Research On Fast Algorithms Of Variable Time Fractional Differential Equations

In recent years,variable-order time-fractional differential equations have been widely used to model complex anomalous diffusion processes.The main challenge of fractional order nonlinear partial differential equations is the nonlocal property of the fractional calculus.Direct time-stepping methods usually require large amount active memory and expensive computational cost.It is significant of developing fast and memory-saving methods for variable-order time-fractional differential equations.This paper aims to provide efficient and fast algorithms for variable-order timefractional differential equations.We illustrate the effectiveness of the proposed methods via comparison with the direct methods.The basic idea of fast calculation is to find a suitable sum-of-exponentials to approximate the kernel functions.We first review several methods to approximate the kernel function,including the contour quadrature methods,the multi-pole approximation method,the Laguerre-Gauss quadrature method,and the trapezoidal rule.Numerical simulations show that the trapezoidal rule performs better than the other methods.Then,we extend the contour quadrature methods and the trapezoidal rule to the approximation of variable-order kernel function.Numerical experiments show the superiority of the trapezoidal rule in long time calculation for the variable-order kernel function.Finally,two fast methods for solving variable-order time-fractional differential equations are proposed based on the trapezoidal rule,namely:the linear interpolation method and the generating function method.In both methods,the discrete convolutions to the approximation of the variable-order fractional integral and derivative operators is divided into two parts,the historical part and the local part.The local part is directly calculated,and the historical part is calculated by the trapezoidal rule.We illustrate the effectiveness of two fast algorithms through three types of numerical experiments,(1)time-fractional Allen-Cahn equation,(2)variable-order timefractional convection diffusion equation,(3)variable-order fractional sub-diffusion equation.The simulation results show that the proposed methods are fast and memory saving.Thus these methods are efficiency for solving variable-order time-fractional differential equations.