分数阶弱奇异积分微分方程数值解的Legendre小波方法

Legendre Wavelets Method for the Numerical Solution of Fractional Integro-differential Equations with a Weakly Singular Kernel

为了求分数阶变系数带弱奇异积分核的Volterra-Fredholm积分微分方程数值解,提出了Legendre小波配点法.利用平移的Legendre多项式解析形式,推导了定义在[0,1]区间上Legendre小波函数的任意阶积分求积公式.利用高斯求积公式来近似定积分项和Legendre小波函数的任意阶积分公式,将原积分微分方程转化为求代数方程组的解.数值算例验证了该方法的有效性.

In order to solve the numerical solution of the Volterra-Fredholm integral differential equation with weakly singular integral kernels,a Legendre wavelet collocation method is proposed in this paper.The fractional integral of a single Legendre wavelet defined in the interval[0,1]is derived from the definition by means of the shifted Legendre polynomial.The original equation is converted to a system of algebraic equation by using Gauss-Legendre quadrature formula to approximate definite integral and the fractional integral to handle the weakly kernel.Some numerical examples are shown to illustrate the efficiency of the proposed method.