The authors consider a nonstationary heat conduction problem for the contacting thermosensitive half-space and layer under the conditions of complex heat exchange with a medium of constant temperature. To construct the solution, an analytical-numerical approach is used, which employs a version of the method of successive approximations, linearizing parameters, the Laplace integral transform and its numerical inversion by means of the Prudnikov formula adapted to heat conduction problems. The developed algorithm is used to study the thermoelastic state of such a thermosensitive piecewise homogeneous structure.