Construction and Utilization of Orthogonal Polynomial for the Fractional Order Integro- Volterra-Fredholm Differential Equations

Abstract

In this article, a novel orthogonal polynomial is introduced and it is taken as a basis function to solve fractional-order Volterra-Fredholm integro-differential equations (FVFIDEs) using standard and perturbed collocation techniques. We then solve the FVFIDEs by approximating the solution with the constructed orthogonal polynomials, substituting the approximation into the FVFIDEs to generate collocation equations at uniformly spaced interior points, yielding a system of linear algebraic equations. Using Gaussian elimination, we solve this system of equations to find the unknown coefficients back into the assumed solution. To validate the efficiency of the proposed techniques, we present four numerical examples. The results indicate that proposed collocation methods are easy to implement, efficient, and produce results that agree well with existing methods in the literature. This work highlights the robustness and potential of these methods for solving the FVFIDEs with high precision, offering valuable insights into the numerical solutions of the fractional-order Volterra-Fredholm integrodifferential equations that occur in applied mathematics.