Inverse problem for forward-backward parabolic equation with generalized conjugation conditions

An inverse problem of finding the solution and the right-hand side member of a second order forward-backward parabolic equation with generalized conjugation conditions is considered. Generalized conjugation conditions ensure the symmetry of the problem and provide an opportunity to apply the Hilbert–Schmidt theorem. The system of eigenfunctions is complete and orthogonal. All the eigenvalues of this operator are real and are found by solving a transcendental equation. Using expansion series, we prove the existence and uniqueness of classical solutions of this problem.