Sinc Method For Partial Integro-differential Equations With A Weakly Singular Kernel

The partial integro-differential equation of parabolic type is widely used in applications such as heat conduction in materials with memory,compression of visco-elastic media,nuclear reactor dynamics,population dynamics,etc.Many experts and scholars at home and abroad have studied the numerical so-lution of this kind of equation,such as J.M.Sanz-Serna[30],G.Fairweather[5],E.G.Yanik[16],K.Adoifsson[31]in overseas and Chuanmiao Chen[12],Da X-u[24],Chunaju Xu[32]in home.They use finite-element methods,finite-difference methods,orthogonal spline collocation methods,but few of them solve the partial integro-differential equation with a weakly singular kernel by Sine collocation methods.In this paper,the numerical solution of partial integro-differential equa-tions with a weakly singular kernel by Sine collocation method is mainly dis-cussed.The Sine collocation method is proposed to handle the spatial deriva-tives while the second-order backward difference method is used to discretize the temporal derivatives.This thesis is divided into six chapters.In chapter 1,we introduce the development of the partial integro-differential equation,the research status at home and abroad as well as the significance of research.The stability and error estimates of the time semi-discrete scheme of a class of partial integro-differential equations are proven in chapter 2.In chapter 3,we introduce the properties of Sine function and Sine numerical methods.The fourth-order partial integro-differential equation of boundless space is solved by Sine collocation method,and the corresponding numerical examples are given in chapter 4.In chapter 5,we use the Sine collocation method to solve the second order partial integro-differential equation of bounded space,and verify the validity and reliability of the proposed method to solve this equa-tion by numerical examples.This paper ends with a conclusion in chapter 6.