Numerical solution of fractional integro-differential equations using ultraspherical polynomials

In this work we will discuss the fractional integro-differential (FIDE) equations of the Fredholm and Volterra type, where we begin by presenting some special concepts about the definition of the fractional integral and the fractional derivative of Riemann-Liouville and Caputo, while addressing the Gegenbauer (Ultraspherical) polynomials and its special differential equation, mentioning the primary polynomials, in addition to presenting some theories such as the Krasnoselskii fixed point theorem, which guarantees the existence and uniqueness of solutions to fractional integro-differential equations (FIDE). To find the approximate solution of fractional integro-differential equations, we relied on orthogonal polynomials of the ultraspherical type (Gegenbauer), which have a great advantage in solving integral equations and integro-differential equations. These polynomials, with the application of fractional Caputo derivatives with the series method, in addition to fixing the values and changing the degree of the polynomials used with fixing the values of , lead to the equation becoming a linear algebraic system of equations. By solving this linear system, the approximate solution is obtained in the form of a Gegenbauer (ultraspherical) series. The change in the degree of the polynomials used will inevitably show the effectiveness and speed of the method through the examples that will be presented, where the approximate solutions obtained are compared with the exact solutions.