Legendre Wavelets And Orthogonal Spline Collocation Method For Fractional Partial Differential Equations

As a generalization of partial differential equations with integer-order,frac-tional partial differential equations can effectively describe various materials and physical processes with memory and genetic properties.It has been widely used in many fields such as biology,material science,chemical kinetics,elec-tromagnetics,transmission and diffusion,automatic control and so on.Since the analytical solutions of fractional partial differential equations are usual-ly difficult to obtain,it has become an urgent and important research topic to solve these equations by numerical methods and attracted more and more attention from scholars.In this paper,two feasible and effective collocation methods are applied to numerically solve several types of fractional partial d-ifferential equations,such as time-fractional telegram equations and fractional reaction-diffusion equations,which are common in engineering fieldsThis paper mainly studies the applications of Legendre wavelet collocation method and orthogonal spline collocation method in seeking numerical solution of three different kinds of time-fractional partial differential equations.Chapter 1 briefly introduces the research background,significance and basic knowledge of fractional calculus.Chapters 2,3 and 4 are the main contents of this paper.which are the main research work during authors's studies towards the doctoral degree.Chapter 5 is the summary and prospectIn Chapter 2?Legendre wavelet collocation method is proposed for solv-ing a class of fourth-order partial integro-differential equations?PIDEs?with a weakly singular kernel under three different boundary conditions:compact boundary,simple support boundary and transverse support boundary.This type of equation is essentially a time-fractional partial differential equation.In this method,second-order backward difference scheme is used to discretize the integer-order time derivative,the L1 formula of Caputo derivative is used to approximate the integral term,and the Legendre wavelet collocation method is applied for the spatial directional derivative.We rigorously prove the sta-bility and convergence of the semi-discrete scheme,and give some numerical examples to verify the feasibility and validity of the proposed algorithm.Based on the Chapter 2,Chapter 3 applies Legendre wavelet basis func-tions to approximate the temporal and spatial directional derivatives.We consider the time-fractional telegraph equation with two types of initial con-ditions and Dirichlet boundary conditions,and transform the problem into linear algebraic equation by collocation method.Moreover,the convergence and error estimation of the algorithm are given.The validity of the proposed algorithm is verified by comparing with the results in literatures.In Chapter 4,we consider the two-dimensional time-fractional reaction-diffusion equation,and approximate the time fractional derivatives of order??0<?<1?by using the weighted and shifted Gr???nwald difference?WSGD?operator with third-order accuracy.The spatial directional derivative is dis-cretized by the orthogonal spline collocation method.We give the stability and convergence analysis of fully discrete schemes,of which the convergence order is O(?³+h^r+1).Several numerical examples with one-and two-dimensional variables are presented to validate our theoretical analysis.Also a numerical example with Neumann boundary condition is given.From the numerical ex-amples,it can be found that the proposed method in this paper is effective and the numerical results are in agreement with the theoretical results.