Rothe's method in problems of non-stationary heat radiation (Q2768780)

This paper deals with the problem of non-stationary heat radiation described by the equation \(\partial T /\partial t=a^2\Delta T\), \(P\in\Omega\), \(t>0\) with the initial-boundary conditions \(T(P,0)=T_0(P)\), \(P\in\overline\Omega\), \(\partial T/\partial n+h(T-T_{c})=0\), \(P\in S_1, t>0\); \(\partial T/\partial n=-\kappa(T^4-T_{c}^4)\), \(P\in S_2, t>0\), where \(a^2=\lambda/c\rho\); \(\lambda, c, \rho\) are coefficients of heat conduction, heat capacity and density of solid, respectively; \(h=\alpha/\lambda\); \(\alpha\) is a coefficient of heat transfer on the part \(S_1\) of the surface of the body; \(\kappa=\varepsilon\sigma/\lambda\); \(\varepsilon\) is a degree of blackness of the complementary part \(S_2\) of the surface \((\partial\Omega=S_1\cup S_2)\); \(\sigma\) is the Stefan-Boltzmann constant; \(T_{c}\) is the temperature of the medium; \(T_0(P)\) is the initial temperature of the body. Using the Rothe method and Green's function the considered problem is reduced to an integral equation. In the one-dimensional case an explicit solution is obtained. In the two-dimensional case a method of approximate solving is proposed.