Study Of Some High Order Numerical Methods For Fractional Differential Equations

Fractional differential equations can more precisely describe memory and inheritance characters of various materials and physical processes due to nonlocal properties of frac-tional operators.In recent decades,the subject of fractional differential equations have been applied extensively to basic science and engineering fields.On the other side,this nonlocal feature makes the mathematical analysis and design of numerical methods dif-ficult.First,it is usually hard to find the analytical solutions of fractional differential equations.Even though an explicit expression of the solution is known,most of them of-ten contain special functions or infinite series that are quite hard to calculate.Therefore,more and more researchers turn to investigate efficient numerical methods for this kind of equations.Our work is focused on the numerical computation of fractional differential equations.The outline of the dissertation is as follows:In Chapter 1,we review the brief history of fractional calculus and the recent progress on fractional diffusion equations.We also give research motivation of our dissertation and summarize the main contribution of this work.In Chapter 2,we propose and analyze a deferred correction method for the fractional differential equation of order ?.The proposed method is based on a well-known finite difference method of?2-??-order for prediction of the numerical solution,which is then corrected through a deferred correction method.In order to derive the convergence rate of the prediction-correction iteration,we first derive an error estimate for the?2-??-order finite difference method on some non-uniform meshes.Then the convergence rate of orders O(?^?2-???p+1?)and O(?^?2-??+p)of the overall scheme is demonstrated numerically for the uniform mesh and the Gauss-Lobatto mesh respectively,where ? is the maximal time step size and p is the number of correction steps.The performed numerical test confirms the efficiency of the proposed method.In Chapter 3,we first consider the numerical method that Lin and Xu proposed and analyzed in JCP 2007 for the time-fractional diffusion equation.It is a method basing on the combination of a finite different scheme in time and spectral method in space.The numerical analysis carried out in that paper showed that the scheme is of?2-??-order convergence in time and spectral accuracy in space for smooth solutions,where ? is the time-fractional derivative order.The main purpose of this chapter consists in refining the analysis and providing a sharper estimate for both time and space errors.Then the theoretical results are validated by a number of numerical tests.In Chapter 4,we cnsider a numerical method for the time-fractional diffusion equa-tion.The method makes use of a high order finite difference method to approximate the fractional derivative in time,resulting in a time stepping scheme for the time-fractional diffusion equation.Then the resulting equation is discretized in space by using a spectral method based on the Legendre polynomials.The main body of this chapter is devoted to carry out a rigorous analysis for the stability and convergence of the time stepping scheme.As a by-product and direct extension of our previous work,an error estimate for the spatial discretization is also provided.The key contribution of the chapter is the proof of the?3-??-order convergence of the time scheme,where ? is the order of the time-fractional derivative.Then the theoretical result is validated by a number of nu-merical tests.To the best of our knowledge,this is the first proof for the stability of the?3-??-order scheme for the time-fractional diffusion equation.