Heterogeneous Media Heat Transfer Simulations Based on 3D-Fractional Parametric Laplace Kernel

This paper introduces a new Mittag–Leffler–Laplace memory kernel defined by Φ˜μ,ν,κα,Ï s=∫0∞EÏ âˆ’Î¼Î¾Îº/κξνα−1e−sξdξ, s>0, and develops a unified framework for modeling heat transfer in heterogeneous media with nonlocal temporal memory. The proposed kernel combines algebraic singularity, stretched attenuation, and fractional relaxation through independent parameters, enabling precise control of heterogeneity, memory depth, and relaxation strength. A nonlocal-in-time heterogeneous heat equation driven by Φ˜μ,ν,κα,Ï is formulated, and its well-posedness, energy stability, and thermodynamic admissibility are established using complete monotonicity arguments. Sharp long-time polynomial decay rates are derived via Tauberian techniques and fractional Grönwall inequalities, revealing the emergence of fractional heat dynamics as a natural asymptotic regime. Fully discrete numerical schemes are analyzed, yielding unconditional energy stability and optimal convergence rates of order OΔt+hp. Numerical experiments confirm the theoretical decay rates and demonstrate the distinct roles of the parameters α, κ, ν, and μ in regulating thermal relaxation and heterogeneity. The proposed kernel unifies classical Fourier, Caputo, and Atangana–Baleanu heat models within a single integral formulation and provides a flexible and physically admissible tool for simulating heat transfer in complex heterogeneous systems.