Analytical and numerical solutions to some one-dimensional relativistic heat equations

Analytical and numerical solutions to a family of one-dimensional (nonlinear) relativistic heat equations in finite domains are presented. The analytical solutions correspond to steady state conditions in the absence of source terms and have been obtained as functions of the (absolute) temperature at one of the boundaries, a characteristic exponent, a Peclet number based on the speed of light and the heat flux, while the numerical ones correspond to an initial Gaussian temperature distribution, adiabatic boundary conditions and different values of the Peclet number and a characteristic exponent. It is shown that, for steady conditions, the difference between the (nondimensional) temperature of the relativistic heat equation and that corresponding to Fourier law is very large for large values of both the coefficient and the exponent of the nonlinearity that characterize the relativistic contribution to the heat flux, small values of the temperature at one of the boundaries and large heat fluxes. Travelling-wave solutions of the wave-front type are reported for odd values of the nonlinearity exponent in infinite domains and in the absence of source terms. For an initial Gaussian distribution, it is shown that the relativistic contribution to heat transfer results in the formation of two triangular corner regions where the temperature is equal to the initial one, and the formation of two temperature fronts that propagate towards the domain's boundaries. The amplitude and steepness of these fronts increase whereas their width and speed decrease as the Peclet number is decreased. It is also shown that the effects of the characteristic exponent are small provided that its value is greater than about two, and that, in the absence of source terms, the temperature becomes uniform in space and constant in time for adiabatic boundary conditions. In the presence of source terms and for adiabatic boundary conditions, it is shown that, soon after the temperature fronts hit the boundaries, the temperature becomes uniform in space but may either increase or decrease with time until it reaches a stable fixed point of the source term. For a cubic source term that exhibits bistability, it is shown that the temperature tends to the attractor of lowest temperature. (C) 2017 Elsevier Inc. All rights reserved.