A Hybrid Collocation Method for Volterra Integral Equations with Weakly Singular Kernels

Abstract

The commonly used graded piecewise polynomial collocation method for weakly singular Volterra integral equations may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors. The singularity preserving (nonpolynomial) collocation method is known to have only local convergence. To overcome the shortcoming of these well-known methods, we introduce a hybrid collocation method for solving Volterra integral equations of the second kind with weakly singular kernels. In this hybrid method we combine a singularity preserving (nonpolynomial) collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain. We prove the optimal order of global convergence for this method. The convergence analysis of this method is based on a singularity expansion of the exact solution of the equations. We prove that the solutions of such equations can be decomposed into two parts, with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function. A numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method.