The study of an inverse boundary problem for the heat conduction equation

This paper is concerned with investigating and solving the mixed initial boundary value problem for the heat conduction equation. The statement of the problem includes the three intervals: the first one (from $0\to T_1$) describes heating the combustion chamber, the second (from $T_1\to T_2$) — cooling the chamber and a slower cooling of its wall. Finally, the third interval describes natural cooling of the chamber wall when the chamber has the temperature coinciding with that of environment. The validity of the application of the Fourier transform with respect to this problem has been proved. This made possible to transform the governing equation to the ordinary differential equation. By using the resulting equation, the inverse boundary value problem for the heat conduction equation by applying the nonlinear method of projection regularization was solved and the error of approximate solution was obtained.