小波方法求一类变系数分数阶微分方程数值解

Wavelet method for numerical solution of a class of fractional differential equation with variable coefficients

为了解决分数阶微分方程数值解的问题,采用Haar小波算子矩阵的方法,研究了一类变系数分数阶微分方程的数值解.将Haar小波与算子矩阵思想有效结合,得到了Haar小波的分数阶微分算子矩阵,并对分数阶微分方程的变系数进行恰当的离散.把变系数分数阶微分方程转化为线性代数方程组,使得计算更简便,同时证明上述算法的收敛性.最后给出数值算例验证了该方法的可行性和有效性.数值计算结果表明:随着取点数的增多,数值解与精确解的近似度越来越高.

In order to solve the problem associated with the numerical solution of fractional differential equation,this paper investigates the numerical solution of a class of fractional differential equation with variable coefficients using the operational matrix of Haar wavelet method.By combining Haar wavelet with operational matrix,the operational matrix of Haar wavelet of fractional order is obtained,and the coefficients of fractional differential equation are efficaciously discretized.Therefore,the original problem is transformed into a system of algebraic equations and the computation is simplified.In addition,the convergence of this method is proved.The numerical examples show that the method is feasible and effective.The numerical results demonstrate that the degree of approximation between the numerical solution and the exact solution is higher with the increase of points selected.