Regularization of the Initial-Boundary Value Problem for a Inhomogeneous Parabolic Equation with Changing Time Direction

In this paper, the Cauchy problem with boundary conditions for an inhomogeneous parabolic equation with changing time direction is investigated. To construct the solution, the method of separation of variables (Fourier method) is employed, leading to a spectral problem where the eigenvalues and eigenfunctions are determined. An a priori estimate for the solution is obtained, theorems of uniqueness and conditional stability in the correctness set are proved. A regularized solution corresponding to the approximate initial data is constructed. The regularization parameter is chosen based on the efficiency estimate derived from the norm of the difference between the exact and the regularized solutions. Numerical experiments were performed for several cases of given data, and the results are presented in tables and graphs. The results show that the regularized solution corresponding to the approximate data closely regularized solution for the exact data, confirming the robustness of the method to errors in the input data.