Tempered FDTM–bell Framework with Hybrid Laplace–sumudu Rules for Fractional Delay Systems
Saiganesh R. Yadav *
Sharadchandra ACS College, Naigaon, Nanded, India.
Rajkumar N. Ingle
Bahirji Smarak Mahavidyalaya, Basmath, Hingoli, India.
*Author to whom correspondence should be addressed.
Abstract
This work introduces a semi-analytical approach for nonlinear fractional delay differential equations governed by tempered Caputo memory. The proposed framework combines the Fractional Differential Transform Method (FDTM) with partial ordinary Bell polynomials to efficiently manage composite nonlinear terms, while a hybrid Laplace–Sumudu formulation ensures that tempering enters the coefficient recursion as an analytic multiplier. On the theoretical side, we establish existence and uniqueness results with bounds independent of the tempering parameter , and further prove geometric convergence of the truncabridhyted FDTM–Bell expansions under mild analyticity assumptions, uniformly valid in both and on compact subsets of . From an algorithmic perspective, we design a dynamic-programming Bell engine and propose an adaptive truncation criterion that balances accuracy with efficiency. Numerical experiments on three benchmark problems—a proportional delay, a time-varying delay, and a two-dimensional neutral-type system—demonstrate the stabilizing influence of tempering over long time intervals and verify the theoretically predicted error–truncation trends. The framework is modular in design and can be extended to tempered Caputo–Fabrizio kernels with only minor modifications.
Keywords: Tempered fractional calculus, Caputo derivative, delay differential systems, FDTM, Bell polynomials, Laplace transform, Sumudu transform, convergence, adaptive truncation