Optimization-Based Modified Laplace Transform Techniques for Addressing Fuzzy Fractional Advection-Diffusion Problems

Abstract

This paper presents an optimization–enhanced Laplace transform homotopy perturbation framework for solving fuzzy time–fractional advection–diffusion equations with memory effects and parametric uncertainty. The model is formulated in a double–parametric fuzzy setting, allowing uncertainty in system parameters to be consistently propagated through the fractional dy-
namics. The Atangana–Baleanu–Caputo fractional derivative is extended to fuzzy–valued functions via the generalized Hukuhara difference.To improve the convergence and accuracy of the classical LT–HPM, two systematic parameter identification strategies are incorporated. The first employs a residual error collocation point (RECP) technique, while the second introduces a least–squares optimization of the fuzzy residual, leading to an optimized variant referred to as OMLT–HPM. Both strategies preserve the original Laplace–homotopy structure while enabling robust and reproducible calibration of auxiliary parameters.Rigorous theoretical results are established, including existence, uniqueness, stability, and convergence of the fuzzy solution in the double–parametric framework. Numerical experiments for a benchmark fuzzy fractional advection–diffusion problem demonstrate that the proposed methods significantly outperform the standard LT–HPM and classical finite difference schemes, with the optimized approach achieving high accuracy using only a small number of perturbation terms. The results highlight the effectiveness and computational efficiency of the proposed framework for applied fractional models under uncertainty.