Numerical solution to loaded difference scheme for time-fractional diffusion equation with temporal loads

IF 1.7 3区化学 Q3 CHEMISTRY, MULTIDISCIPLINARY

Journal of Mathematical Chemistry Pub Date : 2024-07-31 DOI:10.1007/s10910-024-01658-w

Shweta Kumari, Mani Mehra

引用次数: 0

Abstract

This paper investigates the temporally loaded time-fractional diffusion equation with initial and Dirichlet-type boundary conditions. To begin with, a solution form is established using the method of eigenfunction expansions, and its existence and uniqueness are examined along with some apriori estimates. Thereafter, a finite difference approximation is performed using the so-called L1 method for the Caputo fractional derivative, resulting in a loaded difference scheme. The superposition property of systems of linear algebraic equations is applied to solve the loaded difference scheme by appointing an appropriate solution representation. The unique solvability of the proposed scheme is set up. The stability and convergence of the proposed difference scheme are analysed by the discrete energy method with an order of accuracy \(\mathcal {O}(\tau ^{2-\alpha }+h^2)\). Numerical results via two test problems are presented to validate the theoretical findings of the proposed scheme by observing the errors.

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带时间负荷的时间分数扩散方程的加载差分方案数值解法

本文研究了具有初始条件和迪里希勒型边界条件的时间负载时间分数扩散方程。首先，利用特征函数展开法建立了求解形式，并考察了其存在性和唯一性以及一些先验估计。之后，使用所谓的 L1 方法对卡普托分导数进行有限差分逼近，从而得到一个加载差分方案。应用线性代数方程组的叠加特性，通过指定适当的求解表示来求解加载差分方案。建立了所提方案的唯一可解性。通过离散能量法分析了所提差分方案的稳定性和收敛性，精确度为 \(\mathcal {O}(\tau ^{2-\alpha }+h^2)\)。给出了两个测试问题的数值结果，通过观察误差验证了所提方案的理论结论。

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

Journal of Mathematical Chemistry 化学-化学综合

CiteScore

3.70

自引率

17.60%

发文量

105

审稿时长

6 months

期刊介绍： The Journal of Mathematical Chemistry (JOMC) publishes original, chemically important mathematical results which use non-routine mathematical methodologies often unfamiliar to the usual audience of mainstream experimental and theoretical chemistry journals. Furthermore JOMC publishes papers on novel applications of more familiar mathematical techniques and analyses of chemical problems which indicate the need for new mathematical approaches. Mathematical chemistry is a truly interdisciplinary subject, a field of rapidly growing importance. As chemistry becomes more and more amenable to mathematically rigorous study, it is likely that chemistry will also become an alert and demanding consumer of new mathematical results. The level of complexity of chemical problems is often very high, and modeling molecular behaviour and chemical reactions does require new mathematical approaches. Chemistry is witnessing an important shift in emphasis: simplistic models are no longer satisfactory, and more detailed mathematical understanding of complex chemical properties and phenomena are required. From theoretical chemistry and quantum chemistry to applied fields such as molecular modeling, drug design, molecular engineering, and the development of supramolecular structures, mathematical chemistry is an important discipline providing both explanations and predictions. JOMC has an important role in advancing chemistry to an era of detailed understanding of molecules and reactions.