A Unified Transform–Polynomial Framework for Tempered Fractional Delay Systems with Caputo, Caputo–Fabrizio, and Atangana–Baleanu Kernels

Abstract

Fractional models incorporating memory and delay have become essential for capturing complex processes in which the present state depends not only on recent dynamics but also on accumulated historical effects. Standard fractional kernels, however, may lose accuracy over extended time horizons. To overcome this, tempered operators are employed, introducing exponential damping while retaining the inherent nonlocal behavior of fractional calculus.

In this study, we present a unified analytical and computational framework for tempered fractional delay differential systems involving the Caputo, Caputo–Fabrizio, and Atangana–Baleanu kernels. The proposed method combines the Fractional Differential Transform Method (FDTM) with generalized Bell polynomials, enabling efficient treatment of nonlinear terms and delay expansions. To incorporate tempered operators in closed form, we construct a hybrid pipeline based on Laplace, Sumudu, and Mellin transforms, which provides analytic multipliers for each kernel and leads to a systematic recursive scheme for computing solution coefficients without additional adjustments.

On the theoretical side, we prove existence, uniqueness, and stability with bounds uniform in the tempering parameter, and establish geometric and spectral convergence of the FDTM–Bell truncations under mild analyticity, uniformly across Caputo, Caputo–Fabrizio, and Atangana–Baleanu kernels. We also introduce an adaptive truncation rule that balances cost and accuracy. Numerical tests—covering proportional, time-dependent, and distributed neutral delays—confirm the predicted rates, highlight the stabilizing effect of tempering, and outperform standard predictor–corrector schemes, yielding a practical toolkit for nonlinear tempered fractional delay systems.