Finite Difference Method For Integro-Differential Equations With Weakly Singular Kernel

In recent years,fractional partial differential equations have attracted more and more attention from experts and scholars.Fractional partial differential equations play an increasingly important role in many scientific fields,such as mathematics,physics,finance,biological engineering,mechanical engineering,electronic engineering and so on,because the equations can describe the genetic and memory properties of various substances extremely effectively.Unfortunately,experts and scholars find that it is very difficult to solve fractional partial differential equations,and even many solutions cannot be expressed explicitly,which urges us to consider effective numerical methods.In this paper,alternate direction implicit(ADI)finite difference method and second order BDF finite difference method are used to study the integro-differential equations with weakly singular kernel.The thesis consists of four chapters,among which the second and third chapters are the core rese arch content and the main research work of the author.The main research contents are as follows:In chapter 2,the stability and error analysis of ADI fully discrete scheme for two-dimensional fractional evolution equations are given.The backward Euler method and finite difference method are used for time and space directions respectively.Finally,two numerical examples are given.The numerical results show that the convergence order of the numerical solution is order 1 in time direction and order 2 in space direction,which is consistent with the theoretical results.In chapter 3,the stability and error analysis of the fully discrete scheme of the fourth order integro-differential equation with several Riemann-Liouvile fractional integral terms are given.The second order BDF method and the second order convolution quadrature method are used to deal with time derivatives and Riemann-Liouville fractional integral terms respectively.The discretization of spatial derivatives is done using the standard central difference method.Finally,two numerical examples are given.