Integral Representation of Solutions to Initial‐Boundary Value Problems on a Finite Interval in the Framework of the Hyperbolic Heat Equation

ABSTRACT We consider initial‐boundary value problems on a finite interval for the system of the energy balance equation and modified Fourier's law (constitutive equation, commonly called the Cattaneo equation) describing temperature (or internal energy) and heat flux. This hyperbolic system of first‐order partial differential equations is the simplest model of non‐Fourier heat conduction. Boundary conditions include various models of behavior of a physical system at the boundaries, including boundary conditions describing Newton's law, which states that the heat flux at the boundary is directly proportional to the difference in the temperature of the physical system and ambient temperature. In this case, the boundary conditions express the relationship of unknown functions (temperature or internal energy and heat flux) with each other. These problems cannot be solved by the Fourier method and the Laplace transform. To solve the problems, we apply the Fokas unified transform method. To illustrate the final formula, we consider numerical examples for special cases in which heat exchange at one of the boundaries obeys Newton's law.