Initialisation of the adaptive Huber method for solving the first kind Abel integral equation

Abstract

In the previous work of this author (Bieniasz in Computing 83:25–39, 2008) an adaptive numerical method for solving the first kind Abel integral equation was described. It was assumed that the starting value of the solution was known and equal zero. This is a frequent situation in some applications of the Abel equation (for example in electrochemistry), but in general the starting solution value is unknown and non-zero. The presently described extension of the method allows one to automatically determine both the starting solution value and the estimate of its discretisation error. This enables an adaptive adjustment of the first integration step, to achieve a pre-defined accuracy of the starting solution. The procedure works most satisfactorily in cases when the solution possesses all, or at least several of the lowest, derivatives at the initial value of the independent variable. Otherwise, a discrepancy between the true and estimated errors of the starting solution value may occur. In such cases one may either start integration with as small step as possible, or use a smaller error tolerance at the first step.