Non-stationary heat conductivity problem for the multilayers orthotropic wedge-shaped parachute spaces (Q2739819)

The article deals with construction in the region \(\{(t,r,\phi,\mu):t\geq0; r\in I_n=\bigcup_{i=1}^{n+1}(R_{i-1},R_{i}), R_{0}=0, R_{n+1}=\infty; \phi\in[{}0,2\psi_0]{};\phi_0\in(0,2\pi);\mu\in[{}0,\mu_{0}]{}, \mu_{0}\in[{}-1,1]{}, \mu=\cos\theta, \mu_{0}=\cos\theta_{0}, \theta\in[{}0,\pi]{}\}\) a bounded solution of a separate system of the heat equation of the parabolic type NEWLINE\[NEWLINE{\partial T_{i}\over\partial t}-\left[{}a_{ri}^{2}\left( {\partial^{2}\over\partial r^{2}}+{2\over r}{\partial \over\partial r} \right)T_{i}+{a_{\mu i}^{2}\over r^{2}}{\partial \over\partial \mu} \left[{}(1-\mu^{2}){\partial T_{i}\over\partial \mu}\right]{} +{a_{\phi i}^{2}\over r^{2}(1-\mu^{2})}{\partial^{2} T_{i}\over \partial \phi^{2}}\right]{}+\chi_{i}^{2}T_{i}NEWLINE\]NEWLINE NEWLINE\[NEWLINE=f_{i}(t,r,\phi,\mu), i=\overline{1,n+1}, NEWLINE\]NEWLINE with the initial conditions NEWLINE\[NEWLINET_{i}(0,r,\phi,\mu)=g_{i}(r,\phi,\mu), i=\overline{1,n+1},NEWLINE\]NEWLINE the boundary value conditions NEWLINE\[NEWLINE\lim_{r\to0} (\sqrt{r}T_{1})=0, \lim_{r\to\infty} {\partial T_{n+1}\over\partial r}=0, T_{i}(t,r,\phi,\mu)|_{\mu=\mu_{0}} =h_{i}(t,r,\phi),i=\overline{1,n+1},NEWLINE\]NEWLINE conditions of the imperfect thermal contact NEWLINE\[NEWLINE\left.\left[{}\left(b_{k}{\partial \over\partial r}+1\right)T_{k}- T_{k+1}\right]{}\right|_{r=R_{k}}=0, \left.\left( {\partial T_{k}\over\partial r}-e_{k}{\partial T_{k+1}\over\partial r} \right)\right|_{r=R_{k}}=0, k=\overline{1,n}NEWLINE\]NEWLINE and some boundary conditions with respect to \(\phi\).NEWLINENEWLINESolutions are constructed by using the method of fundamental functions and Green's function.