Numerical discretization of initial–boundary value problems for PDEs with integer and fractional order time derivatives

IF 3.4 2区数学 Q1 MATHEMATICS, APPLIED

Communications in Nonlinear Science and Numerical Simulation Pub Date : 2024-09-03 DOI:10.1016/j.cnsns.2024.108331

Zaid Odibat

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引用次数: 0

Abstract

This paper is mainly concerned with introducing a numerical method for solving initial–boundary value problems with integer and fractional order time derivatives. The method is based on discretizing the considered problems with respect to spatial and temporal domains. With the help of finite difference methods, we transformed the studied problem into a set of fractional differential equations. Then, we implemented the fractional Adams method to solve this set in order to provide approximate solutions to the main problem. This combination results in an algorithm that can efficiently and accurately solve a general class of integer and fractional order initial–boundary value problems, such that it does not need to solve large systems of linear equations. In addition, we discussed the stability of the proposed scheme. Three illustrative examples are numerically solved to reveal the effectiveness and validity of the proposed technique.

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具有整数阶和分数阶时间导数的 PDE 初始边界值问题的数值离散化

本文主要介绍一种数值方法，用于求解具有整数阶和分数阶时间导数的初始边界值问题。该方法基于对所考虑问题的空间域和时间域离散化。在有限差分法的帮助下，我们将所研究的问题转化为一组分数微分方程。然后，我们采用分数亚当斯法来求解这组方程，以便为主要问题提供近似解。这种组合产生了一种算法，它可以高效、准确地解决一般的整数阶和分数阶初边界值问题，从而无需求解大型线性方程组。此外，我们还讨论了所提方案的稳定性。通过对三个示例的数值求解，揭示了所提技术的有效性和正确性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Communications in Nonlinear Science and Numerical Simulation MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

6.80

自引率

7.70%

发文量

378

审稿时长

78 days

期刊介绍： The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity. The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged. Topics of interest: Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity. No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.