Analysis Of Numerical Methods For Several Time-space Fractional Partial Differential Equations

Since the fractional differential operator is often used to describe materials andprocesses with memory and genetic characteristics, it is subjected to the concern ofmany mathematicians. A large number of fractional partial differential equationshave occurred since the fractional differential operator is widely used in variousfields. However, generally speaking, the analytical solutions of such equations is noteasy to obtain, so the study of the numerical solution becoming an interesting topicin computational mathematics. In this paper, we study how to solve time-spacefractional partial differential equations utilizing finite difference and finite elementmethods.On the one hand, we study how to use a finite difference method to solve thetime-space fractional supdiffusion equation with the initial and boundary valueconditions. In the aspect of time, Diethelm Fractional backward difference methodis used to discrete Caputo fractional derivative. Then we present a shifted Grünwaldalgorithm for space discretization. Consequently, we get the numerical scheme.Furthermore, the stability and convergence are discussed for the full-discretizationnumerical scheme, then the stability conditions and order of convergence estimatesare received. Finally, numerical examples are given to verify theoretical results ofthe method.On the other hand, using finite element method, we discuss the problem ofnumerical solution for generalized time-space fractional advection-diffusionequation under initial and boundary conditions. Firstly, we give the weakformulation in the framework of Sobolev space, then prove the existence anduniqueness of variational solutions using Lax-Milgram theorem. Furthermore,Diethelm fractional backward difference method is used to discrete the derivative intime aspect, for the spatial discretization, we use finite element method to prove theconvergence of the method, and give corresponding convergence order, and discussthe stability of the scheme. Meanwhile, numerical examples are given todemonstrate the conclusion.