分数次Chebyshev小波结合SA算法求解分数阶微分方程数值解  

Numerical solution fractional-order Chebyshev wavelets combining SA algorithm for solving fractional differential equations

作　　者：许小勇 何通森 楼钦艺 朱婷 XU Xiaoyong;HE Tongsen;LOU Qinyi;ZHU Ting(School of Science,East China University of Technology,Nanchang 330013,China)

机构地区：[1]东华理工大学理学院,江西南昌330013

出　　处：《广西大学学报（自然科学版）》2024年第4期907-916,共10页Journal of Guangxi University（Natural Science Edition）

基　　金：国家自然科学基金项目(12061008);江西省自然科学基金项目(2020BABL201006);东华理工大学博士科研启动项目(DHBK2019213)。

摘　　要：为了求解分数阶微分方程,提出了一种结合分数次第二类Chebyshev小波(FOCWs)配置法与模拟退火(SA)算法的有效数值方法。首先,构造了分数次的第二类Chebyshev小波函数,利用正则化的Beta函数,推导了分数次Chebyshev小波函数在Riemann-Liouville分数阶积分定义下的积分计算公式。其次,利用分数次小波函数及积分公式并结合配置法,将分数阶微分方程转化为线性或非线性代数方程,给出了算法的误差估计。由于分数次小波函数中涉及分数次参数α,解的结果依赖于参数α的选择,考虑使用SA算法寻找最优参数。最后,通过数值算例验证了该方法的可行性和有效性。An effective numerical method is developed based on the collocation method of fractional-order Chebyshev wavelets(FOCWs)of the second kind,combining simulated annealing(SA)algorithm for the numerical solution of fractional differential equations.First,the fractional-order Chebyshev wavelets of the second kind were constructed.Then,using the regularized Beta function,the exact formulas of FOCWs were derived under the definition of Riemann Liouville fractional integral.By the properties of FOCWs and the exact formulas together with the collocation method,the problem under consideration was simplified into algebraic equations.The error analysis of the proposed method is studied.Due to the involvement of parameterαin FOCWs method,the accuracy of the solution depends on the selection of parameterα.SA algorithm was considered to find the optimal parameterα.Finally,the effectiveness and applicability of the suggested method are verified by some numerical examples.

关 键 词：分数次Chebyshev小波 分数阶微分方程 配置法 模拟退火算法 

分 类 号：O241.5[理学—计算数学]