Determination of an Unknown Coefficient in the Third Order Pseudoparabolic Equation with Non-Self-Adjoint Boundary Conditions

The solvability of the inverse boundary problemwith an unknown coefficient dependent on time for the third order pseudoparabolic equation with non-self-adjoint boundary conditions is investigated in the present paper. Here we have introduced the definition of the classical solution of the considered inverse boundary value problem, which is reduced to the system of integral equations by the Fourier method. At first, the existence and uniqueness of the solution of the obtaining system of integral equations is proved by the method of contraction mappings; then the existence and uniqueness of the classical solution of the stated problem is proved.


Introduction
Contemporary problems of natural sciences lead to the need for statement and investigation of the qualitative new problems.As an example we can consider a class of nonlocal problems for the partial differential equations.Researching such kind of problems aroused both theoretical interest and practical necessity and they are still studied actively today.The problems with both nonlocal boundary and initial conditions had previously been studied by many scientists.Classes of nonlocal problems with integral terms in boundary conditions are of great importance in the theory of heat conductivity, thermoelasticity, chemical engineering, underground water flow, population dynamics, and plasma physics.
The questions of solvability of the nonlocal problems with integral terms in boundary conditions had been studied by Samarskii [1].Auxiliary information for investigation of the solution of such kind of problems can be found in [2][3][4][5][6][7].Inverse problems with integral condition of override for pseudoparabolic type of equations had been studied in [8][9][10].
Existence and uniqueness of the solution of an inverse boundary value problem for the third order pseudoparabolic equation with the integral condition of override is proved in the present paper.

Statement of the Problem and Reducing It to Equivalent
in the domain   = {(, ) : 0 ≤  ≤ 1, 0 ≤  ≤ } with initial condition periodical condition Neumann boundary condition and the additional condition where  > 0,  > 0 are the given numbers, (, ), (), ℎ() are the given functions, and (, ) and () are the unknown functions.
The following lemma takes place.
Let us denote the system of eigen and adjoint functions of problem (11) in the following way: where The system of function (13) forms a Riesz basis in the space  2 (0, 1).
Let us choose the system of eigen and adjoint functions of the conjugated problem as follows: From this it follows that for systems (13) and (15) the biorthogonality condition in  2 (0, 1) is satisfied.Here,   is the Kronecker symbol.
Then the arbitrary function () ∈  2 (0, 1) is expanded in biorthogonal series: and the following estimate is true: where Under the assumptions the following estimates hold: Further, under the assumptions the validity of the estimates is proved: 1) . (24) Similarly, under the assumptions the estimations hold: In order to investigate problem ( 1)-( 4), ( 6), consider the following spaces.
So, the following theorem can be proved.) . (55) Then taking into account (53) in (55), it follows that the operator Φ acts in the ball  =   and is contractive.Therefore, in the ball  =   the operator Φ has a unique fixed point {, } that is a unique solution of (54) in the ball  =   ; that is, it is a unique solution of system (42), (45) in the ball  =   .
The function (, ) as an element of the space  3 2, is continuous and has continuous derivatives   (, ) and   (, ) in   .