Generalized temperature field of thermosensitive half-space caused by irradiation of concentrated energy flux (Q2759468)

The surface \(z=0\) of the half-space is heated by the radially-symmetric energy flow \(q= q_0\exp(-kr^2)\). The initial temperatures of the solid and its environment are equal. In the formulation of the heat conduction problem, a finite speed of heat propagation and a thermal nonlinearity of the material are taken into account. Mathematically the problem is reduced to a wave equation for the Kirchhoff variable (normalized integral over temperature of the thermal conductivity coefficient) under the proper boundary conditions. A formal solution is found by the Hankel (with respect to radial coordinate) and the Laplace (with respect to time) transforms. The inversion of the Kirchhoff variable is performed using a numerical approximation of the thermal conductivity coefficient as a function of the temperature. Example calculations are presented.