| Singular integral equations are widely used in many scientific and engineering problems,such as mathematical physics,fluid flow,and so on.It s necessary to solve the integral equations for these practical problems,but it’s difficult to give the exact solutions of integral equations in most cases.Hence,we need to design numerical algorithms to approximate the solutions of the integral equations.In this paper we consider the second kind weakly singular linear and nonlinear Volterra integral equations,which have the typical features that the solutions are singular about derivative at zero.We aim to derive the Puiseux expansion of solution at zero.from which we can accurately demonstrate the behavior of solution at its singularity.Based on the Puisuex expansion,we design efficient algorithms.The paper includes four chapters.In chapter 1,we review the development of the Volterra integral equation.The research advances about linear and nonlinear Volterra integral equation are introduced.The objective and the outline of the paper are also stated.In chapter 2,some preliminaries are provided,including the Puiseux expansion of a function at it s singularity,Laplace transform and its inversion,the modified trapezoidal rule and the existence and uniqueness theorem of nonlinear Volterra integral equation.In chapter 3,we study the second kind linear weakly singular Volterra integral equation.Since the equation involves a convolution kernel.we solve it by Laplace transform.Supposed that the forcing term has Puiseux expansion at zero.we obtain the Puiseux expansion of solution at zero by using inverse Laplace transform.Since the expansion of the solution is very complicated,we only discuss specific forms of the Puiseux expansion via the asymptotic expansion of the forcing term for four simple cases.which explicitly illustrate the relationships between the solution and the forcing term.Because the series expansion has only high accuracy near the zero.we obtain the numerical solutions of the equation by using Laplace inversion formula.We also extend the algorithms to Volterra intcgro-diffcreiitial equations and system of Volterra integral equations in this chapter.At the last of this chapter.some examples are provided to demonstrate that the algorithm is highly accurate.In chapter 4.the nonlinear weakly singular Volterra integral equation of the second kind is considered.We derive the general truncated Puiseux expansion of the solution at its singularity by Picard iteration and series expansion,and design modified trapezoidal integration methods for the equation by using the Puiseux expansion.By extending the Grownwall inequality,the error estimate is conducted and the convergence of the scheme is proved.Numerical examples verify the correctness and efficiency of the methods. |
