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Journal articles on the topic 'Differential equations - Numerical methods'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Jankowski, Tadeusz, and Marian Kwapisz. "Convergence of numerical methods for systems of neutral functional-differential-algebraic equations." Applications of Mathematics 40, no. 6 (1995): 457–72. http://dx.doi.org/10.21136/am.1995.134307.

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2

Leversha, Gerry, G. Evans, J. Blackledge, and P. Yardley. "Numerical Methods for Partial Differential Equations." Mathematical Gazette 84, no. 501 (November 2000): 567. http://dx.doi.org/10.2307/3620819.

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3

T., V., and William F. Ames. "Numerical Methods for Partial Differential Equations." Mathematics of Computation 62, no. 205 (January 1994): 437. http://dx.doi.org/10.2307/2153426.

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4

Morton, K. W. "NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS." Bulletin of the London Mathematical Society 26, no. 5 (September 1994): 507–8. http://dx.doi.org/10.1112/blms/26.5.507.

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5

März, Roswitha. "Numerical methods for differential algebraic equations." Acta Numerica 1 (January 1992): 141–98. http://dx.doi.org/10.1017/s0962492900002269.

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6

Kloeden, Peter, and Eckhard Platen. "Numerical methods for stochastic differential equations." Stochastic Hydrology and Hydraulics 5, no. 2 (June 1991): 172. http://dx.doi.org/10.1007/bf01543058.

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7

Herdiana, Ratna. "NUMERICAL SIMULATION OF STOCHASTIC DIFFERENTIAL EQUATIONS USING IMPLICIT MILSTEIN METHOD." Journal of Fundamental Mathematics and Applications (JFMA) 3, no. 1 (June 10, 2020): 72–83. http://dx.doi.org/10.14710/jfma.v3i1.7416.

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Stiff stochastic differential equations arise in many applications including in the area of biology. In this paper, we present numerical solution of stochastic differential equations representing the Malthus population model and SIS epidemic model, using the improved implicit Milstein method of order one proposed in [6]. The open source programming language SCILAB is used to perform the numerical simulations. Results show that the method is more accurate and stable compared to the implicit Euler method.

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8

Ababneh, Osama Y. "New Numerical Methods for Solving Differential Equations." JOURNAL OF ADVANCES IN MATHEMATICS 16 (January 31, 2019): 8384–90. http://dx.doi.org/10.24297/jam.v16i0.8280.

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In this paper, we present new numerical methods to solve ordinary differential equations in both linear and nonlinear cases. we apply Daftardar-Gejji technique on theta-method to derive anew family of numerical method. It is shown that the method may be formulated in an equivalent way as a RungeKutta method. The stability of the methods is analyzed.

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9

Murphy, R. V. W. "Differential equations - practical methods of numerical solution." Mathematical Gazette 91, no. 521 (July 2007): 227–34. http://dx.doi.org/10.1017/s0025557200181562.

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The most basic problem with differential equations is that of being given an equation that can be put into the formand one pair of solution values x = x0, y = y0, from which to find either an algebraic form of its solution or the numerical value of y corresponding to a particular value of x. This can most conveniently be interpreted as finding, out of the infinity of solution curves to (1.1), an equation for the unique curve that passes through the point (x0, y0) or calculating the coordinates of one particular point on that curve.

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10

Kim, A. V., and V. G. Pimenov. "Multistep numerical methods for functional differential equations." Mathematics and Computers in Simulation 45, no. 3-4 (February 1998): 377–84. http://dx.doi.org/10.1016/s0378-4754(97)00117-1.

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11

Li, Changpin, and An Chen. "Numerical methods for fractional partial differential equations." International Journal of Computer Mathematics 95, no. 6-7 (July 4, 2017): 1048–99. http://dx.doi.org/10.1080/00207160.2017.1343941.

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12

Arter, W. "Numerical methods for differential equations and applications." Computer Physics Communications 34, no. 3 (January 1985): 335. http://dx.doi.org/10.1016/0010-4655(85)90011-6.

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13

Sukale, Yogita, and Varsha Daftardar-Gejji. "New Numerical Methods for Solving Differential Equations." International Journal of Applied and Computational Mathematics 3, no. 3 (October 25, 2016): 1639–60. http://dx.doi.org/10.1007/s40819-016-0264-6.

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14

Alzaid, Sara Salem, and Badr Saad T. Alkahtani. "Modified numerical methods for fractional differential equations." Alexandria Engineering Journal 58, no. 4 (December 2019): 1439–47. http://dx.doi.org/10.1016/j.aej.2019.11.015.

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15

FALCONE, M. "NUMERICAL METHODS FOR DIFFERENTIAL GAMES BASED ON PARTIAL DIFFERENTIAL EQUATIONS." International Game Theory Review 08, no. 02 (June 2006): 231–72. http://dx.doi.org/10.1142/s0219198906000886.

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In this paper we present some numerical methods for the solution of two-persons zero-sum deterministic differential games. The methods are based on the dynamic programming approach. We first solve the Isaacs equation associated to the game to get an approximate value function and then we use it to reconstruct approximate optimal feedback controls and optimal trajectories. The approximation schemes also have an interesting control interpretation since the time-discrete scheme stems from a dynamic programming principle for the associated discrete time dynamical system. The general framework for convergence results to the value function is the theory of viscosity solutions. Numerical experiments are presented solving some classical pursuit-evasion games.

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16

Dimarco, G., and L. Pareschi. "Numerical methods for kinetic equations." Acta Numerica 23 (May 2014): 369–520. http://dx.doi.org/10.1017/s0962492914000063.

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In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

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17

Dhamacharoen, Ampon. "Efficient Numerical Methods for Solving Differential Algebraic Equations." Journal of Applied Mathematics and Physics 04, no. 01 (2016): 39–47. http://dx.doi.org/10.4236/jamp.2016.41007.

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18

Cheng, Yan, and Cuilian You. "Convergence of numerical methods for fuzzy differential equations." Journal of Intelligent & Fuzzy Systems 38, no. 4 (April 30, 2020): 5257–66. http://dx.doi.org/10.3233/jifs-191856.

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19

A. Connolly, Joseph, and Neville J. Ford. "Comparison of numerical methods for fractional differential equations." Communications on Pure & Applied Analysis 5, no. 2 (2006): 289–307. http://dx.doi.org/10.3934/cpaa.2006.5.289.

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20

Hairer, E., and P. Maass. "Numerical methods for singular nonlinear integro-differential equations." Applied Numerical Mathematics 3, no. 3 (June 1987): 243–56. http://dx.doi.org/10.1016/0168-9274(87)90051-1.

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21

Jankowski, T. "Monotone and numerical-analytic methods for differential equations." Computers & Mathematics with Applications 45, no. 12 (June 2003): 1823–28. http://dx.doi.org/10.1016/s0898-1221(03)90003-4.

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22

de Raynal, Paul-Éric Chaudru, Gilles Pagès, and Clément Rey. "Numerical methods for Stochastic differential equations: two examples." ESAIM: Proceedings and Surveys 64 (2018): 65–77. http://dx.doi.org/10.1051/proc/201864065.

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The goal of this paper is to present a series of recent contributions arising in numerical probability. First we present a contribution to a recently introduced problem: stochastic differential equations with constraints in law, investigated through various theoretical and numerical viewpoints. Such a problem may appear as an extension of the famous Skorokhod problem. Then a generic method to approximate in a weak way the invariant distribution of an ergodic Feller process by a Langevin Monte Carlo simulation. It is an extension of a method originally developed for diffusions and based on the weighted empirical measure of an Euler scheme with decreasing step. Finally, we mention without details a recent development of a multilevel Langevin Monte Carlo simulation method for this type of problem.

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23

Czernous, W., and Z. Kamont. "Numerical methods for Hamilton Jacobi functional differential equations." Computational Mathematics and Mathematical Physics 52, no. 3 (March 2012): 330–50. http://dx.doi.org/10.1134/s0965542512030050.

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24

Boruah, Khiord, Bipan Hazarika, and A. E. Bashirov. "Solvability of bigeometric differential equations by numerical methods." Boletim da Sociedade Paranaense de Matemática 39, no. 2 (January 1, 2021): 203–22. http://dx.doi.org/10.5269/bspm.39444.

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The objective of this paper is to derive and analyze Bigeometric-Euler, Taylor's Bigeometric-series and Bigeometric-Runge-Kutta methods of different orders for the approximation of initial value problems of Bigeometric-differential equations.

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25

Xiu, Dongbin, and Daniel M. Tartakovsky. "Numerical Methods for Differential Equations in Random Domains." SIAM Journal on Scientific Computing 28, no. 3 (January 2006): 1167–85. http://dx.doi.org/10.1137/040613160.

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26

Burrage, Kevin, Ian Lenane, and Grant Lythe. "Numerical Methods for Second‐Order Stochastic Differential Equations." SIAM Journal on Scientific Computing 29, no. 1 (January 2007): 245–64. http://dx.doi.org/10.1137/050646032.

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27

Pimenov, V. G. "Multistep numerical methods for functional-differential-algebraic equations." Proceedings of the Steklov Institute of Mathematics 259, S2 (December 2007): S201—S212. http://dx.doi.org/10.1134/s0081543807060144.

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28

Douglas, Jim, Jin Ma, and Philip Protter. "Numerical methods for forward-backward stochastic differential equations." Annals of Applied Probability 6, no. 3 (August 1996): 940–68. http://dx.doi.org/10.1214/aoap/1034968235.

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29

Hairer, Ernst. "Book Review: Numerical methods for evolutionary differential equations." Mathematics of Computation 79, no. 269 (January 1, 2010): 613. http://dx.doi.org/10.1090/s0025-5718-09-02321-7.

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30

Yazıcı, Muhammet, and Harun Selvitopi. "Numerical methods for the multiplicative partial differential equations." Open Mathematics 15, no. 1 (November 22, 2017): 1344–50. http://dx.doi.org/10.1515/math-2017-0113.

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Abstract We propose the multiplicative explicit Euler, multiplicative implicit Euler, and multiplicative Crank-Nicolson algorithms for the numerical solutions of the multiplicative partial differential equation. We also consider the truncation error estimation for the numerical methods. The stability of the algorithms is analyzed by using the matrix form. The result reveals that the proposed numerical methods are effective and convenient.

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31

El-Borai, Mahmoud M., Khairia El-said El-Nadi, Osama L. Mostafa, and Hamdy M. Ahmed. "Numerical methods for some nonlinear stochastic differential equations." Applied Mathematics and Computation 168, no. 1 (September 2005): 65–75. http://dx.doi.org/10.1016/j.amc.2004.08.015.

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32

Butcher, J. C. "Trees and numerical methods for ordinary differential equations." Numerical Algorithms 53, no. 2-3 (March 14, 2009): 153–70. http://dx.doi.org/10.1007/s11075-009-9285-0.

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33

Torelli, Lucio. "Stability of numerical methods for delay differential equations." Journal of Computational and Applied Mathematics 25, no. 1 (January 1989): 15–26. http://dx.doi.org/10.1016/0377-0427(89)90071-x.

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34

Liu, X., M. H. Song, and M. Z. Liu. "Linear Multistep Methods for Impulsive Differential Equations." Discrete Dynamics in Nature and Society 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/652928.

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This paper deals with the convergence and stability of linear multistep methods for impulsive differential equations. Numerical experiments demonstrate that both the mid-point rule and two-step BDF method are of orderp=0when applied to impulsive differential equations. An improved linear multistep method is proposed. Convergence and stability conditions of the improved methods are given in the paper. Numerical experiments are given in the end to illustrate the conclusion.

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35

Roche, Michel. "Implicit Runge–Kutta Methods for Differential Algebraic Equations." SIAM Journal on Numerical Analysis 26, no. 4 (August 1989): 963–75. http://dx.doi.org/10.1137/0726053.

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36

LI, CHANGPIN, and FANHAI ZENG. "FINITE DIFFERENCE METHODS FOR FRACTIONAL DIFFERENTIAL EQUATIONS." International Journal of Bifurcation and Chaos 22, no. 04 (April 2012): 1230014. http://dx.doi.org/10.1142/s0218127412300145.

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In this review paper, the finite difference methods (FDMs) for the fractional differential equations are displayed. The considered equations mainly include the fractional kinetic equations of diffusion or dispersion with time, space and time-space derivatives. In some way, these numerical methods have similar form as the case for classical equations, some of which can be seen as the generalizations of the FDMs for the typical differential equations. And the classical tools, such as the von Neumann analysis method, the energy method and the Fourier method are extended to numerical methods for fractional differential equations accordingly. At the same time, the techniques for improving the accuracy and reducing the computation and storage are also introduced.

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37

Eriksson, Kenneth, Don Estep, Peter Hansbo, and Claes Johnson. "Introduction to Adaptive Methods for Differential Equations." Acta Numerica 4 (January 1995): 105–58. http://dx.doi.org/10.1017/s0962492900002531.

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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).

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38

Methi, Giriraj, and Anil Kumar. "Numerical Solution of Linear and Higher-order Delay Differential Equations using the Coded Differential Transform Method." Computer Research and Modeling 11, no. 6 (December 2019): 1091–99. http://dx.doi.org/10.20537/2076-7633-2019-11-6-1091-1099.

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39

Ishak, Fuziyah, and Najihah Chaini. "Numerical computation for solving fuzzy differential equations." Indonesian Journal of Electrical Engineering and Computer Science 16, no. 2 (November 1, 2019): 1026. http://dx.doi.org/10.11591/ijeecs.v16.i2.pp1026-1033.

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Fuzzy differential equations (FDEs) play important roles in modeling dynamic systems in science, economics and engineering. The modeling roles are important because most problems in nature are indistinct and uncertain. Numerical methods are needed to solve FDEs since it is difficult to obtain exact solutions. Many approaches have been studied and explored by previous researchers to solve FDEs numerically. Most FDEs are solved by adapting numerical solutions of ordinary differential equations. In this study, we propose the extended Trapezoidal method to solve first order initial value problems of FDEs. The computed results are compared to that of Euler and Trapezoidal methods in terms of errors in order to test the accuracy and validity of the proposed method. The results shown that the extended Trapezoidal method is more accurate in terms of absolute error. Since the extended Trapezoidal method has shown to be an efficient method to solve FDEs, this brings an idea for future researchers to explore and improve the existing numerical methods for solving more general FDEs.

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40

Platen, Eckhard. "An introduction to numerical methods for stochastic differential equations." Acta Numerica 8 (January 1999): 197–246. http://dx.doi.org/10.1017/s0962492900002920.

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This paper aims to give an overview and summary of numerical methods for the solution of stochastic differential equations. It covers discrete time strong and weak approximation methods that are suitable for different applications. A range of approaches and results is discussed within a unified framework. On the one hand, these methods can be interpreted as generalizing the well-developed theory on numerical analysis for deterministic ordinary differential equations. On the other hand they highlight the specific stochastic nature of the equations. In some cases these methods lead to completely new and challenging problems.

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41

Allahviranloo, Tofigh. "Difference Methods for Fuzzy Partial Differential Equations." Computational Methods in Applied Mathematics 2, no. 3 (2002): 233–42. http://dx.doi.org/10.2478/cmam-2002-0014.

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AbstractIn this paper numerical methods for solving ’fuzzy partial differential equations’(FPDE) is considered. Fuzzy reachable set can be approximated by proposed methods with complete error analysis which is discussed in details. The methods are illustrated by solving some linear and nonlinear FPDE.

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42

kaur, Dr Jatinder. "A REVIEW ON NUMERICAL METHODS FOR NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS." Journal of University of Shanghai for Science and Technology 23, no. 07 (August 1, 2021): 1342–52. http://dx.doi.org/10.51201/jusst/21/07230.

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Progression in innovation and engineering presents us with numerous difficulties, comparably to conquer such engineering difficulties with the assistance of various numerical models, equations are taken. Since in the first place Mathematicians, Designers and Engineers make progress toward accuracy what’s more, exactness while addressing equations Differential equations, specifically, hold an enormous application in engineering and numerous different areas. One such sort of Differential equation is known as partial differential equation. The range of application of partial differential equations comprises of recreation, calculation age, and investigation of higher request PDE and wave equations. Adjusting diverse numerical methods prompts an assortment of answers and contrast among them, subsequently the determination of the method of addressing is one of the urgent boundaries to produce exact outcomes. Our work centres’ around the survey of various numerical methods to settle Non-linear differential equations based on exactness and effectiveness, in order to diminish the emphases. These would orchestrate rules to existing numerical methods of nonlinear partial differential equations.[1]

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43

Bildik, Necdet, and Mustafa Inc. "A Comparison between Adomian Decomposition and Tau Methods." Abstract and Applied Analysis 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/621019.

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We present a comparison between Adomian decomposition method (ADM) and Tau method (TM) for the integro-differential equations with the initial or the boundary conditions. The problem is solved quickly, easily, and elegantly by ADM. The numerical results on the examples are shown to validate the proposed ADM as an effective numerical method to solve the integro-differential equations. The numerical results show that ADM method is very effective and convenient for solving differential equations than Tao method.

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44

Ascher, Uri M., and Linda R. Petzold. "Projected Implicit Runge–Kutta Methods for Differential-Algebraic Equations." SIAM Journal on Numerical Analysis 28, no. 4 (August 1991): 1097–120. http://dx.doi.org/10.1137/0728059.

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45

Ascher, Uri M., Steven J. Ruuth, and Brian T. R. Wetton. "Implicit-Explicit Methods for Time-Dependent Partial Differential Equations." SIAM Journal on Numerical Analysis 32, no. 3 (June 1995): 797–823. http://dx.doi.org/10.1137/0732037.

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46

Lin, Tao, Yanping Lin, Ming Rao, and Shuhua Zhang. "Petrov--Galerkin Methods for Linear Volterra Integro-Differential Equations." SIAM Journal on Numerical Analysis 38, no. 3 (January 2000): 937–63. http://dx.doi.org/10.1137/s0036142999336145.

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47

Kamont, Z., and W. Czernous. "Implicit difference methods for Hamilton-Jacobi functional differential equations." Numerical Analysis and Applications 2, no. 1 (January 2009): 46–57. http://dx.doi.org/10.1134/s1995423909010054.

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48

T., V., and G. D. Smith. "Numerical Solution of Partial Differential Equations, Finite Difference Methods." Mathematics of Computation 48, no. 178 (April 1987): 834. http://dx.doi.org/10.2307/2007849.

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49

Seferi, Ylldrita. "COMPARISON OF DIFFERENT NUMERICAL METHODS FOR FRACTIONAL DIFFERENTIAL EQUATIONS." Математички билтен/BULLETIN MATHÉMATIQUE DE LA SOCIÉTÉ DES MATHÉMATICIENS DE LA RÉPUBLIQUE MACÉDOINE, no. 2 (2018): 61–74. http://dx.doi.org/10.37560/matbil18200061s.

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50

Iserles, A., and G. D. Smith. "Numerical Solution of Partial Differential Equations: Finite Difference Methods." Mathematical Gazette 70, no. 454 (December 1986): 330. http://dx.doi.org/10.2307/3616228.

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