Method for solving nonstationary heat problems with mixed discontinuous boundary conditions on the boundary of a half-space (Q1603143)

This paper is devoted to investigate the solution of the differential nonstationary \((\tau>0)\) heat equation \[ \theta_{rr} (r,z,\tau)+ r^{-1}\theta_r (r,z,\tau)+ \theta_{zz}(r,x, \tau)=a^{-1} \theta_\tau (r,z,\tau), \tag{1} \] in cylindrical coordinates \((r,z>0)\) for a half-space with the homogeneous initial data \(\theta(r,z,0)=0\) and the mixed boundary conditions \[ \theta(r,0,\tau)= f_1(r)\cdot f_2(r),\;0<r<R,\quad \theta_z(r,0,\tau)=0,\;R<r< \infty, \tag{2} \] on the surface \(z=0\) in the case of the axial symmetry \[ \theta_r (0,z,\tau) =0.\tag{3} \] Using the Laplace integral transform the authors prove an existence theorem for (1)--(3).