Finite Element And Finite Difference Methods For Several Nonlinear Fractional Partial Differential Equations

In the past few years,fractional calculus and associated fractional differential equations have been powerful tools to simulate phenomena in multiple physical fields.They are also hot topics in the field of science and engineering.This doctoral dissertation focuses on the numerical methods of nonlinear fractional differential equations as well as their convergence analysis and conservation laws.According to the type of equations,the dissertation contains the finite element method for solving time nonlinear fractional differential equations,the finite element method for solving space fractional differential equation and the finite difference method for this kind of equations.The whole dissertation contains the following six parts:In Chapter 1,we introduce the development of fractional calculus and fractional differential equations and their applications in science and engineering.At the same time,we also introduce the background and development of finite difference and finite element methods for fractional differential equations.Moreover,we give the research status of numerically solving three kinds of fractional differential equations considered in this dissertation.Motivation and main content of this dissertation are also included.In Chapter 2,we recall some definitions of fractional derivatives and their basic properties.Fractional derivative space,fractional Sobolev space and some useful lemmas are also included in this chapter.In Chapter 3,we study the nonlinear time fractional Klein-Gordon equation with a cubic nonlinear term.Based on a weighted combination of L2-1? formula presented by Lyu and Vong(2018)in time,Galerkin finite element method in space,and an explicit treatment to the nonlinear term,a linear implicit discrete scheme is constructed.For the reason that the nonlinear term does not satisfy the global Lipschitz condition and is explicit in the scheme,it is difficult to obtain the bound estimate of the numerical solution alone.Combined the technique of estimating the error and bound of the numerical solution synchronously and the basic idea of energy method,the convergence analysis of the proposed scheme is accomplished.The scheme has second order accuracy in time,which performs more efficient than those fully implicit and linearized methods equipped with L1 formula.In Chapter 4,we will consider the nonlinear coupled spatial fractional Schrodinger equation.By employing Galerkin finite element method for special discretization and CrankNicolson(CN)difference scheme to discrete time derivative,we propose a fully discrete scheme.We strictly analyze the conservation law of the system,where the definitions of discrete mass and energy in the conservation law are analogs of those in the underlying system.Subsequently,the unique solvability of the system is proved.We also show the unconditional convergence of the scheme,i.e.,the error analysis is completed under the condition of no temporal and spatial grid ratio limitation.Both the L2-norm and L?-norm error estimates of the proposed scheme are given.The theoretical results are illustrated by numerical examples at last.In Chapter 5,we study the spatial fractional Zakharov equation.By utilizing the fractional central difference approximation to discrete the fractional Laplacian,central difference method for integer order derivatives,and using an explicit discrete method for the second part of the coupled system,we develop a semi-explicit fully discrete finite difference scheme.The technique proposed by Chang et al(1995)provides us an idea to prove the conservation law of the semi-explicit scheme.Moreover,the boundedness of the numerical solution is proved.At the same time,we combine this technique with the basic idea of energy method to derive the error analysis.The conservation law and error estimate of the scheme are also illustrated by some numerical examples.In Chapter 6,we summarize the whole dissertation and give the future work.