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 β (z − x) −α g(y(x)), where α ∈ (0, 1), β > 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 L ∞ and the L 2 norms. Some numerical examples are presented to illustrate the exponential decay of the errors in the spectral approximation.

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