An Efficient Numerical Solution of Multi-Term Fractional Pantograph Differential Equations via Generalized Bell Functions

Abstract

In this study, we introduce a novel method for addressing multi-term fractional pantograph-type differential equations (FPDE) involving Caputo fractional derivative operators. Our technique relies on a specially crafted operational matrix of fractional integration, constructed through a generalized Bell basis vector. The pivotal objective is to transform the multi-term FPDE into a concise set of algebraic equations characterized by Bell coefficients. By solving this system, we ascertain the unknown coefficients, allowing for the efficient derivation of the approximate solution using both coefficients and generalized Bell polynomials. Rigorous analysis, encompassing convergence and error assessments, underpins our approach. The efficacy of our method is vividly demonstrated through diverse illustrative examples, showcasing its applicability and precision. We further substantiate the superiority of our multi-term approach by rigorously comparing it with existing techniques in the literature, affirming its high efficiency.