Product Integration Method for Generalized Fractional Differential Equations

Abstract

This work aims to develop a numerical method based on Newton-Cotes quadrature rule to approximate the solution of fractional differential equations using the Lagrange poly nomial approximations, which is followed by the exact integration. A formula for ap proximating the fractional integrals is established using the product integration strategy. This formula is then utilized to develop schemes for solutions of three classes of frac tional differential equation, that differ slightly. Numerical experiments are conducted to illustrate the applicability of all of the proposed schemes. Upper bound of error in the approximations for the proposed product integration method are also worked out. After dealing with FDEs, a novel numerical method for approximating the solutions of φ−FDEs is introduced. The method employs a generalization of the Lagrange polynomial to approximate the unknown function in the differential equations and a product inte gration strategy to approximate φ−fractional integrals. The product integration formula developed for φ−fractional integrals yields two numerical schemes for solving φ−FDEs. We also provide an error estimate for the product integration method. To validate the applicability and effectiveness of the presented numerical schemes, we provide several numerical examples.