A Spectral Method For A Semi-linear Fractional Convection Diffusion Equations

Fractional partial differential equations are highly attentional in the last three decades, they can be used to describe the anomalous phenomenon, different from the Brownian motion. Semi-Linear time fractional convection diffusion equations(TFCDEs) describe some anomaly phenomena, such as porous diffusion, viscous convection, solid-liquid and solid-gas diffusion, and gas-liquid diffusion. In this paper, we propose a new space-time spectral Galerkin method for finding the numerical solution. This paper is made by four chapters.In chapter 1, we introduce the origin of the fractional order calculus, the developing history of fractional partial differential equations, the domestic and international research and the main part we study in this paper. As a new subject, the first chapter introduces the main research contents.In chapter 2, we prove the existence and uniqueness of this class of semi-linear TFCDEs with suitable initial-boundary conditions we study. At first we introduce the relationship between the fractional Sobolev space as well as the corresponding norm, then we lead in some lemmas in order to make calculation convenient. Finally we prove the existence and uniqueness of this problem by Galerkin method.In chapter 3, we verify the convergence of this new space-time spectral Galerkin method, construct the numerical format and calculate the numerical result. We firstly calculate error estimate by semi discrete scheme, and for the special f, we get the error estimate by fully discrete scheme. For the time direction with fractional order, we use standard space interpolation technique to calculate the error, and get error estimates of the space-time spectral Galerkin method in the whole domain. Based on the theory of error analysis, we finally give a numerical format constructed by orthogonal polyamines in spectral spaces, and get the solution of the problem via the variational formulation.In chapter 4, we offer some numerical results to verify the convergence of this new space-time spectral Galerkin method. Depending on the relationship with the convergence of the new space-time spectral Galerkin method and the regularity of the solution, we give an infinite smooth solution and the solution with a certain regularity respectively, and verify the convergence order we proved in chapter 3.