Finite Difference Methods For Two Kinds Of The Space Fractional Partial Differential Equations

Fractional partial differential equations is the generalization and extension of traditional integral partial differential equations,which are widely used in simulation engineering,mechanical engineering,electronic engineering,physics,chemistry,biology and other scientific fields and have attracted more and more attention from scholars at home and abroad.In general,most fractional partial differential equations are difficult to give explicit analytic solutions,such as fractional partial differential equations with variable coefficients.Therefore,it is very important to study the numerical solution of fractional partial differential equations.In this dissertation,we study the practical numerical methods to solve the fractional advection-dispersion equation with Robin boundary condition and the fractional diffusion equation with the fractional boundary condition,We propose a finite difference scheme based on the Grünwald formula and the shifted Grünwald formula to discrete Riemann-Liouville fractional derivative.We use the classical Grünwald formula,the shifted Grünwald formula and the weighted Grünwald formula to discretize Riemann-Liouville fractional derivative and solve two kinds of the space fractional partial differential equations with the finite difference methods.In this dissertation,we mainly study the finite difference methods for two kinds of the space fractional partial differential equations.This dissertation consists four chapters.In Chapter 1,we introduce the research background and the current research situation,which are related to the numerical solution of the fractional partial differential equations,and summarize the work done by predecessors,expound the main work and give the preparatory knowledge to be used in this dissertation.In Chapter 2,we study the practical numerical methods to solve the fractional advection-dispersion equation with Robin boundary condition.We propose an implicit finite difference scheme based on the shifted Grünwald formula to discrete Riemann-Liouville fractional derivative.Existence and uniqueness of numerical solutions are derived.It is proved that the implicit finite difference scheme is unconditionally stable and convergent.Finally,numerical simulations show that the method is efficient.In Chapter 3,an implicit finite difference method based on a class of second order approximations which called the weighted and shifted Grünwald-Letnikov difference operators is developed for the fractional diffusion equation with the fractional boundary condition.When (?),the stability and convergence are proven and the convergence order is (?).Finally,numerical examples are performed to confirm the accuracy and efficiency of our method.In Chapter 4,we summarize the main works and shortcomings of this dissertation,and prospect the future research.