Analytical Solution of Generalized Fractional Integro-Differential Equations via Shifted Gegenbauer Polynomials

In this paper, we proposed an analytical solution for generalized fractional order integro-differential equations with non-local boundary conditions via shifted Gegenbauer polynomials as an approximating polynomial using the Galerkin method and collocation techniques involving operational matrix that make use of the Liouville-Caputo operator of differentiation in combination with Gegenbauer polynomials. Shifted Gegenbauer polynomial properties were exploited to transform fractional order integro-differential equation and its non-local boundary conditions into an algebraic system of equations. Shifted Gegenbauer polynomial Cm(α)(x) was used in order to generate and generalize the results of some other orthogonal polynomials by varying the value of parameter α. The accuracy and effectiveness of the proposed method are tested on some selected examples from the literature. We observed that, when the exact solution is in polynomial form, the approximate solution gives a closed form solution, and non-polynomial exact solution, also give better results compared to the existing results in the literature.