Numerical solutions of integral equations using shifted fractional Vieta-Fibonacci polynomials

Abstract

In this paper, we propose a numerical technique to find approximate solutions of generalized Abel’s integral equations, GAIEs, of the first and second kinds, based on the use of shifted fractional Vieta-Fibonacci polynomials. This possibility is created by establishing a relationship between the appearance of Abel’s integral equations and the definition of fractional derivatives. The method reduces the numerical solutions of the Abel’s integral equations to a system of algebraic equations. Convergence analysis and error bound of the proposed method are studied. The applicability and efficiency of the given methodology are demonstrated by a considerable number of examples. These examples show the remarkable superiority of our method.