Parameter-Uniform Convergent Numerical Approach for Time-Fractional Singularly Perturbed Partial Differential Equations With Large Time Delay

IF 0.9 Q3 MATHEMATICS, APPLIED
Habtamu Getachew Kumie, Awoke Andargie Tiruneh, Getachew Adamu Derese
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Abstract

In this study, we consider a parameter-uniform convergent numerical approach for a class of time-fractional singularly perturbed partial differential equations (TF-SPDPDEs) with large delay in time that exhibits a regular exponential boundary layer on the right side of the spatial domain. An arbitrary very small parameter ε(0 < ε < <1) multiplies the highest-order derivative term of these singularly perturbed problems. The time-fractional derivative is considered in the Caputo sense with order α ∈ (0, 1). The numerical scheme comprises the L1 scheme and nonstandard finite difference method (FDM) for discretizing the time and space variables, respectively, on a uniform mesh. To show the parameter uniform convergence of the proposed method, the truncation error and stability analysis are discussed. The method is shown to be parameter-uniform convergent of order O((Δt)2−α + Δx), where Δt and Δx are the step sizes in the time and space directions, respectively. In order to confirm the theoretical predictions, two numerical examples are presented, and the numerical results support the theoretical concepts discussed. Finally, to show the advantage of the proposed scheme, we made comparisons with the existing numerical methods in the literature, and the numerical results reveal that the present scheme is more accurate.

Abstract Image

大时延时分数奇异扰动偏微分方程的参数统一收敛数值方法
在本研究中,我们考虑了一种参数均匀收敛数值方法,适用于一类具有大时间延迟的时间分数奇异扰动偏微分方程(TF-SPDPDEs),该方程在空间域右侧表现出规则的指数边界层。一个任意的极小参数ε(0 < ε < <1)乘以这些奇异扰动问题的最高阶导数项。卡普托意义上的时分导数阶数 α∈ (0, 1)。数值方案包括 L1 方案和非标准有限差分法(FDM),分别用于在均匀网格上离散时间变量和空间变量。为了说明所提方法的参数均匀收敛性,讨论了截断误差和稳定性分析。该方法的参数均匀收敛性为 O((Δt)2-α+Δx),其中 Δt 和 Δx 分别为时间和空间方向的步长。为了证实理论预测,介绍了两个数值示例,数值结果支持所讨论的理论概念。最后,为了说明所提方案的优势,我们与文献中现有的数值方法进行了比较,数值结果表明本方案更为精确。
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CiteScore
2.20
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