A Jacobi spectral collocation scheme for solving Abel's integral equations

Abdelkawy M. A., Ezz-Eldien S. S., Amin A. Z. M. A.

Progress in Fractional Differentiation and Applications, vol.1, no.3, pp.187-200, 2015 (Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 1 Issue: 3
  • Publication Date: 2015
  • Doi Number: 10.12785/pfda/010304
  • Journal Name: Progress in Fractional Differentiation and Applications
  • Journal Indexes: Scopus
  • Page Numbers: pp.187-200
  • Keywords: Abel's integral equation, Gauss quadrature, Jacobi polynomials, Riemann-Liouville integral, Spectral collocation method
  • Istanbul Gelisim University Affiliated: No

Abstract

In this article, we construct a numerical technique for solving the first and second kinds of Abel's integral equations. Using the spectral collocation method, the properties of fractional calculus and the Gauss-quadrature formula, we can reduce such problems into those of a system of algebraic equations which greatly simplifies the problem. The proposed numerical technique is based on the shifted Jacobi polynomials and the fractional integral is described in the sense of Riemann-Liouville. In addition, our numerical technique is applied also to solve the system of generalized Abel's integral equations. For testing the accuracy and validity of the proposed numerical techniques, we apply them to solve several numerical examples.