A Hybrid Fibonacci Polynomial–Block-Pulse Method for Fractional Pantograph Differential Equations

This paper proposes a numerical method for solving fractional pantograph differential equations based on generalized fractional-order hybrid functions constructed from blockpulse functions and Fibonacci polynomials. The fractional derivatives are considered in the Caputo sense. The hybrid basis is obtained by introducing the transformation t→tα into the Fibonacci polynomial representation. An exact expression for the action of the Riemann–Liouville fractional integral operator on the resulting hybrid basis functions is derived. Using this representation, the fractional pantograph differential equation is reduced to a system of algebraic equations, which is solved using Newton’s iterative method. Error estimates for the proposed approximation are also discussed. Five numerical examples are presented to illustrate the applicability and accuracy of the method, including comparisons with existing numerical approaches.