Multistep Collocation Methods For Volterra Integral Equations Of The Third Kind

Volterra integral equations of the third-kind frequently arise in acoustic scattering,heat transfer models,population prediction models and so on.Generally,it is difficult to obtain analytical solutions of Volterra integral equations of the third-kind.Therefore,numerically solving the Volterra integral equation of the third-kind is a hot topic of computational mathematics.This thesis studies efficient solutions to the weakly singular and highly oscillatory Volterra integral equation of the third-kind via the collocation boundary value method.The main achievements are as follows:(1)a class of non-polynomial approximation methods for solving the weakly singular Volterra integral equation of the third-kind is studied.This method is particularly feasible for approximating functions in the form of tη with the real number η>0.The weakly singular Volterra integral equation of the third-kind is discretized into a linear system by designing a fractional collocation boundary value method on the graded mesh.Based on the weak singularity of solutions,the solvability of the system is studied.The convergence of fractional collocation boundary value method is analyzed by using Peano’s theorem and Gronwall’s inequality.(2)The collocation boundary value method for the weakly singular Volterra integral equation of the third-kind is developed via the graded mesh and the classical Lagrange interpolation.The solvability and convergence of this method are analyzed.(3)With the help of the collocation boundary value method and highly oscillatory numerical integration,the numerical solution of the highly oscillatory Volterra integral equation of the third-kind is studied.Theoretical and numerical results manifest the fractional collocation boundary value method and the collocation boundary value method with the graded mesh are able to efficiently and stably compute the numerical solution to the weakly singular Volterra integral equation of the third-kind,and the proposed collocation boundary value method has the property that the higher the oscillation,the better the computational accuracy.