Asymptotic Expansions And Integration Method With Piecewise Linear Interpolation For Weakly Singular Volterra Integral Equation Of The First Kind

Integral equation,as an important branch of modern mathematics,is widely used in mathematics,physics,and mechanics.Singular integral equation is a kind of integral equation whose analytic solution is difficult to obtain,so it is very important to study its numerical algorithms.In this paper,the first kind linear weakly singular Volterra integral equation is studied.Because the solution may have a singularity at the origin,the accuracy may be deteriorated when the equation is solved by standard numerical methods.First,the asymptotic expansion of the solution at the origin is obtained by using the Laplace transform and undetermined coefficient method.Second,this expansion is used to separate the singularities,and then the singular integral equation is transformed into a regular integral equation with perturbation.Finally,an integration method with piecewise linear interpolation is designed to solve the equation on the regular interval.By separating the singularity of the solution,the convergence order of the approximate solution does not decrease,so the method is called singularity separated piecewise linear interpolation method.Chapter 1 briefly describes the research background,summarizes the research progress of numerical methods for the Volterra integral equations of the first kind,and then outlines the research objectives and arrangements of this paper.In chapter 2,the preliminary knowledge is introduced,including the existence and uniqueness,regularity,stability of solutions of Abel integral equations,asymptotic expansions of Laplace transform and its inversion,as well as Pade approximation.Chapter 3 studies the singular expansions of the solution at the origin and infinity(algebraic singularity only)for the first kind convolution Volterra integral equation with weakly singular algebraic and logarithmic kernel.The Laplace transform is used to obtain the asymptotic expansions of the solution at the origin and infinity.The asymptotic expansion of the solution at the origin has high precision when the independent variable becomes small,while the asymptotic expansion of the solution at infinity becomes more and more accurate as the independent variable tends to larger.Numerical examples confirm the effectiveness of the proposed method.In chapter 4,we study the generalized Abel integral equation.First,the asymptotic expansion of the solution at the origin is generated by designing a method of undetermined coefficients.Then,this expansion is used to construct a piecewise linear interpolation method by separating the singularity to obtain the numerical solution on the regular interval.The detailed construction process of the computational scheme is given and the convergence analysis is conducted to show that the computational scheme has second order accuracy by choosing a suitable point to splitting the interval.