An integral discretization scheme on a graded mesh for a fractional differential equation with integral boundary conditions

IF 1.7 3区化学 Q3 CHEMISTRY, MULTIDISCIPLINARY

Journal of Mathematical Chemistry Pub Date : 2024-04-08 DOI:10.1007/s10910-024-01596-7

Zhongdi Cen, Jian Huang, Aimin Xu

引用次数: 0

Abstract

In this paper, a fractional differential equation with integral conditions is studied. The fractional differential equation is transformed into an integral equation with two initial values, where the initial values needs to ensure that the exact solution satisfies the integral boundary conditions. A graded mesh based on a priori information of the exact solution is constructed and the linear interpolation is used to approximate the functions in the fractional integral. The rigorous analysis about the convergence of the discretization scheme is derived by using the truncation error estimate techniques and the generalized Grönwall inequality. A quasi-Newton method is used to determine the initial values so that the numerical solution satisfies two integral boundary conditions within a prescribed precision. It is shown that the scheme is second-order convergent, which improves the results on the uniform mesh.

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带积分边界条件的分数微分方程的梯度网格积分离散方案

本文研究了带积分条件的分微分方程。分数微分方程被转化为具有两个初值的积分方程，其中初值需要确保精确解满足积分边界条件。根据精确解的先验信息构建分级网格，并使用线性插值来逼近分数积分中的函数。利用截断误差估计技术和广义格伦瓦不等式对离散化方案的收敛性进行了严格分析。利用准牛顿方法确定初始值，从而使数值解在规定精度内满足两个积分边界条件。结果表明，该方案具有二阶收敛性，从而改善了均匀网格的结果。

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.