Finite Element Methods For Some Types Of Fractional Partial Differential Equations

Thanks to the nonlocal characteristic of the fractional derivative,the models involving fractional derivatives are very suitable to describe materials and physics process.In the past twenty years,fractional partial differential equations(FPDEs),as the extension of the integer ones,have been applied widely to many areas,such as highenergy physics,viscoelastic me-chanics,system control and finance,etc.However,it is always difficult to derive the analytical solutions to FPDEs.Even if we can find out the analytical solutions,they always contain some special functions,such as the Mittag-Leffler function,Wright function,H function and Hypergeometric function.Therefore,many scholars have paid more and more attentions to find the numerical.solutions of the FPDEs.This paper focuses on numerically solving some types of FPDEs by finite element methods,including the multi-term time fractional diffu-sion equation in higher spatial dimensions,two-dimensional nonlinear fractional diffusion-wave equation,nonlinear space fractional Schrodinger equation and nonlinear space fractional Ginzburg-Landau equation.The whole dissertation contains the following six parts:In Chapter ?,we briefly introduce some development history,applicable background and current research states of the fractional calculus.Meanwhile,we introduce the current states of using finite element methods to numerically solve the fractional partial differential equations.Finally,we give the research motlivation and the work summary of this paper.In Chapter ?,we introduce some types of the definitions and properties of fractional derivatives,and then present the fractional derivative spaces in 1D and 2D spaces.In Chapter ?,we mainly discuss a Galerkin finite element method for higher dimensional multi-term time fractional diffusion equation on non-uniform meshes,and prove the stability and convergence of the scheme.The validity of the scheme is verified by both theoretical and numerical results.In Chapter ?,we study a class of two-dimensional nonlinear time-space fractional diffusion-wave equations.To overcome the difficulty of higher dimensional problem,we de?couple in time direction and then construct a Crank-Nicolson ADI Galerkin finite element scheme.We prove the stability and convergence of the scheme,and then check the theoretical results by some numerical examples.In Chapter ?,we investigate the nonlinear space fractional Schrodinger equations.We construct the semi-discrete and fully discrete schemes which satisfy both the mass and en-ergy conservation.Then,we prove the solvability,conservation and convergence results of the schemes.In the numerical simulation,we verify the correctness of the theory and the feasibility of the numerical method.In Chapter ?,we mainly consider an implicit midpoint Galerkin finite element method for numerically solving nonlinear space fractional Ginzburg-Landau equations.We show the boundedness,solvability 'and unconditional L2-norm error estimates of the scheme.Finally,numerical tests are given to verify the the correctness of theoretical analysis.