一种基于微分求积的空间分数阶扩散方程求解方法

A numerical approach for space-fractional diffusion equation by differential quadrature

通过微分求积建立求解变系数空间分数阶扩散方程的一种有效直接数值方法。基于Reciprocal Multiquadric和Thin-Plate Spline径向基函数推导两种逼近分数阶导数的微分求积公式,将所考虑的模型问题转化成易求解的常微分方程组,并采用Crank-Nicolson格式进行离散。给出5个数值算例,计算结果表明,只要径向基函数的形状参数选择恰当,本文方法在精度和效率上均优于一些现有算法。

This article is devoted to develop a direct numerical approach for the space-fractional diffusion equation with variable coefficients by differential quadrature(DQ)technique.Two DQ approximations of fractional derivatives based on Reciprocal Multiquadric and Thin-Plate Spline radial basis functions(RBFs)are introduced and applied to turn the equation in consideration into a set of easily solvable ordinary differential equations,which are discretized by the Crank-Nicolson scheme.The presented methods are verified by five numerical examples and the numerical results illustrate that they outperform some existing algorithms in term of both accuracy and efficiency as long as the shape parameters of RBFs are properly chosen.