Collocation method for Generalized Abel's integral equations

The purpose of this paper is to present an approximate method for solving the Generalized Abel's integral equations. The approximate method is based on the collocation method for solving Volterra integral equations. Generalized Abel's integral equations could be considered as a more general form of Volterra integral equations. Collocation method in sense of Atkinson's approach (Atkinson, 2016) is applied to get the approximate solution of Generalized Abel's integral equations. The convergence analysis of the presented method is also established. The different polynomials such as (1) Jacobi polynomials (2) Legendre polynomials (3) Chebyshev polynomials and (4) Gegenbauer polynomials are considered to get the numerical solution of the Generalized Abel's integral equations. Illustrative examples with different solutions are considered to show the validity and applicability of the proposed method. Numerical results show that the proposed method works well and achieve good accuracy even for less number of polynomials. Further, the performance of the proposed method is compared under the effect of different polynomials. (C) 2016 Elsevier B.V. All rights reserved.