Inverse problems for the heat equation

The inverse problem of finding initial conditions and the right-hand side had been studied for the inhomogeneous heat equation on the basis of formulas for the solution of the first initial-boundary value problem. A criterion of uniqueness of solution of the inverse problem for finding the initial condition was found with Spectral analysis. The right side of the heat equation is represented as a product of two functions, one of which depends on the spatial coordinates and the other from time. In one task, along with an unknown solution is sought factor on the right side, depending on the time, and in another - a factor that depends on the spatial coordinates. For these tasks, we prove uniqueness theorems, the existence and stability of solution.

heat equation, first initial-boundary value problem, inverse problems, spectral method, integral equation, uniqueness, existence, stability