High Order Difference Schemes For Several Integro-Differential Equations

Integro-differential equations are very suitable for describing many processes with memory.They are widely used as mathematical models,whose situations depend on the current and history information.In practical applications,more and more integro-differential equations are used to simulate some natural phenomena in the fields of economy,biology,control,physics,etc.The analytical solutions of integro-differential equations are often difficult to be obtained directly.Therefore,it is of great significance to study the numerical solutions for these models.This thesis mainly focuses on the constructions of high order numerical solutions for several types of time fractional differ-ential equations and integro-differential equations with distributed delay,analyzes the unique solvability,convergence,stability of the schemes,and verifies the convergence results by numerical experiments.The thesis consists of 6 Chapters:In Chapter 1,we introduce the definitions of Riemann-Liouville and Caputo frac-tional derivative,the properties of the fractional derivative and the Alternating Direc-tion Implicit（ADI）schemes.Then,we present the background of numerical methods to solve integro-differential equations.This part is an important basis for the following chapters.In Chapter 2,several linearized numerical schemes are developed for solving the nonlinear time-fractional sub-diffusion equations.The numerical schemes are con-structed as follows:the L1 discretization is applied to discrete time-fractional part,the compact finite difference scheme is applied to discretize the diffusion term,and several different linearized techniques are employed to approximate the nonlinear term.After that,the unique solvability of numerical solutions and the convergence are inves-tigated.Furthermore,by applying the recent derived discrete fractional inequality,it can obtain that the linearized numerical methods are of order 1 or 2-α（αis the order of fractional derivative）in temporal direction,and of order 4 in spatial directions.In addition,we apply the sum-of-exponentials approximation,and develop the correspond-ing accelerated methods,with which the computational cost is reduced from O（N²）to O（log N）or O（log²N）（N is the number of temporal mesh points）.Finally,several numerical experiments are conducted to verify all the theoretical results.In Chapter 3,a transformed L1 scheme is proposed to solve nonlinear fractional sub-diffusion equations.Generally speaking,the solution of the nonlinear sub-diffusion equation has the initial layer and its initial energy may decay very fast.Therefore,it is important to investigate the evolution of the solution at the beginning.The transformed L1 scheme is based on the change of variable,an L1-type finite difference method is proposed for solving the changed fractional differential equation,and nonlinear term is handled using Newton’s linearization method.In this chapter,a discrete fractional Gr?nwall inequality is obtained,the convergence of the scheme is proved by this in-equality.It is proved that the temporal error of the new method is O(τ^2-α),whereτis the temporal stepsize and 0<α<1.The error estimate holds even when t→0.In contrast,the maximum errors of the uniform L1 scheme,the convolution quadrature（CQ）Euler method,CQ BDF method,and their corrected forms are usually O（τ^α）at the beginning.The new method can capture the initial dramatic evolution,and the proposed time discretization is particularly effective for models with the smallα.Fi-nally,numerical comparison are provided with widely used L1-type methods,the CQ methods,and their corrected forms.In Chapter 4,a compact ADI method is established for the fractional sub-diffusion equation with Neumann boundary conditions.The time fractional derivative is approxi-mated by the L1 scheme on graded meshes,the spatial discretization is done by using the compact finite difference methods on uniform grids.By adding some corrected terms,the fully discrete ADI method is obtained.Convergence of the scheme is obtained under the assumptions of the weak singularity of solutions.With the obtained properties of the coefficient matrix and the recent discrete fractional Gr?nwall type inequality,the optimal error estimates of the methods are obtained.The convergence order reaches the 4th order in spatial direction.The scheme converts two-dimensional equation to two separate one-dimensional equations,it reduces the computational complexity.Nu-merical results are given to confirm the results.In Chapter 5,an effective compact ADI scheme is developed for solving the semi-linear parabolic problems with distributed delay.The Crank-Nicolson method and the extrapolation method are combined for the temporal discretization,the spatial dis-cretization is done by using the compact finite difference method,and the compound trapezoidal formula is used to calculate the integral term.By adding some corrected terms,the compact ADI scheme is obtained.Solvability,convergence and stability of the proposed linearized schemes are studied.The extension of numerical scheme to the three-dimensional case is presented.After that,the schemes are applied to simulate several biological models.It is shown that the obtained numerical schemes have better convergent results than the usual ADI method,without increasing extra computational cost.The numerical results fit well with the theoretical findings.In Chapter 6,the conclusions of the thesis and the future plans are presented.