Singularity Separation Chebyshev Collocation Method For Nonlinear Weakly Singular Volterra Integral Equations Of The Second Kind

Singular integral equations are widely used in natural science and engineering problems,such as mathematics,physics,fluid mechanics.Since it is difficult to find the analytical solution of the integral equation,it is of great practical significance to study its numerical algorithm.This paper studies the nonlinear algebra and logarithmic weakly singular Volterra integral equation of the second kind.The typical feature of this equation is that the derivative of the solution is singular at the initial point,which makes the standard numerical methods have very low accuracy.In this paper,the singular integral equation is transformed into a regular equation by using the psi series expansion of the solution at origin,and then Chebyshev collocation method is designed to solve this kind of equation in the regular interval.We call this method as singularity separation Chebyshev collocation method.In chapter 1,the development of Volterra integral equation is briefly introduced,and the advances of spectral method and singular Volterra integral equation are introduced.The research objective and outline of the paper are also given.In chapter 2,the preliminary knowledge about the psi series expansion of the solution for weakly singular Volterra integral equation about origin is introduced.The properties of Chebyshev polynomial and Chebyshev interpolation,and some lemmas to be used in convergence analysis are also included in this chapter.In chapter 3,the singularity separation Chebyshev collocation method is developed for solving nonlinear weakly singular Volterra integral equations.The derivation of the computationd scheme is discussed in detail.The convergence analysis is conducted for three cases,which are linear equation with algebraic singularity,nonlinear equation with algebraic singularity and nonlinear equation with algebraic and logarithmic singularity.The analysis shows the numerical solution is convergent to the accurate one with respect to maximum norm.In chapter 4,numerical examples are provided to confirm that Chebyshev collocation method with singularity separation has high computational accuracy for solving the second kind of linear and nonlinear weakly singular Volterra integral equations,and the results are also compared with those obtained by directly applying collocation method,which shows that the algorithm in this paper has obvious advantages for the calculation of these kinds of singular problems.In chapter 5,we study the asymptotic expansion of the solution at infinity for a class of weakly singular Volterra integral equations in chemical reaction kinetics,which can accurately describe the long term behavior of the solution to the equation.When this expansion is combined with the singularity separation Chebyshev collocation method,we can obtain high-precision approximate solutions in semi-infinite interval for these kinds of singular problems.