Power Series Variational Iteration Method for Fractional Order Boundary Value Integro-Differential Equations with certain Orthogonal Polynomials

The motivation behind this work is the recent advances in literature for seeking numerical techniques for fractional order boundary value integro-differential equations. The Power Series Approximation Method (PSAM) is a new approach for the numerical solution of generalized Nth-order boundary value problems. The proposed method is structurally simple with well-posed mathematical formulas. It involves transforming the given boundary value problems into a system of Ordinary Differential Equations together with the boundary conditions prescribed. Thereafter, the coefficients of the power series solution are uniquely obtained with a well-posed recurrence relation along the boundary, which leads to the solution. The unknown parameters in the solution are determined at the other boundary. This finally leads to a system of algebraic equations, which, upon solving, yields the required approximate series solution. We hence extend the Power Series Variational Iteration Method through systematic modification for the solution of fractional order boundary value integro- differential equations with Mamadu-Njoseh polynomials as basis functions. Two examples of the Fredholm type with resulting numerical evidence show that the method is accurate and reliable with an excellent convergence rate for both illustrations considered, with results presented in graphs and tables.