An efficient algorithm based on the Pseudospectral method for solving Abel's integral equation using Hermite cubic spline scaling bases

Abstract

This research is dedicated to the analysis and construction of an efficient method based on the Pseudospectral method for the second type of Abel's integral equation using Hermite cubic spline scaling bases (HCSSb). After expressing the Abel operator based on the HCSSb, the equation will be reduced to a system of linear or nonlinear algebraic equations. To solve this system, we apply Newton's method and the generalized minimal residual (GMRES) method for the nonlinear and linear types, respectively. The results demonstrate that the speed of convergence for this method is O(2−J) where J∈N0. One of the advantages of the method is that it is not too complicated in form, and it can be used to solve a variety of problems. Furthermore, the method is computationally desirable and gives very accurate results.