Electrical Analogy Approach to Fractional Heat Conduction Models

Fractional heat conduction models extend classical formulations by incorporating fractional differential operators that capture multiscale relaxation effects. In this work, we introduce an electrical analogy that represents the action of these operators via generalized longitudinal impedance and admittance elements, thereby clarifying their physical role in energy transfer: fractional derivatives account for the redistribution of heat accumulation and dissipation within micro-scale heterogeneous structures. This analogy unifies different classes of fractional models—diffusive, wave-like, and mixed—as well as distinct fractional operator types, including the Caputo and Atangana–Baleanu forms. It also provides a general computational methodology for solving heat conduction problems through the concept of thermal impedance, defined as the ratio of surface temperature variations (relative to ambient equilibrium) to the applied heat flux. The approach is illustrated for a semi-infinite sample, where different models and operators are shown to generate characteristic spectral patterns in thermal impedance. By linking these spectral signatures of microstructural relaxation to experimentally measurable quantities, the framework not only establishes a unified theoretical foundation but also offers a practical computational tool for identifying relaxation mechanisms through impedance analysis in microscale thermal transport.