The Jacobi Collocation Method for a Class of Nonlinear Volterra Integral Equations with Weakly Singular Kernel

Abstract

A Jacobi spectral collocation method is proposed for the solution of a class of nonlinear Volterra integral equations with a kernel of the general form $x^{\beta }\, (z-x)^{-\alpha } \, g(y(x))$, where $\alpha \in (0,1), \beta >0$ and g(y) is a nonlinear function. Typically, the kernel will contain both an Abel-type and an end point singularity. The solution to these equations will in general have a nonsmooth behaviour which causes a drop in the global convergence orders of numerical methods with uniform meshes. In the considered approach a transformation of the independent variable is first introduced in order to obtain a new equation with a smoother solution. The Jacobi collocation method is then applied to the transformed equation and a complete convergence analysis of the method is carried out for the $\displaystyle L^{\infty }$ and the $L^2$ norms. Some numerical examples are presented to illustrate the exponential decay of the errors in the spectral approximation.