时间分数阶扩散波方程的无单元Galerkin法分析

An Element-Free Galerkin Method for Time-Fractional Diffusion-Wave Equations

利用无单元Galerkin法,对Caputo意义下的时间分数阶扩散波方程进行了数值求解和相应误差理论分析.首先用L1逼近公式离散该方程中的时间变量,将时间分数阶扩散波方程转化成与时间无关的整数阶微分方程;然后采用罚函数方法处理Dirichlet边界条件,并利用无单元Galerkin法离散整数阶微分方程;最后推导该方程无单元Galerkin法的误差估计公式.数值算例证明了该方法的精度和效果.

Numerical solution and theoretical error analysis of the element-free Galerkin(EFG) method were presented for the time-fractional diffusion-wave equations in the sense of Caputo. Through discretization of the time variables in the equation with the L1 approximate formula, the time-fractional diffusion-wave equation was transformed into a series of time-independent integer-order differential equations. Then, the penalty method was used to deal with the Dirichlet boundary condition and the EFG method was used to discretize the integer-order differential equations. Error estimates of the EFG method for the time-fractional diffusion-wave equations were derived theoretically. Finally, several numerical examples show the accuracy and effectiveness of the proposed meshless method.