AN INVERSE PROBLEM OF RECOVERING THE RIGHT HAND SIDE OF 1D PSEUDOPARABOLIC EQUATION

Abstract

Inverse problems of finding the right-hand side of a partial differential equations arise when an external source is unknown or impossible to measurement, for example, the source is in a high-temperature environment or underground. The partial differential equations with mixed derivatives by time and space variables are usually called pseudo-parabolic equations or Sobolev type equations. Pseudo-parabolic equations occur in mathematical modeling of many physical phenomena such as the motion of non-Newtonian fluids, thermodynamic processes, filtration in a porous medium, unsteady flow of second-order fluids, etc. This paper is devoted to investigate the unique solvability of two inverse problems for a linear one dimensional pseudoparabolic equation. The inverse problems consist of recovering the right hand side of the equation depending on the space variable. An additional information for the first inverse problem is given by an final overdetermination condition and for the second is given by an integral overdetermination condition. Under the suitable conditions on the initial data of the problem, the existence and uniqueness of a classical solution to these inverse problems are established. By using the Fourier method, the explicit formulas of a solutions are presented in the form of a series, which make it possible to perform the necessary numerical calculations with a given accuracy.