to their dimensionless form by simple mathematical manipulations. In this work, a formal protocol is proposed for applying the nondimensionalization process to ordinary differential equations, linear or not, leading to dimensionless normalized equations from which the resulting dimensionless groups have two inherent properties: In one hand, they are physically interpreted as balances between counteracting quantities in the problem, and on the other hand, they are of the order of magnitude unity. The solutions provided by nondimensionalization are more precise in every case than those from dimensional analysis, as it is illustrated by the applications studied in this work.

  2. 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.

  3. Advanced-Retarded Differential Equations in Quantum Photonic Systems

    NASA Astrophysics Data System (ADS)

    Alvarez-Rodriguez, Unai; Perez-Leija, Armando; Egusquiza, Iñigo L.; Gräfe, Markus; Sanz, Mikel; Lamata, Lucas; Szameit, Alexander; Solano, Enrique

    2017-02-01

    We propose the realization of photonic circuits whose dynamics is governed by advanced-retarded differential equations. Beyond their mathematical interest, these photonic configurations enable the implementation of quantum feedback and feedforward without requiring any intermediate measurement. We show how this protocol can be applied to implement interesting delay effects in the quantum regime, as well as in the classical limit. Our results elucidate the potential of the protocol as a promising route towards integrated quantum control systems on a chip.

  4. Advanced-Retarded Differential Equations in Quantum Photonic Systems

    PubMed Central

    Alvarez-Rodriguez, Unai; Perez-Leija, Armando; Egusquiza, Iñigo L.; Gräfe, Markus; Sanz, Mikel; Lamata, Lucas; Szameit, Alexander; Solano, Enrique

    2017-01-01

    We propose the realization of photonic circuits whose dynamics is governed by advanced-retarded differential equations. Beyond their mathematical interest, these photonic configurations enable the implementation of quantum feedback and feedforward without requiring any intermediate measurement. We show how this protocol can be applied to implement interesting delay effects in the quantum regime, as well as in the classical limit. Our results elucidate the potential of the protocol as a promising route towards integrated quantum control systems on a chip. PMID:28230090

  5. Error estimation in the neural network solution of ordinary differential equations.

    PubMed

    Filici, Cristian

    2010-06-01

    In this article a method of error estimation for the neural approximation of the solution of an Ordinary Differential Equation is presented. Some examples of the application of the method support the theory presented. Copyright 2010. Published by Elsevier Ltd.

  6. Numerical solution of nonlinear partial differential equations of mixed type. [finite difference approximation

    NASA Technical Reports Server (NTRS)

    Jameson, A.

    1976-01-01

    A review is presented of some recently developed numerical methods for the solution of nonlinear equations of mixed type. The methods considered use finite difference approximations to the differential equation. Central difference formulas are employed in the subsonic zone and upwind difference formulas are used in the supersonic zone. The relaxation method for the small disturbance equation is discussed and a description is given of difference schemes for the potential flow equation in quasi-linear form. Attention is also given to difference schemes for the potential flow equation in conservation form, the analysis of relaxation schemes by the time dependent analogy, the accelerated iterative method, and three-dimensional calculations.

  7. 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…

  8. Building Context with Tumor Growth Modeling Projects in Differential Equations

    ERIC Educational Resources Information Center

    Beier, Julie C.; Gevertz, Jana L.; Howard, Keith E.

    2015-01-01

    The use of modeling projects serves to integrate, reinforce, and extend student knowledge. Here we present two projects related to tumor growth appropriate for a first course in differential equations. They illustrate the use of problem-based learning to reinforce and extend course content via a writing or research experience. Here we discuss…

  9. 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

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

    NASA Astrophysics Data System (ADS)

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

    2007-04-01

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

  11. Accelerating numerical solution of stochastic differential equations with CUDA

    NASA Astrophysics Data System (ADS)

    Januszewski, M.; Kostur, M.

    2010-01-01

    Numerical integration of stochastic differential equations is commonly used in many branches of science. In this paper we present how to accelerate this kind of numerical calculations with popular NVIDIA Graphics Processing Units using the CUDA programming environment. We address general aspects of numerical programming on stream processors and illustrate them by two examples: the noisy phase dynamics in a Josephson junction and the noisy Kuramoto model. In presented cases the measured speedup can be as high as 675× compared to a typical CPU, which corresponds to several billion integration steps per second. This means that calculations which took weeks can now be completed in less than one hour. This brings stochastic simulation to a completely new level, opening for research a whole new range of problems which can now be solved interactively. Program summaryProgram title: SDE Catalogue identifier: AEFG_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEFG_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Gnu GPL v3 No. of lines in distributed program, including test data, etc.: 978 No. of bytes in distributed program, including test data, etc.: 5905 Distribution format: tar.gz Programming language: CUDA C Computer: any system with a CUDA-compatible GPU Operating system: Linux RAM: 64 MB of GPU memory Classification: 4.3 External routines: The program requires the NVIDIA CUDA Toolkit Version 2.0 or newer and the GNU Scientific Library v1.0 or newer. Optionally gnuplot is recommended for quick visualization of the results. Nature of problem: Direct numerical integration of stochastic differential equations is a computationally intensive problem, due to the necessity of calculating multiple independent realizations of the system. We exploit the inherent parallelism of this problem and perform the calculations on GPUs using the CUDA programming environment. The GPU's ability to execute

  12. Analytical approximate solutions for a general class of nonlinear delay differential equations.

    PubMed

    Căruntu, Bogdan; Bota, Constantin

    2014-01-01

    We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.

  13. Stability of an abstract system of coupled hyperbolic and parabolic equations

    NASA Astrophysics Data System (ADS)

    Hao, Jianghao; Liu, Zhuangyi

    2013-08-01

    In this paper, we provide a complete stability analysis for an abstract system of coupled hyperbolic and parabolic equations = -Au + γ A^{α} θ, quad θ_t = -γ A^{α}u_t - kA^{β}θ, u(0) = u_0, quad u_t(0) = v_0, quad θ(0) = θ_0 where A is a self-adjoint, positive definite operator on a Hilbert space H. For {(α,β) in [0,1] × [0,1]} , the region of exponential stability had been identified in Ammar-Khodja et al. (ESAIM Control Optim Calc Var 4:577-593,1999). Our contribution is to show that the rest of the region can be classified as region of polynomial stability and region of instability. Moreover, we obtain the optimality of the order of polynomial stability.

  14. 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.

  15. 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.

  16. On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order

    NASA Astrophysics Data System (ADS)

    Tu, Jin; Yi, Cai-Feng

    2008-04-01

    In this paper, the authors investigate the growth of solutions of a class of higher order linear differential equationsf(k)+Ak-1f(k-1)+...+A0f=0 when most coefficients in the above equations have the same order with each other, and obtain some results which improve previous results due to K.H. Kwon [K.H. Kwon, Nonexistence of finite order solutions of certain second order linear differential equations, Kodai Math. J. 19 (1996) 378-387] and ZE-X. Chen [Z.-X. Chen, The growth of solutions of the differential equation f''+e-zf'+Q(z)f=0, Sci. China Ser. A 31 (2001) 775-784 (in Chinese); ZE-X. Chen, On the hyper order of solutions of higher order differential equations, Chinese Ann. Math. Ser. B 24 (2003) 501-508 (in Chinese); Z.-X. Chen, On the growth of solutions of a class of higher order differential equations, Acta Math. Sci. Ser. B 24 (2004) 52-60 (in Chinese); Z.-X. Chen, C.-C. Yang, Quantitative estimations on the zeros and growth of entire solutions of linear differential equations, Complex Var. 42 (2000) 119-133].

  17. 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.

  18. Partial differential equations constrained combinatorial optimization on an adiabatic quantum computer

    NASA Astrophysics Data System (ADS)

    Chandra, Rishabh

    Partial differential equation-constrained combinatorial optimization (PDECCO) problems are a mixture of continuous and discrete optimization problems. PDECCO problems have discrete controls, but since the partial differential equations (PDE) are continuous, the optimization space is continuous as well. Such problems have several applications, such as gas/water network optimization, traffic optimization, micro-chip cooling optimization, etc. Currently, no efficient classical algorithm which guarantees a global minimum for PDECCO problems exists. A new mapping has been developed that transforms PDECCO problem, which only have linear PDEs as constraints, into quadratic unconstrained binary optimization (QUBO) problems that can be solved using an adiabatic quantum optimizer (AQO). The mapping is efficient, it scales polynomially with the size of the PDECCO problem, requires only one PDE solve to form the QUBO problem, and if the QUBO problem is solved correctly and efficiently on an AQO, guarantees a global optimal solution for the original PDECCO problem.

  19. Electron dynamics inside a vacuum tube diode through linear differential equations

    NASA Astrophysics Data System (ADS)

    González, Gabriel; Orozco, Fco. Javier González; Orozco

    2014-04-01

    In this paper we analyze the motion of charged particles in a vacuum tube diode by solving linear differential equations. Our analysis is based on expressing the volume charge density as a function of the current density and coordinates only, i.e. ρ=ρ(J,z), while in the usual scheme the volume charge density is expressed as a function of the current density and electrostatic potential, i.e. ρ=ρ(J,V). We show that, in the case of slow varying charge density, the space-charge-limited current is reduced up to 50%. Our approach gives the well-known behavior of the classical current density proportional to the three-halves power of the bias potential and inversely proportional to the square of the gap distance between electrodes, and does not require the solution of the nonlinear differential equation normally associated with the Child-Langmuir formulation.

  20. Solving Partial Differential Equations in a data-driven multiprocessor environment

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

    Gaudiot, J.L.; Lin, C.M.; Hosseiniyar, M.

    1988-12-31

    Partial differential equations can be found in a host of engineering and scientific problems. The emergence of new parallel architectures has spurred research in the definition of parallel PDE solvers. Concurrently, highly programmable systems such as data-how architectures have been proposed for the exploitation of large scale parallelism. The implementation of some Partial Differential Equation solvers (such as the Jacobi method) on a tagged token data-flow graph is demonstrated here. Asynchronous methods (chaotic relaxation) are studied and new scheduling approaches (the Token No-Labeling scheme) are introduced in order to support the implementation of the asychronous methods in a data-driven environment.more » New high-level data-flow language program constructs are introduced in order to handle chaotic operations. Finally, the performance of the program graphs is demonstrated by a deterministic simulation of a message passing data-flow multiprocessor. An analysis of the overhead in the data-flow graphs is undertaken to demonstrate the limits of parallel operations in dataflow PDE program graphs.« less

  1. Using trees to compute approximate solutions to ordinary differential equations exactly

    NASA Technical Reports Server (NTRS)

    Grossman, Robert

    1991-01-01

    Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.

  2. Simulating Chemical Kinetics Without Differential Equations: A Quantitative Theory Based on Chemical Pathways.

    PubMed

    Bai, Shirong; Skodje, Rex T

    2017-08-17

    A new approach is presented for simulating the time-evolution of chemically reactive systems. This method provides an alternative to conventional modeling of mass-action kinetics that involves solving differential equations for the species concentrations. The method presented here avoids the need to solve the rate equations by switching to a representation based on chemical pathways. In the Sum Over Histories Representation (or SOHR) method, any time-dependent kinetic observable, such as concentration, is written as a linear combination of probabilities for chemical pathways leading to a desired outcome. In this work, an iterative method is introduced that allows the time-dependent pathway probabilities to be generated from a knowledge of the elementary rate coefficients, thus avoiding the pitfalls involved in solving the differential equations of kinetics. The method is successfully applied to the model Lotka-Volterra system and to a realistic H 2 combustion model.

  3. Model identification using stochastic differential equation grey-box models in diabetes.

    PubMed

    Duun-Henriksen, Anne Katrine; Schmidt, Signe; Røge, Rikke Meldgaard; Møller, Jonas Bech; Nørgaard, Kirsten; Jørgensen, John Bagterp; Madsen, Henrik

    2013-03-01

    The acceptance of virtual preclinical testing of control algorithms is growing and thus also the need for robust and reliable models. Models based on ordinary differential equations (ODEs) can rarely be validated with standard statistical tools. Stochastic differential equations (SDEs) offer the possibility of building models that can be validated statistically and that are capable of predicting not only a realistic trajectory, but also the uncertainty of the prediction. In an SDE, the prediction error is split into two noise terms. This separation ensures that the errors are uncorrelated and provides the possibility to pinpoint model deficiencies. An identifiable model of the glucoregulatory system in a type 1 diabetes mellitus (T1DM) patient is used as the basis for development of a stochastic-differential-equation-based grey-box model (SDE-GB). The parameters are estimated on clinical data from four T1DM patients. The optimal SDE-GB is determined from likelihood-ratio tests. Finally, parameter tracking is used to track the variation in the "time to peak of meal response" parameter. We found that the transformation of the ODE model into an SDE-GB resulted in a significant improvement in the prediction and uncorrelated errors. Tracking of the "peak time of meal absorption" parameter showed that the absorption rate varied according to meal type. This study shows the potential of using SDE-GBs in diabetes modeling. Improved model predictions were obtained due to the separation of the prediction error. SDE-GBs offer a solid framework for using statistical tools for model validation and model development. © 2013 Diabetes Technology Society.

  4. 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)

  5. Asymptotic and spectral analysis of the gyrokinetic-waterbag integro-differential operator in toroidal geometry

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

    Besse, Nicolas, E-mail: Nicolas.Besse@oca.eu; Institut Jean Lamour, UMR CNRS/UL 7198, Université de Lorraine, BP 70239 54506 Vandoeuvre-lès-Nancy Cedex; Coulette, David, E-mail: David.Coulette@ipcms.unistra.fr

    2016-08-15

    Achieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov–Poisson and Vlasov–Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to themore » VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phase-space. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magnetic-field-induced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested one-dimensional problems: an integral equation in the poloidal variable, followed by a one-dimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finite-dimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, “Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry” (submitted)] and were found to be surprisingly close to those for the original

  6. Asymptotic and spectral analysis of the gyrokinetic-waterbag integro-differential operator in toroidal geometry

    NASA Astrophysics Data System (ADS)

    Besse, Nicolas; Coulette, David

    2016-08-01

    Achieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov-Poisson and Vlasov-Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to the VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phase-space. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magnetic-field-induced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested one-dimensional problems: an integral equation in the poloidal variable, followed by a one-dimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finite-dimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, "Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry" (submitted)] and were found to be surprisingly close to those for the original gyrokinetic

  7. 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.

  8. Solving Second-Order Ordinary Differential Equations without Using Complex Numbers

    ERIC Educational Resources Information Center

    Kougias, Ioannis E.

    2009-01-01

    Ordinary differential equations (ODEs) is a subject with a wide range of applications and the need of introducing it to students often arises in the last year of high school, as well as in the early stages of tertiary education. The usual methods of solving second-order ODEs with constant coefficients, among others, rely upon the use of complex…

  9. Control of functional differential equations with function space boundary conditions.

    NASA Technical Reports Server (NTRS)

    Banks, H. T.

    1972-01-01

    The results of various authors dealing with problems involving functional differential equations with terminal conditions in function space are reviewed. The review includes not only very recent results, but also some little known results of Soviet mathematicians prior to 1970. Particular attention is given to results concerning controllability, existence of optimal controls, and necessary and sufficient conditions for optimality.

  10. Solution of partial differential equations on vector and parallel computers

    NASA Technical Reports Server (NTRS)

    Ortega, J. M.; Voigt, R. G.

    1985-01-01

    The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed.

  11. Laplace and Z Transform Solutions of Differential and Difference Equations With the HP-41C.

    ERIC Educational Resources Information Center

    Harden, Richard C.; Simons, Fred O., Jr.

    1983-01-01

    A previously developed program for the HP-41C programmable calculator is extended to handle models of differential and difference equations with multiple eigenvalues. How to obtain difference equation solutions via the Z transform is described. (MNS)

  12. 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.

  13. A note on a corrector formula for the numerical solution of ordinary differential equations

    NASA Technical Reports Server (NTRS)

    Chien, Y.-C.; Agrawal, K. M.

    1979-01-01

    A new corrector formula for predictor-corrector methods for numerical solutions of ordinary differential equations is presented. Two considerations for choosing corrector formulas are given: (1) the coefficient in the error term and (2) its stability properties. The graph of the roots of an equation plotted against its stability region, of different values, is presented along with the tables that correspond to various corrector equations, including Hamming's and Milne and Reynolds'.

  14. A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

    NASA Astrophysics Data System (ADS)

    Camporesi, Roberto

    2016-01-01

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

  15. Variation of Parameters in Differential Equations (A Variation in Making Sense of Variation of Parameters)

    ERIC Educational Resources Information Center

    Quinn, Terry; Rai, Sanjay

    2012-01-01

    The method of variation of parameters can be found in most undergraduate textbooks on differential equations. The method leads to solutions of the non-homogeneous equation of the form y = u[subscript 1]y[subscript 1] + u[subscript 2]y[subscript 2], a sum of function products using solutions to the homogeneous equation y[subscript 1] and…

  16. Student Perceptions of Writing Projects in a University Differential-Equations Course

    ERIC Educational Resources Information Center

    Latulippe, Christine; Latulippe, Joe

    2014-01-01

    This qualitative study surveyed 102 differential-equations students in order to investigate how students participating in writing projects in university-level mathematics courses perceive the benefits of writing in the mathematics classroom. Based on previous literature on writing in mathematics, students were asked specifically about the benefits…

  17. 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.

  18. Direct application of Padé approximant for solving nonlinear differential equations.

    PubMed

    Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Garcia-Gervacio, Jose Luis; Huerta-Chua, Jesus; Morales-Mendoza, Luis Javier; Gonzalez-Lee, Mario

    2014-01-01

    This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant. 34L30.

  19. 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.

  20. A neutral functional differential equation of Lurie type. [on asymptotic stability of feedback control

    NASA Technical Reports Server (NTRS)

    Chukwu, E. N.

    1980-01-01

    The problem of Lurie is posed for systems described by a functional differential equation of neutral type. Sufficient conditions are obtained for absolute stability for the controlled system if it is assumed that the uncontrolled plant equation is uniformly asymptotically stable. Both the direct and indirect control cases are treated.

  1. Fragmentation Under the Scaling Symmetry and Turbulent Cascade with Intermittency

    DTIC Science & Technology

    2003-12-01

    distribution. In Gorokhovski & Saveliev (2003), the fragmentation under the scaling symmetry has been reviewed as a continuous evolution process with... Saveliev (2003). Second, the paper shows a new application of the fragmentation theory under the scale invariance. This application concerns the tur...following integro-differential equation (see, for example, Gorokhovski & Saveliev (2003)): of T- = (1+ - 1W (2.1) f University of Rouen 198 M

  2. On a perturbed Sparre Andersen risk model with multi-layer dividend strategy

    NASA Astrophysics Data System (ADS)

    Yang, Hu; Zhang, Zhimin

    2009-10-01

    In this paper, we consider a perturbed Sparre Andersen risk model, in which the inter-claim times are generalized Erlang(n) distributed. Under the multi-layer dividend strategy, piece-wise integro-differential equations for the discounted penalty functions are derived, and a recursive approach is applied to express the solutions. A numerical example to calculate the ruin probabilities is given to illustrate the solution procedure.

  3. Existence and uniqueness of solution for a class of stochastic differential equations.

    PubMed

    Cao, Junfei; Huang, Zaitang; Zeng, Caibin

    2013-01-01

    A class of stochastic differential equations given by dx(t) = f(x(t))dt + g(x(t))dW(t),  x(t 0) = x 0,  t 0 ≤ t ≤ T < +∞, are investigated. Upon making some suitable assumptions, the existence and uniqueness of solution for the equations are obtained. Moreover, the existence and uniqueness of solution for stochastic Lorenz system, which is illustrated by example, are in good agreement with the theoretical analysis.

  4. 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.

  5. A note on the generation of phase plane plots on a digital computer. [for solution of nonlinear differential equations

    NASA Technical Reports Server (NTRS)

    Simon, M. K.

    1980-01-01

    A technique is presented for generating phase plane plots on a digital computer which circumvents the difficulties associated with more traditional methods of numerical solving nonlinear differential equations. In particular, the nonlinear differential equation of operation is formulated.

  6. Control problem for a system of linear loaded differential equations

    NASA Astrophysics Data System (ADS)

    Barseghyan, V. R.; Barseghyan, T. V.

    2018-04-01

    The problem of control and optimal control for a system of linear loaded differential equations is considered. Necessary and sufficient conditions for complete controllability and conditions for the existence of a program control and the corresponding motion are formulated. The explicit form of control action for the control problem is constructed and a method for solving the problem of optimal control is proposed.

  7. Homotopy perturbation method with Laplace Transform (LT-HPM) for solving Lane-Emden type differential equations (LETDEs).

    PubMed

    Tripathi, Rajnee; Mishra, Hradyesh Kumar

    2016-01-01

    In this communication, we describe the Homotopy Perturbation Method with Laplace Transform (LT-HPM), which is used to solve the Lane-Emden type differential equations. It's very difficult to solve numerically the Lane-Emden types of the differential equation. Here we implemented this method for two linear homogeneous, two linear nonhomogeneous, and four nonlinear homogeneous Lane-Emden type differential equations and use their appropriate comparisons with exact solutions. In the current study, some examples are better than other existing methods with their nearer results in the form of power series. The Laplace transform used to accelerate the convergence of power series and the results are shown in the tables and graphs which have good agreement with the other existing method in the literature. The results show that LT-HPM is very effective and easy to implement.

  8. Introducing Differential Equations Students to the Fredholm Alternative--In Staggered Doses

    ERIC Educational Resources Information Center

    Savoye, Philippe

    2011-01-01

    The development, in an introductory differential equations course, of boundary value problems in parallel with initial value problems and the Fredholm Alternative. Examples are provided of pairs of homogeneous and nonhomogeneous boundary value problems for which existence and uniqueness issues are considered jointly. How this heightens students'…

  9. 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.

  10. Steady, Oscillatory, and Unsteady Subsonic and Supersonic Aerodynamics, production version (SOUSSA-P 1.1). Volume 1: Theoretical manual. [Green function

    NASA Technical Reports Server (NTRS)

    Morino, L.

    1980-01-01

    Recent developments of the Green's function method and the computer program SOUSSA (Steady, Oscillatory, and Unsteady Subsonic and Supersonic Aerodynamics) are reviewed and summarized. Applying the Green's function method to the fully unsteady (transient) potential equation yields an integro-differential-delay equation. With spatial discretization by the finite-element method, this equation is approximated by a set of differential-delay equations in time. Time solution by Laplace transform yields a matrix relating the velocity potential to the normal wash. Premultiplying and postmultiplying by the matrices relating generalized forces to the potential and the normal wash to the generalized coordinates one obtains the matrix of the generalized aerodynamic forces. The frequency and mode-shape dependence of this matrix makes the program SOUSSA useful for multiple frequency and repeated mode-shape evaluations.

  11. Modeling of glow discharge in a gas flow

    NASA Astrophysics Data System (ADS)

    Galeev, I. G.; Asadullin, T. Ya

    2017-11-01

    The discharge plasma of positive column in electronegative gas flow has been described by two-dimensional integro-differential system of equations written in the approximation of “narrow channel”. An efficient algorithm of solving this system of equations is suggested. In this work an implicit method is used of solution with selection gradient of the electric field along the channel. The simulation of discharge characteristics was conducted under various boundary conditions at the inlet of the discharge area and with various full discharge currents.

  12. Using the Homotopy Method to Find Periodic Solutions of Forced Nonlinear Differential Equations

    ERIC Educational Resources Information Center

    Fay, Temple H.; Lott, P. Aaron

    2002-01-01

    This paper discusses a result of Li and Shen which proves the existence of a unique periodic solution for the differential equation x[dots above] + kx[dot above] + g(x,t) = [epsilon](t) where k is a constant; g is continuous, continuously differentiable with respect to x , and is periodic of period P in the variable t; [epsilon](t) is continuous…

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

    NASA Astrophysics Data System (ADS)

    Arshad, Muhammad; Lu, Dianchen; Wang, Jun

    2017-07-01

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

  14. a Non-Overlapping Discretization Method for Partial Differential Equations

    NASA Astrophysics Data System (ADS)

    Rosas-Medina, A.; Herrera, I.

    2013-05-01

    Mathematical models of many systems of interest, including very important continuous systems of Engineering and Science, lead to a great variety of partial differential equations whose solution methods are based on the computational processing of large-scale algebraic systems. Furthermore, the incredible expansion experienced by the existing computational hardware and software has made amenable to effective treatment problems of an ever increasing diversity and complexity, posed by engineering and scientific applications. The emergence of parallel computing prompted on the part of the computational-modeling community a continued and systematic effort with the purpose of harnessing it for the endeavor of solving boundary-value problems (BVPs) of partial differential equations. Very early after such an effort began, it was recognized that domain decomposition methods (DDM) were the most effective technique for applying parallel computing to the solution of partial differential equations, since such an approach drastically simplifies the coordination of the many processors that carry out the different tasks and also reduces very much the requirements of information-transmission between them. Ideally, DDMs intend producing algorithms that fulfill the DDM-paradigm; i.e., such that "the global solution is obtained by solving local problems defined separately in each subdomain of the coarse-mesh -or domain-decomposition-". Stated in a simplistic manner, the basic idea is that, when the DDM-paradigm is satisfied, full parallelization can be achieved by assigning each subdomain to a different processor. When intensive DDM research began much attention was given to overlapping DDMs, but soon after attention shifted to non-overlapping DDMs. This evolution seems natural when the DDM-paradigm is taken into account: it is easier to uncouple the local problems when the subdomains are separated. However, an important limitation of non-overlapping domain decompositions, as that

  15. Xcas as a Programming Environment for Stability Conditions for a Class of Differential Equation Models in Economics

    NASA Astrophysics Data System (ADS)

    Halkos, George E.; Tsilika, Kyriaki D.

    2011-09-01

    In this paper we examine the property of asymptotic stability in several dynamic economic systems, modeled in ordinary differential equation formulations of time parameter t. Asymptotic stability ensures intertemporal equilibrium for the economic quantity the solution stands for, regardless of what the initial conditions happen to be. Existence of economic equilibrium in continuous time models is checked via a Symbolic language, the Xcas program editor. Using stability theorems of differential equations as background a brief overview of symbolic capabilities of free software Xcas is given. We present computational experience with a programming style for stability results of ordinary linear and nonlinear differential equations. Numerical experiments on traditional applications of economic dynamics exhibit the simplicity clarity and brevity of input and output of our computer codes.

  16. Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells.

    PubMed

    Sundnes, J; Lines, G T; Tveito, A

    2001-08-01

    The contraction of the heart is preceded and caused by a cellular electro-chemical reaction, causing an electrical field to be generated. Performing realistic computer simulations of this process involves solving a set of partial differential equations, as well as a large number of ordinary differential equations (ODEs) characterizing the reactive behavior of the cardiac tissue. Experiments have shown that the solution of the ODEs contribute significantly to the total work of a simulation, and there is thus a strong need to utilize efficient solution methods for this part of the problem. This paper presents how an efficient implicit Runge-Kutta method may be adapted to solve a complicated cardiac cell model consisting of 31 ODEs, and how this solver may be coupled to a set of PDE solvers to provide complete simulations of the electrical activity.

  17. The geometric approach to sets of ordinary differential equations and Hamiltonian dynamics

    NASA Technical Reports Server (NTRS)

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

    1975-01-01

    The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field V in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along V: scalar fields, forms, vectors, and integrals over subspaces. It is shown that to any field V can be associated a Hamiltonian structure of forms if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added. Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton's variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms must include the characteristics of the set.

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

    ERIC Educational Resources Information Center

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

    2008-01-01

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

  19. 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.

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

    NASA Technical Reports Server (NTRS)

    Klein, Vladislav; Noderer, Keith D.

    1994-01-01

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

  1. Lines of Eigenvectors and Solutions to Systems of Linear Differential Equations

    ERIC Educational Resources Information Center

    Rasmussen, Chris; Keynes, Michael

    2003-01-01

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

  2. Introduction of the Notion of Differential Equations by Modelling Based Teaching

    ERIC Educational Resources Information Center

    Budinski, Natalija; Takaci, Djurdjica

    2011-01-01

    This paper proposes modelling based learning as a tool for learning and teaching mathematics. The example of modelling real world problems leading to the exponential function as the solution of differential equations is described, as well as the observations about students' activities during the process. The students were acquainted with the…

  3. Algorithm refinement for stochastic partial differential equations: II. Correlated systems

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

    Alexander, Francis J.; Garcia, Alejandro L.; Tartakovsky, Daniel M.

    2005-08-10

    We analyze a hybrid particle/continuum algorithm for a hydrodynamic system with long ranged correlations. Specifically, we consider the so-called train model for viscous transport in gases, which is based on a generalization of the random walk process for the diffusion of momentum. This discrete model is coupled with its continuous counterpart, given by a pair of stochastic partial differential equations. At the interface between the particle and continuum computations the coupling is by flux matching, giving exact mass and momentum conservation. This methodology is an extension of our stochastic Algorithm Refinement (AR) hybrid for simple diffusion [F. Alexander, A. Garcia,more » D. Tartakovsky, Algorithm refinement for stochastic partial differential equations: I. Linear diffusion, J. Comput. Phys. 182 (2002) 47-66]. Results from a variety of numerical experiments are presented for steady-state scenarios. In all cases the mean and variance of density and velocity are captured correctly by the stochastic hybrid algorithm. For a non-stochastic version (i.e., using only deterministic continuum fluxes) the long-range correlations of velocity fluctuations are qualitatively preserved but at reduced magnitude.« less

  4. Unsteady three-dimensional marginal separation caused by surface-mounted obstacles and/or local suction

    NASA Astrophysics Data System (ADS)

    Braun, Stefan; Kluwick, Alfred

    2004-09-01

    Earlier investigations of steady two-dimensional marginally separated laminar boundary layers have shown that the non-dimensional wall shear (or equivalently the negative non-dimensional perturbation displacement thickness) is governed by a nonlinear integro-differential equation. This equation contains a single controlling parameter Gamma characterizing, for example, the angle of attack of a slender airfoil and has the important property that (real) solutions exist up to a critical value Gamma_c of Gamma only. Here we investigate three-dimensional unsteady perturbations of an incompressible steady two-dimensional marginally separated laminar boundary layer with special emphasis on the flow behaviour near Gamma_c. Specifically, it is shown that the integro differential equation which governs these disturbances if Gamma_c {-} Gamma {=} O(1) reduces to a nonlinear partial differential equation known as the Fisher equation as Gamma approaches the critical value Gamma_c. This in turn leads to a significant simplification of the problem allowing, among other things, a systematic study of devices used in boundary-layer control and an analytical investigation of the conditions leading to the formation of finite-time singularities which have been observed in earlier numerical studies of unsteady two-dimensional and three-dimensional flows in the vicinity of a line of symmetry. Also, it is found that it is possible to construct exact solutions which describe waves of constant form travelling in the spanwise direction. These waves may contain singularities which can be interpreted as vortex sheets. The existence of these solutions strongly suggests that solutions of the Fisher equation which lead to finite-time blow-up may be extended beyond the blow-up time, thereby generating moving singularities which can be interpreted as vortical structures qualitatively similar to those emerging in direct numerical simulations of near critical (i.e. transitional) laminar separation

  5. Conservation properties of numerical integration methods for systems of ordinary differential equations

    NASA Technical Reports Server (NTRS)

    Rosenbaum, J. S.

    1976-01-01

    If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.

  6. Discretization and Numerical Solution of a Plane Problem in the Mechanics of Interfacial Cracks

    NASA Astrophysics Data System (ADS)

    Khoroshun, L. P.

    2017-01-01

    The Fourier transform is used to reduce the linear plane problem of the tension of a body with an interfacial crack to a system of dual equations for the transformed stresses and, then, to a system of integro-differential equations for the difference of displacements of the crack faces. After discretization, this latter system transforms into a system of algebraic equations for displacements of the crack faces. The effect of the bielastic constant and the number of discretization points on the half-length of the crack faces and the distribution of stresses at the interface is studied

  7. 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.

  8. 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.

  9. Extending the Constant Coefficient Solution Technique to Variable Coefficient Ordinary Differential Equations

    ERIC Educational Resources Information Center

    Mohammed, Ahmed; Zeleke, Aklilu

    2015-01-01

    We introduce a class of second-order ordinary differential equations (ODEs) with variable coefficients whose closed-form solutions can be obtained by the same method used to solve ODEs with constant coefficients. General solutions for the homogeneous case are discussed.

  10. Exact differential equation for the density and ionization energy of a many-particle system

    NASA Technical Reports Server (NTRS)

    Levy, M.; Perdew, J. P.; Sahni, V.

    1984-01-01

    The present investigation is concerned with relations studied by Hohenberg and Kohn (1964) and Kohn and Sham (1965). The properties of a ground-state many-electron system are determined by the electron density. The correct differential equation for the density, as dictated by density-functional theory, is presented. It is found that the ground-state density n of a many-electron system obeys a Schroedinger-like differential equation which may be solved by standard Kohn-Sham programs. Results are connected to the traditional exact Kohn-Sham theory. It is pointed out that the results of the current investigations are readily extended to spin-density functional theory.

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

    ERIC Educational Resources Information Center

    Camporesi, Roberto

    2011-01-01

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

  12. Coupled latent differential equation with moderators: simulation and application.

    PubMed

    Hu, Yueqin; Boker, Steve; Neale, Michael; Klump, Kelly L

    2014-03-01

    Latent differential equations (LDE) use differential equations to analyze time series data. Because of the recent development of this technique, some issues critical to running an LDE model remain. In this article, the authors provide solutions to some of these issues and recommend a step-by-step procedure demonstrated on a set of empirical data, which models the interaction between ovarian hormone cycles and emotional eating. Results indicated that emotional eating is self-regulated. For instance, when people do more emotional eating than normal, they will subsequently tend to decrease their emotional eating behavior. In addition, a sudden increase will produce a stronger tendency to decrease than will a slow increase. We also found that emotional eating is coupled with the cycle of the ovarian hormone estradiol, and the peak of emotional eating occurs after the peak of estradiol. The self-reported average level of negative affect moderates the frequency of eating regulation and the coupling strength between eating and estradiol. Thus, people with a higher average level of negative affect tend to fluctuate faster in emotional eating, and their eating behavior is more strongly coupled with the hormone estradiol. Permutation tests on these empirical data supported the reliability of using LDE models to detect self-regulation and a coupling effect between two regulatory behaviors. (c) 2014 APA, all rights reserved.

  13. Model Validation for a Noninvasive Arterial Stenosis Detection Problem

    DTIC Science & Technology

    2013-06-09

    simplicity in the final model, at the expense of some minor notational confusion at the current stage . Note that (9) implies that ur(r, t) = − A rmax...which was an improvement computationally since it led to purely differential equations in the model rather than inclusion of an integro-differential...values (c.f. (15)), this gives us θ0gls = (3.6532, 0.6990, 4.4472,−1.2218,−3.7696,−2) T . 4.1.2 Residuals We will also include residual plots to assist

  14. Introduction to Adaptive Methods for Differential Equations

    NASA Astrophysics Data System (ADS)

    Eriksson, Kenneth; Estep, Don; Hansbo, Peter; Johnson, Claes

    Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method (Gottfried Wilhelm von Leibniz, 1646-1719).When, several years ago, I saw for the first time an instrument which, when carried, automatically records the number of steps taken by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only addition and subtraction, but also multiplication and division, could be accomplished by a suitably arranged machine easily, promptly and with sure results. For it is unworthy of excellent men to lose hours like slaves in the labour of calculations, which could safely be left to anyone else if the machine was used. And now that we may give final praise to the machine, we may say that it will be desirable to all who are engaged in computations which, as is well known, are the managers of financial affairs, the administrators of others estates, merchants, surveyors, navigators, astronomers, and those connected with any of the crafts that use mathematics (Leibniz).

  15. Matrix Solution of Coupled Differential Equations and Looped Car Following Models

    ERIC Educational Resources Information Center

    McCartney, Mark

    2008-01-01

    A simple mathematical model for the behaviour of how vehicles follow each other along a looped stretch of road is described. The resulting coupled first order differential equations are solved using appropriate matrix techniques and the physical significance of the model is discussed. A number possible classroom exercises are suggested to help…

  16. 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.

  17. Reflecting Solutions of High Order Elliptic Differential Equations in Two Independent Variables Across Analytic Arcs. Ph.D. Thesis

    NASA Technical Reports Server (NTRS)

    Carleton, O.

    1972-01-01

    Consideration is given specifically to sixth order elliptic partial differential equations in two independent real variables x, y such that the coefficients of the highest order terms are real constants. It is assumed that the differential operator has distinct characteristics and that it can be factored as a product of second order operators. By analytically continuing into the complex domain and using the complex characteristic coordinates of the differential equation, it is shown that its solutions, u, may be reflected across analytic arcs on which u satisfies certain analytic boundary conditions. Moreover, a method is given whereby one can determine a region into which the solution is extensible. It is seen that this region of reflection is dependent on the original domain of difinition of the solution, the arc and the coefficients of the highest order terms of the equation and not on any sufficiently small quantities; i.e., the reflection is global in nature. The method employed may be applied to similar differential equations of order 2n.

  18. 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.

  19. Undergraduate Students' Perceptions of Collaborative Learning in a Differential Equations Mathematics Course

    ERIC Educational Resources Information Center

    Hajra, Sayonita Ghosh; Das, Ujjaini

    2015-01-01

    This paper uses collaborative learning strategies to examine students' perceptions in a differential equations mathematics course. Students' perceptions were analyzed using three collaborative learning strategies including collaborative activity, group-quiz and online discussion. The study results show that students identified both strengths and…

  20. Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras.

    PubMed

    Gainetdinova, A A; Gazizov, R K

    2017-01-01

    We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.