Stochastic non-stationary temperature fields in semi-spherical solids (Q2768768)

Let \(Q=\{(r,\varphi,\mu): r\in(0,R)\), \(\varphi\in[0,2\pi)\), \(\mu\in[0,1]\); \(\mu=\cos\theta\}\) be a semi-spherical solid. The author studies the probabilistic characteristics, such as mathematical expectation and correlation function, of solution in the domain \(D=\{(t,r,\varphi,\mu): t\in(0,\infty)\), \((r,\varphi,\mu)\in Q\}\) of the heat-transfer equation NEWLINE\[NEWLINE{1\over a^2}{\partial T\over\partial t}+{\gamma^2\over a^2}T-{\partial^2 T\over\partial r^2}+{2\over r}{\partial T\over \partial r}+{1\over r^2}(1-\mu^2)^{-1}{\partial^2 T \over\partial\varphi^2}+{1\over r^2}{\partial\over\partial\mu}\left[(1-\mu^2){\partial T\over\partial\mu}\right]=f(t,r,\varphi,\mu);NEWLINE\]NEWLINE \(a>0\), \(\gamma^2\geq 0\), with zero initial condition and boundary conditions \(\lim_{r\to 0}{\partial T\over\partial r}=0\), \((h_{21}{\partial\over\partial r}+h_{22}) T|_{r=R}=g(t,\varphi,\mu)\), \(T(t,r,\varphi+2\pi,\mu)=T(t,r,\varphi,\mu)\), \(T(t,r,\varphi,\mu)|_{\mu=0}=\omega(t,r,\varphi)\). Here \(h_{2j}\geq 0\), \(h_{21}+h_{22}\neq 0\), \(f=f_1(r,\varphi,\mu)v_1(t)\), \(g=g_1(\varphi,\mu)v_2(t)\), \(\omega=\omega_1(r,\varphi)v_3(t)\), \(f_1\), \(g_1, \omega_1\) are non-stochastic functions, \(v_{i}(t)\), \(i=1,2,3\) are stationary random processes.