Numerical Solution via Operational Matrix for Solving Prabhakar Fractional Differential Equations

Abstract

In this work, we apply the operational matrix based on shifted Legendre polynomials for solving Prabhakar fractional differential equations. The Prabhakar derivative is defined in three-parameter Mittag-Leffler function. We achieve this by first deriving the analytical expression for Prabhakar derivative of xp where p is positive integer, via integration. Hence, for the first time, the operational matrix method for Prabhakar derivative is derived by using the properties of shifted Legendre polynomials. Hence, we transform the Prabhakar fractional differential equations into a system of algebraic equations. By solving the system of algebraic equations, we were able to obtain the numerical solution of fractional differential equations defined in Prabhakar derivative. Only a few terms of shifted Legendre polynomials are needed for achieving the accurate solution.