An efficient numerical solution of multi-term fractional pantograph differential equations via generalized Bell functions

Abstract

In this study, we propose an efficient numerical method for multi-term fractional pantograph differential equations (FPDEs) with Caputo derivatives, covering both linear and nonlinear models. The method builds an operational matrix of fractional integration in a generalized Bell basis and reduces the FPDE to a compact algebraic system in the Bell coefficients. Nonlinear terms (powers/products, including delayed interactions) are treated by projection-based (pseudo-operational) matrices within the same framework. Solving the resulting system determines the coefficients and yields the approximate solution in explicit Bell form. We provide convergence and error analyses, and numerical experiments demonstrate high accuracy—often near machine precision—with modest truncation and consistently low CPU time on standard hardware. Relative to results reported in the literature, the proposed scheme attains comparable or better accuracy at reduced computational cost, highlighting its practicality for multi-term FPDEs with delays. © 2026, University of Nis. All rights reserved.