时间分数阶偏微分方程的数值算法研究

A Study on the Numerical Algorithm for Solving Time Fractional Order Partial Differential Equations

提出了一种求解时间分数阶偏微分方程的数值算法.根据方程的初值条件和边界条件构造辅助函数,应用辅助函数将方程转化为具有零初值条件和零边界条件的方程.应用差分算法计算方程中的分数阶导数,然后求解出方程的数值解.方程的分数阶可以为任意正实数,无论分数阶取何值,此算法均采用同样的差分格式求解方程.给出的两个计算实例说明此算法是有效的.实例中的方程具有解析解,可以作为检验同类算法的测试问题.

A numerical algorithm is proposed for solving time fractional order partial differential equations.An auxiliary function is constructed based on the initial conditions and boundary conditions of the equation,the auxiliary function is used to transform the equation into the equation with zero initial conditions and zero boundary conditions.The fractional order derivative in the equation is calculated by difference method,and the numerical solution of the equation is obtained.The fractional order of the equation can be arbitrary positive real number.The same difference scheme is used to solve the equation,no matter what value the fractional order of the equation is.Two calculation examples are given to illustrate the effectiveness of the algorithm.The equations in the examples have analytical solutions,which can be used as benchmark problems to test the similar algorithms.