Heterogeneous Media Heat Transfer Simulations with Modified AB-Fractional Calculus

This paper introduces a novel extension of the Atangana-Baleanu-Caputo (ABC) fractional operator via a generalized Laplace-type memory kernel constructed from a three-parameter deformed Gamma function Γ μ ,ν ,κ(· ) Γ μ , ν , κ ( · ) . The resulting operator captures a wide spectrum of nonlocal memory behaviors with tunable decay rates and heterogeneity control, enabling enhanced modeling of physical and biological processes across multi-layered complex domains. The mathematical formulation accommodates nonsingular and non-power-law kernels, addressing longstanding issues in standard ABC models related to initial conditions and long-time accuracy. Applications are presented in composite heat conduction with discontinuous diffusivity and epidemic dynamics with region-specific memory fading. Numerical simulations using Talbot inversion validate the proposed framework, and a new class of analytical solutions under piecewise diffusion and generalized forcing is established. The proposed operator sets a foundation for new classes of fractional models in control, imaging, epidemiology, and soft matter physics.