时空分数扩散方程前向和后向问题的数值方法

IF 1.4 2区数学 Q1 MATHEMATICS

Calcolo Pub Date : 2024-02-21 DOI:10.1007/s10092-024-00567-3

Xiaoli Feng, Xiaoyu Yuan, Meixia Zhao, Zhi Qian

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引用次数: 0

摘要

本文考虑了时空分数扩散方程的正向和反向问题的数值方法。对于二维正向问题，我们提出了一种有限差分法。给出了方案的稳定性和相应的快速预处理共轭梯度算法。对于后向问题，由于它是一个求解困难的问题，我们采用了一种准界值方法来处理它。基于傅立叶变换，我们利用先验正则化参数选择规则和后验正则化参数选择规则，得到了两种阶最优收敛率。前向问题和后向问题的数值实例表明，所提出的数值方法效果良好。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

$Numerical methods for the forward and backward problems of a time-space fractional diffusion equation$

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Numerical methods for the forward and backward problems of a time-space fractional diffusion equation

In this paper, we consider the numerical methods for both the forward and backward problems of a time-space fractional diffusion equation. For the two-dimensional forward problem, we propose a finite difference method. The stability of the scheme and the corresponding Fast Preconditioned Conjugated Gradient algorithm are given. For the backward problem, since it is ill-posed, we use a quasi-boundary-value method to deal with it. Based on the Fourier transform, we obtain two kinds of order optimal convergence rates by using an a-priori and an a-posteriori regularization parameter choice rules. Numerical examples for both forward and backward problems show that the proposed numerical methods work well.

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来源期刊

Calcolo 数学-数学

CiteScore

2.40

自引率

11.80%

发文量

审稿时长

>12 weeks

期刊介绍： Calcolo is a quarterly of the Italian National Research Council, under the direction of the Institute for Informatics and Telematics in Pisa. Calcolo publishes original contributions in English on Numerical Analysis and its Applications, and on the Theory of Computation. The main focus of the journal is on Numerical Linear Algebra, Approximation Theory and its Applications, Numerical Solution of Differential and Integral Equations, Computational Complexity, Algorithmics, Mathematical Aspects of Computer Science, Optimization Theory. Expository papers will also appear from time to time as an introduction to emerging topics in one of the above mentioned fields. There will be a "Report" section, with abstracts of PhD Theses, news and reports from conferences and book reviews. All submissions will be carefully refereed.