Collocation Methods And Compact Difference Methods For Fractional Partial Integral Differential Equations

Fractional partial integral differential equations have become more and more important in modeling of many actual phenomenon,such as,physics,chemistry,biology,finance,material mechanics,environmental science.Such equations also have weakly singular term,analytical solutions can not be ob-tained explicitly.This motivates us to develop effective numerical methods for fractional partial differential equations.In the present paper,we study orthogonal spline collocation(OSC)method,quasi-wavelet method and compact difference method for three different types of fractional partial differential equations.Firstly,we present OSC method for the multi-term time-fractional sub-diffusion equation.Secondly,quasi-wavelet numerical method is developed for solving the variable-order fractional advection-diffusion equation.Finally,compact difference scheme is formulated and analysed for two-dimensional parabolic integro-differential equation.This thesis consists of five chapters.We introduce some preliminary knowledge in the first chapter.Especially,chapter 2,3 and 4 are the key of my paper.The popularity of OSC method is due in part to its conceptual simplicity,wide applicability and ease of implementation.Another attractive feature of OSC method is their super-convergence.In chapter 2,we propose the OSC method for the two dimension multi-term time-fractional sub-diffusion equa-tion.The OSC method is used for in space,and a finite difference method in time.The stability and convergence analysis are provided.The numeri-cal examples for one and two dimensional problems support our theoretical analysis.We know that the wavelet function is a finite energy function with good local characteristics.The wavelet method can well analyze the local features of the function.In chapter 3,we formulate the quasi-wavelet numerical scheme for solving the space variable-order fractional advection-diffusion equation.Quasi-wavelet numerical method is used for the spatial discretization,for the time stepping,Euler method is considered.Integral term approximated by the compound trapezoidal formula.In order to relax restriction on the time step,we propose double quasi-wavelet numerical scheme in space,Euler method is still considered in time.Discrete formats are given in detail.Some numerical experiments are provided to demonstrate the effectiveness of the numerical schemes.The attraction of alternating direction implicit(ADI)scheme is that they reduce a multidimensional problem to sets of independent one-dimensional problems,thus reducing the computational cost.The coefficient matrix of the compact difference format is the tridiagonal matrix,it is easy to solve by the Thomas algorithm.There is a fourth-order accuracy in space.In chapter 4,compact difference approach for spatial discretization and ADI method in time for two dimensional fractional evolution equation.Integral term approx-imated with second-order fractional quadrature rule.The L2 stability and convergence are derived.Numerical examples with known exact solution are given to support the theoretical results.