Collocation Method And Theoretical Analysis For Fractional Partial Differential Equations

Fractional order partial differential equation can effectively describe the memory and transmissibility of many kinds of material, and plays an incresingly important role in physics, mathematics, biology, electrical engineering, mechanical engineering and finance and other fields. However, many analytical solutions for the fractional order partial differential equation are complicated, and analytic solutions of most fractional equations can not be obtained explicitly. This motivates us to develop effective numerical schemes for the fractional partial differential equations. In the present thesis, we study two collocation methods for solving the fractional order partial differential equations, that is, quasi-wavelet method and orthogonal splice collocation method. This thesis consist of five chapters. Especially, chapter2,3and4are the key of my thesis and the main contents include as follows.In chapter2, we propose the quasi-wavelet method for a class of time dependent fractional partial differential equations. The main idea of quasi-wavelet method is Mallat multiresolution analysis. As we all know, arbitrary wavelet subspace can be standardized by a group of orthogonal wavelet generation, this group of wavelet-base can be regularized by its own corresponding combination of orthogonal scaling func-tion to be normalized. However, standardization of the general orthogonal scaling function of the Fourier transform is not continuous. In order to improve this features, we have its regular treatment. This is the main idea of quasi-wavelet algorithm. In this chapter, we use the second order backward differentiation formula scheme for time discretization, and the quasi-wavelet method is used for spatial discretization. The stability and convergence properties related to the time discretization are dis-cussed and theoretically proven. Numerical examples are obtained to investigate the accuracy and efficiency of the proposed method. Moreover, we also present the dis-cretization formula of the integro-differential equation with two singular kernels and the corresponding numerical results.In chapter3, we formulate and analyze a novel numerical method for solving a time fractional Fokker-Planck equation which models an anomalous sub-diffusion process. In this method, orthogonal spline collocation is employed for the spatial discretization and the time-stepping is done using a backward Euler method based on the L1approximation to the Caputo derivative. The stability and convergence of the method are considered, and the theoretical results are supported by numerical examples, which also exhibit superconvergence.In chapter4, discrete-time orthogonal spline collocation (OSC) methods are p-resented for the two-dimensional fractional cable equation. The proposed scheme is based on the OSC method for space discretization and finite difference method for time, which is proved to be unconditionally stable and convergent with the order O(rmin(2-γ1,2-γ2)+(hr+1-k) in Hk-norm, k=0,1, where r,h and r are the time step size, space step size and polynomial degree, repectively, and γ1and γ2are two differ-ent exponents of fractional derivatives with0<γ1,γ2<1.Numerical experiments are presented to demonstrate the results of theoretical analysis and show the accuracy and effectiveness of the method described herein, and super-convergence phenomena at the partition nodes is also exhibited, which is a characteristic of the OSC methods, namely, the rates of convergence in the maximum norm at the partition nodes in ux and uy are approximately hr+1in our numerical experiment.3chart,18tables,154references.