Application of Chebyshev polynomials for solving Abel’s integral equations of the first and second kind

A numerical implementation of an expansion method is developed for solving Abel’s integral equations. The solution of such equations may demonstrate a singular behaviour in the neighbourhood of the initial point of the interval of integration. The suggested method is based on the use of Taylor series expansion to overcome the singularity which leads to approximating the unknown function and it’s derivatives in terms of Chebyshev polynomials of the first kind. Some numerical examples are included to clarify the accuracy and applicability of the presented method which indicate that proposed method is computationally very attractive.