广西师范大学学报（自然科学版） ›› 2023, Vol. 41 ›› Issue (3): 130-143.doi: 10.16088/j.issn.1001-6600.2022050602

• 研究论文 • 上一篇    下一篇

第六类Chebyshev小波配置法求分数阶微分方程数值解

黄英杰, 周凤英*, 许小勇, 何红梅   

  1. 东华理工大学 理学院, 江西 南昌 330013
  • 收稿日期:2022-05-06 修回日期:2022-07-20 出版日期:2023-05-25 发布日期:2023-06-01
  • 通讯作者: 周凤英(1981—), 女, 湖南岳阳人, 东华理工大学副教授, 博士。E-mail: zhoufengying@ecut.edu.cn
  • 基金资助:
    国家自然科学基金(11601076); 江西省自然科学基金(20202BABL201006); 东华理工大学博士科研启动项目(DHBK2019213)

The Sixth Kind of Chebyshev Wavelet Collocation Method for Numerical Solutions of Fractional Differential Equations

HUANG Yingjie, ZHOU Fengying*, XU Xiaoyong, HE Hongmei   

  1. School of Science, East China University of Technology, Nanchang Jiangxi 330013, China
  • Received:2022-05-06 Revised:2022-07-20 Online:2023-05-25 Published:2023-06-01

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摘要： 基于第六类Chebyshev小波配置法,提出一种求解分数阶微分方程数值解的数值方法。利用平移的第六类Chebyshev多项式,在Riemann-Liouville分数阶定义下,获得了第六类Chebyshev小波函数的分数阶积分公式的精确表达式。利用积分公式,结合有效配置法,将分数阶微分方程的求解问题转化为代数方程组进行求解。同时,给出了第六类Chebyshev小波函数展开逼近的一致收敛性分析和L2范数意义下的误差估计。通过数值算例验证该算法的适用性与有效性。

关键词: 第六类Chebyshev小波, 分数阶微分方程, Riemann-Liouville分数阶积分, Caputo分数阶微分, 配置法

Abstract: Based on the sixth kind of Chebyshev wavelet collocation method, a numerical method for solving fractional differential equations is proposed. By using the shifted sixth kind of Chebyshev polynomials and under the definition of Riemann-Liouville fractional order, the exact expressions of the fractional order integral formulas of the sixth kind of Chebyshev wavelet functions are obtained. Using the integral formulas together with the effective collocation method, the problem of solving fractional differential equations is transformed into algebraic equations. The uniform convergence analysis of the expansion of the sixth kind of Chebyshev wavelet functions and error estimation in the sense of L2 norm are given. Numerical examples show the applicability and effectiveness of the algorithm.

Key words: the sixth kind of Chebyshev wavelet, fractional order differential equation, Riemann-Liouville fractional integration, Caputo fractional differential, collocation method

中图分类号:  O241.8

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