Non-stationary heat problem for the multilayers orthotropic parachute spaces (Q2703317)

The article deals in the region \(\{(t,r,\phi,\mu):t\geq 0\); \(r\in\bigcup\limits_{i=1}^{n+1}(R_{i-1},R_{i}), R_{0}=0, R_{n+1}=\infty; \phi\in[0,2\pi];\mu\in[0,\mu_{0}], \mu_{0}\in[-1,1], \mu=\cos\theta, \mu_{0}=\cos\theta_{0}, \theta\in[0,\pi]\}\) with the construction of a bounded solution of a separate system of heat equations NEWLINE\[NEWLINE{\partial T_{i}\over\partial t}-a_{ri}^{2}\left( {\partial^{2}\over\partial r^{2}}+{2\over r}{\partial \over\partial r} \right)T_{i}-{a_{\mu i}^{2}\over r^{2}}{\partial \over\partial \mu} \left[(1-\mu^{2}){\partial T_{i}\over\partial \mu}\right] -{a_{\phi i}^{2}\over r^{2}(1-\mu^{2})}{\partial^{2} T_{i}\over \partial \phi^{2}}+{\mathcal N}_{i}^{2}T_{i}=f_{i}(t,r,\phi,\mu),NEWLINE\]NEWLINE \(i=1,\dots,n+1\), with the initial conditions \(T_{i}(0,r,\phi,\mu)=g_{i}(r,\phi,\mu), i=1,\dots,n+1\), the boundary value conditions \(\lim\limits_{r\to 0} (\sqrt{r}T_{1})=0, \lim\limits_{r\to\infty} {\partial T_{n+1}\over\partial r}=0\), \(T_{i}(t,r,\phi,\mu)|_{\mu=\mu_{0}} =h_{i}(t,r,\phi),i=1,\dots,n+1\) or \(\left.{\partial T_{i}\over \partial \mu}\right|_{\mu=\mu_{0}}=h_{i}(t,r,\phi),i=1,\dots,n+1\), periodic conditions \(T_{i}(t,r,\phi+2\pi,\mu)=T_{i}(t,r,\phi,\mu), i=1,\dots,n+1\) and conditions of imperfect thermal contact NEWLINE\[NEWLINE\begin{aligned} \left.\left[\left(b_{k}{\partial \over\partial r}+1\right)T_{k}- T_{k+1}\right]\right|_{r=R_{k}}&=0, \\ \left.\left( {\partial T_{k}\over\partial r}-e_{k}{\partial T_{k+1}\over\partial r} \right)\right|_{r=R_{k}}&=0, \end{aligned} \qquad k=1,\dots,n.NEWLINE\]NEWLINE This solution is constructed by using the finite integral Fourier transform of \(\phi\), the finite integral Legendre of 2-nd type transforms of \(\mu\) and the hybrid integral Fourier-Bessel transforms of \(r\).