Analytical solutions to generalized problems of locally nonequilibrium heat transfer: Operational method

This study develops an analytical framework for mathematical modeling of locally nonequilibrium heat transfer within the context of boundary value problems for hyperbolic-type equations with generalized boundary conditions. Nonstandard operational relations based on the Laplace transform and their corresponding originals, which are absent from known handbooks on operational calculus, are presented. The obtained image–original correspondences are characteristic of operational solutions to a broad class of generalized boundary value problems arising in various branches of mathematical physics (heat conduction, diffusion, hydrodynamics, oscillation theory, electrodynamics, thermomechanics). The lack of a developed mathematical apparatus, including complex operational relations, has previously precluded the existence of functional constructs serving as exact analytical solutions for this class of heat transfer problems. The present study proposes an approach to solving this problem and significantly expands the analytical capabilities in the field of generalized locally nonequilibrium heat transfer problems. Solutions for partially bounded and finite domains of canonical shape are provided as illustrations.