Numerical Solutions Of Fractional Partial Differential Equations—Analyses And Algorithms

Fractional partial differential equations are an effective method of describing anomalous diffusion phenomena.Based on different application backgrounds,scholars have proposed different forms of fractional derivatives and fractional models.Since there are no available analytical solutions for fractional problems in general,numerical methods play a decisive role in the actual verification and application of these models.Unlike the integer-order derivatives,fractional derivatives have(weak)singularity,non-locality,and sometimes even involve space-time coupling,which bring many difficulties and challenges to the design,analysis,and implementation of numerical methods.Therefore,although the numerical solutions of fractional partial differential equations have received a lot of attention from researchers and have made a lot of progress in recent decades,it is still a dynamic discipline and there are many problems that need to be resolved.In this paper,we will discuss numerical solutions for some specific models.This paper has the following chapters:In Chapter 1,we outline the research backgrounds and main contents of the dissertation,including the concept of anomalous diffusion,several important fractional derivative models,the research status of numerical solution of fractional partial differential equations,and the research contents and innovations of this paper,etc.In Chapter 2,we discuss the Riesz-based Galerkin methods for the tempered fractional Laplacian equation.The tempered fractional Laplacian operator is a derivative given in the form of integral and a generalization of the fractional Laplacian operator.In statistics,it represents the generator of the symmetric tempered -stable L?evy process,and has an important application in solving the first mean exit time and escape probability of the particles.This chapter will introduce three aspects of work,namely,the definition of Galerkin weak solutions for tempered fractional Laplacian equations and the well-posedness analysis of corresponding variational forms,the error estimation under the Riesz basis and the effective numerical implementation,and the homogeneity of non-homogeneous boundary conditions.In chapter 3,we present the finite difference schemes to solve tempered fractional Laplacian equation,including the constructions of the difference schemes and the preconditioning of the corresponding discrete systems.Our difference schemes can be used to deal with general nonhomogeneous boundary conditions,and their convergence rates depend on the regularity of the exact solution on ? rather than the regularity on the whole line R.In Chapter 4,we develop variational schemes for partial differential equations with the integral-differential tempered fractional derivatives.This class of fractional models is given in [9,17,144],which is a generalization of the corresponding Remiann-Liouville fractional models.The current numerical methods for solving them are the finite difference methods,which have strict stability analyses and convergence estimations.Other methods are mainly focused on some numerical simulations.In this chapter,we first give some variational properties of the tempered fractional derivatives,and then establish the Galerkin and Petrov-Galerkin finite element schemes for a given model,including the rigorous theoretical analyses and the effective numerical implementations.In Chapter 5,we investigate the contour integral or rational approximation schemes of the subdiffusion equation.Contour integrals or rational approximations have a very wide range of applications in solving classical parabolic equations.However,there are few literatures that use them to deal with the time fractional partial differential equations.In fact,the nonlocal property of the time fractional derivative makes the application of this kind of methods appear more natural and has a greater advantage.This chapter first gives the convergence and stability analysis of the model's finite element spatial semi-discrete scheme,and then provides three contour integral or rational approximation schemes for dealing with the time fractional derivative,and gives solutions to the specific problems encountered in each case.These algorithms solve the problem of excessive calculation and storage capacity for long-time computation of the time fractional partial differential equations.In Chapter 6,numerical approximation schemes of the tempered fractional FeynmanKac equation are given.This model is presented in [179] and is called the backward tempered fractional Feynman-Kac equation,which governing the functional distribution of particles' s trajectories.There are two difficulties in dealing with this model.The first is that the time fractional derivative of the equation is time-space coupled,and the second is that the symbol in front of the response term is contrary to the models we usually encounter.In this chapter,we first give a reasonable approximation for the time fractional derivative and the reaction term.Then we prove the stability and convergence of the fully discrete schemes under the time-space coupled norms.Finally,in Chapter 7,we give the summary of this article and our future work.