Stationary temperature fields in multi-layer semi-spherical spaces. Symmetry on azimuth (Q2768770)

Using the method of fundamental functions the author obtains in the domain \(D=\{(r,\mu): r\in\bigcup_{j=1}^{n+1}(R_{j-1},R_{j})\), \(\mu\in[0,1]\); \(R_0=0,R_{n+1}=\infty\); \(\mu=\cos\theta\}\) solution of the separate system of Poisson equations NEWLINE\[NEWLINE\left({\partial^2\over\partial r^2}+{2\over r}{\partial\over\partial r} \right)T_{j}+{1\over r^2}{\partial\over\partial \mu}\left[(1-\mu^2){\partial T_{j} \over\partial \mu}\right]=-f_{j}(r,\mu),\;j=1,\ldots,n+1NEWLINE\]NEWLINE with boundary condition \(T_{j}(r,\mu)|_{\mu=0}=g_{j}(r),\;r\in(R_{j-1},R_{j}),\;j=1,\ldots,n+1\) and conjunction conditions NEWLINE\[NEWLINE\left.\left[\left(b_{m}{\partial\over\partial r}+1\right)T_{m}-T_{m+1}\right]\right|_{r=R_{m}}=0,\;\left.\left[{\partial T_{m}\over\partial r}-e_{m}{\partial T_{m+1} \over\partial r}\right]\right|_{r=R_{m}}=0,\;m=1,\ldots,n.NEWLINE\]NEWLINE Here \(b_{m}\geq 0\) is a thermo-resistance coefficient, \(e_{m}=\lambda_{m+1},\;\lambda_{m}>0\), \(\lambda_{m}\) is a heat conduction coefficient.