基于再生核和有限差分法求解变系数时间分数阶对流扩散方程

数学物理学报 ›› 2025, Vol. 45 ›› Issue (1): 153-164.

基于再生核和有限差分法求解变系数时间分数阶对流扩散方程

吕学琴^1,2, 何松岩³, 王世宇^2,4,*

¹天津中德应用技术大学基础课部天津 300350;
²哈尔滨师范大学数学科学学院哈尔滨 150025;
³东北师范大学数学与统计学院长春 130024;
⁴北京市第一〇一中学昌平实验学校北京 102206

收稿日期:2024-03-12 修回日期:2024-08-02 出版日期:2025-02-26 发布日期:2025-01-08
通讯作者: *王世宇, E-mail: wsyhhhh2023@163.com
基金资助:
天津市高等教育委员会科技发展基金 (2019KJ142) 和天津中德应用技术大学教学质量与教学改革研究计划项目 (A2301)

Combining RKM with FDM for Time Fractional Convection-Diffusion Equations with Variable Coefficients

Lv Xueqin^1,2, He Songyan³, Wang Shiyu^2,4

¹College of Basic Science, Tianjin Sino-German University of Applied Sciences, Tianjin 300350;
²School of Mathematics and Sciences, Harbin Normal University, Harbin 150025;
³School of Mathematics and Statistics, Northeast Normal University, Changchun 130024;
⁴Beijing No.101 Middle School Changping Experimental School, Beijing 102206

Received:2024-03-12 Revised:2024-08-02 Online:2025-02-26 Published:2025-01-08
Supported by:
Science & Technology Development Fund of Tianjin Education Commission for Higher Education (2019KJ142) and the Teaching Quality and Teaching Reform Research Project of Tianjin Sino-German University of Applied Technology

摘要/Abstract

摘要： 针对变系数的时间分数阶对流-扩散方程, 首先, 使用有限差分法, 得到了该方程的半离散格式. 之后再利用再生核方法, 得到了方程的精确解 $u(x,t_{n})$, 将精确解 $u(x,t_{n})$ 取 $m$ 项截断, 可得到近似解 $u_{m}(x,t_{n})$. 通过证明, 得到该方法是稳定的. 最后, 通过三个数值例子, 并与其他文献中的方法在同等条件下进行了比较, 证明该算法有效.

关键词: Caputo 分数阶导数, 再生核方法, 变系数时间分数阶对流扩散方程, 有限差分方法

Abstract: In this paper, we will study the time fractional convection-diffusion equation with variable coefficients. First, we use the finite difference method. The time variable is discretized, and the semi-discrete scheme of the equation is obtained. The exact solution $u(x,t_{n})$ of the equation is obtained by using the theory of reproducing kernel method. Then the exact solution $u(x,t_{n})$ is truncated by $m$ term to obtain the approximate solution $u_{m}(x,t_{n})$. By proving, we know that the method is stable. Moreover, $u_{m}^{(i)}(x,t_{n})$ converge uniformly to $u^{(i)}(x,t_{n})$ $(i=0,1,2)$. Finally, we give several numerical examples and compare them with the methods in other literatures, which show that our algorithm is effective.

Key words: Caputo fractional derivative, reproducing kernel method, variable coefficient time fractional convection-diffusion equation, finite difference method

中图分类号:

吕学琴, 何松岩, 王世宇. 基于再生核和有限差分法求解变系数时间分数阶对流扩散方程[J]. 数学物理学报, 2025, 45(1): 153-164.

Lv Xueqin, He Songyan, Wang Shiyu. Combining RKM with FDM for Time Fractional Convection-Diffusion Equations with Variable Coefficients[J]. Acta mathematica scientia,Series A, 2025, 45(1): 153-164.

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