Analytical solutions for heat diffusion beyond Fourier law

We obtain solutions for differential equations, describing a broad range of physical problems by the operational method with recourse to inverse differential operators, integral transforms and operational exponent. Generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions. The evolutional type problems for heat transfer in various heat conduction models are studied. Exact analytical solutions for Guyer–Krumhansl hyperbolic heat equation are obtained and compared with those of Fourier and Cattaneo equations. Modelling heat pulse propagation from a laser source is performed in the framework of Fourier, Cattaneo and Guyer–Krumhansl heat transfer models. Compliance of obtained solutions with the maximum principle is studied.