Performance of Ritz‐Piecewise Gegenbauer Approach for Two Types of Fractional Pantograph Equations Including Piecewise Fractional Derivative

This paper introduces a novel numerical algorithm for solving pantograph differential equations and Volterra functional integro‐differential equations including the piecewise fractional derivative. The proposed algorithm combines the piecewise Gegenbauer functions, the Ritz method, and operational matrices. These functions have great flexibility to match the problems containing piecewise fractional derivatives. Due to the problems, we calculate the operational matrix of derivatives, the operational matrix of the piecewise fractional derivative, and the pantograph operational matrix. By employing the proposed numerical algorithm, we transform the problems into a system of algebraic equations. Moreover, we discuss the error estimation of the approximate solution and residual error. Finally, we provide details on the implementation of this technique through several numerical examples to demonstrate its applicability.