Applications of finite and infinite Fourier series in heat processes with impulse data

Initial boundary value problems for the heat conduction equation with the constant and piecewise constant coefficients are considered in the paper. The Dirac delta function appears in the initial data. The considered problems are mathematical models of the temperature distribution arising from a pulse point heat source in homogeneous or two composite materials. The Fourier series expansion method is applied for solving these problems. The formal solutions are constructed in the form of the infinite series which are not classical functions. We show that these formal solutions are generalized functions (distributions). We prove that the partial sum of the constructed infinite series is a classical function which can be considered as a regularization of the generalized solution. The method of the computation of this regularized solution is suggested in the paper. The computational examples confirm the robustness of the proposed method.