On some heat transfer problem for a piecewise homogeneous medium with mild boundaries (Q2768789)

The author obtains in the domain \(\{(t,r): t\in (0,\infty)\), \(r\in (R_0,R_1)\cup(R_1,R_2)\cup(R_2,\infty)\); \(R_0\geq 0\}\) the explicit solution of the system of equations NEWLINE\[NEWLINE\begin{aligned} \left({\partial \over\partial t}+\gamma_{1}^2-a_{1}^2{\partial^2 \over\partial r^2}\right)u_1(t,r) &= f_{1}(t,r), \quad r\in (R_{0},R_{1}),\\ \left({\partial \over\partial t}+\gamma_{2}^2-a_{2}^2B_{\nu,\alpha}\right)u_2(t,r) &= f_{2}(t,r), \quad r\in (R_{1},R_{2}),\\ \left({\partial \over\partial t}+\gamma_{3}^2-a_{3}^2\Lambda_{\mu}\right)u_3(t,r) &= f_{3}(t,r),\quad r\in (R_{2},\infty)\end{aligned}NEWLINE\]NEWLINE with the initial conditions \(u_{j}(t,r)|_{t=0}=g_{j}(r)\), \(r\in(R_{j-1},R_{j})\), \(j=1,2,3\), \(R_3=\infty\), the boundary conditions NEWLINE\[NEWLINE\left.\left[\left(\alpha_{11}^{0}+\delta_{11}^{0}{\partial\over\partial t}\right){\partial\over\partial r}+\beta_{11}^{0}+\gamma_{11}^{0}{\partial\over\partial t}\right]u_{1}(t,r)\right|_{r=R_0}=w_1(t),\quad \lim_{r\to\infty}{\partial u\over\partial r}=0,NEWLINE\]NEWLINE and the conjunction conditions NEWLINE\[NEWLINE\begin{multlined}\left\{\left[\left(\alpha_{j1}^{k}+\delta_{j1}^{k}{\partial\over\partial t}\right){\partial\over\partial r}+\beta_{j1}^{k}+\gamma_{j1}^{k}{\partial\over\partial t}\right]u_{k}(t,r)-\right.\\ \left.\left.\left[\left(\alpha_{j2}^{k}+\delta_{j2}^{k}{\partial\over\partial t}\right){\partial\over\partial r}+\beta_{j2}^{k}+\gamma_{j2}^{k}{\partial\over\partial t}\right]u_{k+1}(t,r)\right\}\right|_{r=R_{k}}=w_{jk}(t);\quad j,k=1,2.\end{multlined}NEWLINE\]NEWLINE Here \(B_{\nu,\alpha}=d^2/dr^2+(2\alpha+1)r^{-1}d/dr-(\nu^2-\alpha^2)r^{-2}\), \(\nu\geq\alpha\geq-1/2\), \(\Lambda_{\mu}=d^2/dr^2+cth r d/dr+1/4-\mu^2sh^{-2}r\), \(\mu\geq 0\).