An Hp-Version Legendre-Jacobi Spectral Collocation Method For Volterra Integro-Differential Equations With Smooth And Weakly Singular Kernels*

Volterra integral differential equations have very extensive applications in many fields, for example, biology, population dynamics, control systems and finance, etc.. This kind of equations always arises in some time-dependent practical problems, it not only depends on current states, but also depends on history states or physical phenomena. In scientific and engineering calculations, many practical problems, such as heat conduction problems with memory materials, viscoelastic problems, heat transfer problems in nuclear reactors, etc., can all be boiled down to the problems on Volterra integral differential equations. To compare with the traditional ordinary differential equations and partial differential equations, there exist some essential differences and difficulties in solving Volterra integral differential equations, due to its memory characteristic. Therefore, how to solve Volterra integral differential equations fastly and efficiently is becoming the focus in the field.As an important numerical method for differential equations, spectral method has been widely used in numerical simulations of scientific and engineering problems. Because of its high accu-racy and global properties, spectral method has been gradually applied to numerical simulations of integral and integral differential equations in recent years. However, the existing spectral meth-ods for Volterra integral differential equations with weakly singular kernel are mainly based on single interval schemes, they are not suitable for the singular problems or long-time numerical simulations. This paper is devoted to develop an hp-version Legendre-Jacobi spectral collocation method for Volterra integro-differential equations with smooth and weakly singular kernels. We shall establish several new approximation results of the Legendre/Jacobi polynomial interpola-tions for both smooth and singular functions. As applications of these approximation results, we derive hp-version error bounds of the Legendre-Jacobi collocation method under the H1-norm for the Volterra integro-differential equations with smooth and singular solutions. Numerical experi-ments are included to illustrate the theoretical results.The dissertation consists of the following parts:In Chapter Ⅰ, we briefly introduce some existing numerical methods for Volterra integral dif-ferential equations with weakly singular kernel and the motivation of this study.In Chapter Ⅱ, we introduce some basic properties of the shifted Legendre/Jacobi polynomial interpolations and propose the hp-version Legendre-Jacobi spectral collocation method for the Volterra integral differential equations with weakly singular kernel.In Chapter Ⅲ, We establish several new approximation results of the Legendre/Jacobi poly-nomial interpolations for both smooth and singular functions.In Chapter Ⅳ, we derive hp-version error bounds of the Legendre-Jacobi collocation method under the H1-norm for the Volterra integral-differential equations with smooth and singular solu-tions.In Chapter Ⅴ, we present some numerical results to demonstrate its high accuracy.In Chapter Ⅵ, we give some concluding remarks.