Numerical Solution And Application Of Caputo Derivative With Generalized Memory Kernel

Fractional differential equations,a generalization of integer order partial differen-tial equations,have been widely used in the fields of natural science and social science.Fractional partial differentials have been successfully employed to describe he physi-cal and dynamic system process with the memory and inheritance since the nonlocal property of fractional derivative.In the article,we propose three numerical schemes for Caputo fractional derivative with a generalized memory kernel and estimate the corresponding error.As an application,we develop two difference schemes for the time fractional wave equation with a generalized memory kernel.We prove two schemes are unconditionally stable,Finally,the validity of the formats is verified using numerical examples.This article is divided into three parts.In the first part,we propose a(τ^3-τ)order discrete scheme for Caputo fractional derivative with a generalized memory kernel,calledL1scheme,and that the coefficient decreasing monotonically.As an application,we develop a finite difference scheme based onL1 for time fractional wave equation with a generalized memory kernel,which is unconditionally stable,and that convergence rate is(τ^3-τ)order in temporal direction while second order in spatial direction.Numerical experiments are presented to verify the accuracy of the proposed scheme,which illuminates the scheme is simple and efficient.In the second part,we propose a second order scheme for Caputo fractional deriva-tive with a generalized memory kernel by the order reduction andL2-1_σscheme and that the coefficients is studied in detail.We construct a temporal and spatial sec-ond order for time fractional wave equation with a generalized memory kernel,which is unconditionally stable and convergent.Numerical examples are used to verify the accuracy and efficiency of the scheme.In the third part,we construct a new second-order scheme for（0<≤1）order Caputo fractional derivatives with a generalized memory kernel,and numerical examples are presented to verify the validity of the scheme.Finally,we conclude the artical and provide an outlook on the subsequent research.