Numerical Methods And Fast Implementation For Fractional Diffusion Equations And Fractional Sine-Gordon Equations

A large number of studies have shown that fractional differential operators with nonlocal properties are very suitable for describing materials with memory and genetic properties.Therefore,in recent years,fractional differential equation-s have been widely concerned and applied.However,the analytical solutions of many fractional differential equations are difficult to obtain,so numerical simu-lation has become an important method to study fractional differential equations in practical applications.This dissertation is devoted to the study of efficient numerical schemes and fast algorithms for two-dimensional Riesz space fractional diffusion equations and fractional Sine-Gordon equations.In Chapter 2,the ADI-CN method is applied to reduce the two-dimensional Riesz space fractional diffusion equation into a series of independent one-dimensional problems.Based on the fact that the coefficient matrices of these one-dimensional problems are all real symmetric positive definite Toeplitz matrices,a fast method is developed for the implementation of the numerical scheme.Finally,some nu-merical results are used to verify the correctness of the theoretical analysis and the effectiveness of the fast algorithm in the implementation of the numerical scheme.In Chapter 3,a quasi-compact difference scheme for two-dimensional Riesz space fractional diffusion equation is introduced.By using some new techniques,it is proved that the numerical scheme is convergent with order O(?~2+h_x~4+h_y~4) in the discrete L?-norm.Based on the special form of coefficient matrix of the difference equation system,we propose a new fast algorithm to accelerate the implementation of the difference scheme.Finally,some numerical examples are provided to verify the correctness of the theoretical results and the effectiveness of the fast algorithm in the implementation of the difference scheme.In Chapter 4,a conservative implicit difference scheme is proposed for one-dimensional Riesz space fractional Sine-Gordon equation.And a rigorous theoret-ical analysis of the numerical scheme is carried out.To reduce the computational cost,we introduce a revised Newton method to accelerate the implementation of the difference scheme.Finally,some numerical experiments are carried out to ver-ify the validity of the difference scheme and the effectiveness of the revise Newton method.In Chapter 5,the difference scheme developed in Chapter 4 is extend to solve two-dimensional Riesz space fractional Sine-Gordon equation.The theoretical analysis results of the difference scheme are given.To reduce the computational cost,we adopta fast algorithm to accelerate the implementation of the difference scheme.In Chapter 6,in order to avoid solving the nonlinear equations iteratively,a linearized difference scheme is proposed for time fractional Sine-Gordon equation.And a rigorous theoretical analysis of the numerical scheme is carried out.Finally,some numerical examples are provided to verify the correctness of the theoretical results.