| Integral equations arise in many scientific and engineering problems.A large class of initial and boundary value problems can be converted to integral equations.There are many classes of integral equations,which must be solved by different methods for different type equations.In this paper,we consider the first kind and second kind of weakly singular Volterra equations and design high accuracy algorithms on condition that the solutions have very low regularity.The tool is the Puiseux expansion of a function,which is an extension to Taylor’s series.The Puiseux expansion can accurately demonstrate the property of a function at its singularity.Based on the Puiseux expansion and the highly efficient algorithms of singular integral,we obtain some high accuracy algorithms for Volterra integral equations by modifying the existing methods.The paper includes five chapters.In chapter one,we first briefly introduce the development of Volterra integral equations,then review the various methods for solving the sigular Volterra integral equation.At the end of this chapter,the main objective and schedule of this paper are outlined.In chapter two,some preliminaries are provided,including the Laplace transform,the Laplace inversion,the Puiseux series expansion,the composite Gauss-Legendre quadrature formula,which play a fundamental role in solving weakly singular Volterra integral equations.In chapter three,Abel integral equation is considered.Firstly,integral form of the exact so-lution based on Laplace transform and Laplace inverse transform is given.Secondly,two different Puiseux series solutions are obtained by assuming that the function has two kinds of Puiseux expan-sions.Finally,under weak regularity conditions,high-precision numerical solution for Abel integral equation is obtained by using singular integral quadrature.Numerical examples demonstrate that the method has double-precision accuracy.The fourth chapter studies the second kind of weakly linear singular Volterra integral equa-tions.Usually we can get the image function of solutions by Laplace transform.By expanding the image function in Puiseux series at infinity,we design a new type of Laplace inverse algorithm,which can accurately solve the linear Volterra integral equations.Numerical examples verify the effectiveness of the method.In chapter five,Puiseux series expansion method is studied for nonlinear weakly singular Volterra integral equations.By iterating the Puiseux expansion,we obtain a finite term truncation of the Puiseux series at zero point,which is the approximate solution of the equation.The Puiseux expansion has two roles,one is to possibly get the true solution by analyzing the expansion and the other is constructing effective algorithms for nonlinear Volterra equations by combining numerical integration methods.Examples show the method is feasible. |
