An adaptive Huber method with local error control, for the numerical solution of the first kind Abel integral equations

Abstract

In contrast to the existing plethora of adaptive numerical methods for differential and integro-differential equations, there seems to be a shortage of adaptive methods for purely integral equations with weakly singular kernels, such as the first kind Abel equation. In order to make up this deficiency, an adaptive procedure based on the product-integration method of Huber is developed in this work. In the procedure, an a posteriori estimate of the dominant expansion term of the local discretisation error at a given grid node is used to determine the size of the next integration step, in a way similar to the adaptive solvers for ordinary differential equations. Computational experiments indicate that in practice the control of the local errors is sufficient for bringing the true global errors down to the level of a prescribed error tolerance. The lower limit of the acceptable values of the error tolerance parameter depends on the interference of machine errors, and the quality of the approximations available for the method coefficients specific for a given kernel function.