Numerical Studies for Fractional Pantograph Differential Equations Based on Piecewise Fractional-Order Taylor Function Approximations

Abstract

Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique as a solution is regarded. In this paper, a new numerical method for solving fractional pantograph differential equations is presented. The fractional derivative is described in the Caputo sense. The method is based upon piecewise fractional-order Taylor function approximations. The piecewise fractional-order Taylor function is presented. An operational matrix of fractional order integration is derived and is utilized to reduce the under study problem to the solution of a system of algebraic equations. Using Newton’s iterative method, this system is solved and the solution of fractional pantograph differential equations is achieved. A bound of the error is given in the sense of Sobolev norms. Five examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method.