Research On Higher Order Difference Schemes Of Two Kinds Of Fractional Integral Differential Equations

In the past few decades,fractional integro-differential equations have attracted more and more researchers’attention,and the related theoretical research has also made significant progress.The solution process of most fractional integro-differential equations is very difficult or even impossible to obtain analytical solutions.However,in practical problems,it is usually only necessary to obtain an effective approximate solution,so scholars have turned to the study of numerical solutions of fractional integro-differential equations.In this thesis,two numerical solutions of fourth-order integro-differential equations are mainly studied.Firstly,we use BDF2（second-order backward differential formula）finite difference scheme to solve a fourth-order integral differential equations with the multi-term kernels.The second-order backward differential formula is used to discretize in the time direction,and the finite difference method is used in the space direction.The Riemann-Liouville fractional integral term in the equation is approximated by the second-order convolution quadrature rule.Then,the stability and convergence of the scheme are analyzed and proved,and the accuracy and feasibility of the theoretical results are verified by numerical examples,the numerical results indicated that the formally BDF2 difference scheme is stable and convergent with the convergence of order min{1+α₁,1+α₂} for the time and 2 for the space.Secondly,we consider an effective numerical method for the three-dimensional fractional evolution equation.In our method,the backward Euler formula is used to discretize in the time direction,a fully discrete difference scheme is constructed with space discretization by finite difference method.By using ADI scheme for the three-dimensional problem,the overall computational cost is reduced significantly.The resulting finite difference scheme is unconditional stable and convergent with the convergence order of 1in time and 2 for space in new norm.Finally,the two numerical methods discussed in this thesis are summarized,and the future research work is prospected.