High Order Finite Difference Methods And Spectral Methods For Fractional Or Tempered Fractional Differential Equations

Fractional derivatives are generalizations of the classical integer-order counterparts.Fractional operators can be used more accurately to describe the probability distribution of particles in space and in time.Therefore,they have became the most effective tools to describe the abnormal diffusion problems.Recently,the fractional calculus have been applied in many new areas,such as finance,hydrology,chaos synchronization,multi-directional turbination attractors,etc.The modeling progress on using fractional differential equations has led to increasing interest on numerical schemes for their solutions.While,unlike the classical counterparts,because of the nonlocal properties of fractional operators,obtaining the analytical solutions of the fractional differential equations is more challenging or sometimes even impossible;or the obtained analytical solutions are just expressed by transcendental functions or infinite series.So,efficiently solving the fractional problems naturally becomes an urgent topic.This thesis is composed of five chapters.Chapter 1 elaborately illustrates the background and current research status for solving the involved fractional problems as well as tempered fractional problems,and concisely introduced the main tasks and innovative points.In Chapter 2,based on the superconvergent approximation at some point(depending on the fractional order ,but not belonging to the mesh points)for Grünwald discretization to fractional derivative,we develop a series of high order quasi-compact schemes for space fractional diffusion equations.Because of the quasi-compactness of the derived schemes,no points beyond the domain are used for all of the high order schemes including second order,third order,fourth order,and even higher order schemes;moreover,the algebraic equations for all the high order schemes have completely a same matrix structure.The stability and convergence analysis for some typical schemes are made;the techniques of treating the nonhomogeneous derivatives at boundary point are introduced;and extensive numerical experiments are performed to confirm the theoretical analysis or verify the convergence orders.Chapter 3 introduces how to design high order finite difference methods on non-uniform meshes for space fractional operators.In the past decades,the finite difference methods for space fractional operators develop rapidly;to the best of our knowledge,all the existing finite difference schemes,including the first and high order ones,just work on uniform meshes.The nonlocal property of space fractional operator makes it difficult to design the finite difference scheme on nonuniform meshes.This chapter provides a basic strategy to derive the first and high order discretization schemes on non-uniform meshes for fractional operators.And the obtained first and second schemes on non-uniform meshes are used to solve space fractional diffusion equations.The error estimates and stability analysis are detailedly performed;and extensive numerical experiments confirm the theoretical analysis or verify the convergence orders.In fact,the above finite difference methods for Riemann-Liouville fractional problems can all be generalized into solving tempered fractional problems,without any difficulty.In Chapter 4,we discuss spectral methods for tempered fractional advection/diffusion problems by first introducing fractional integral spaces,which possess some features:(i)when 0 <? < 1,functions in these spaces are not required to be zero on the boundary;(ii)the tempered fractional operators are equivalent to the Riemann-Liouville operator in the sense of the norm.Spectral Galerkin and Petrov-Galerkin methods for tempered fractional advection problems and tempered fractional diffusion problems can be developed as the classical spectral Galerkin and Petrov-Galerkin methods.Error analysis is provided and numerically confirmed for the tempered fractional advection and diffusion problems.Chapter 5 contains a summary of the whole thesis and future research directions,which focus on the spectral methods for the tempered fractional advectiondiffusion problems.