On One Nonlinear Inverse Boundary Value Problem for Linearized Sixth-Order Boussinesq Equation with an Additional Integral Condition.

A huge number of mathematical models are called Boussinesq-type equations; therefore, a wide range of sixth-order Boussinesq-type equations attracts a lot of attention from outside researchers around the world. The paper studies the classical solution of one nonlinear inverse boundary value problem for linearized Boussinesq equation of the sixth order with an additional integral condition. One method is based on the application of the Fourier method. The other method is the application of the method of contracted mappings. The essence of the problem is that it is required, together with the solution, to determine the unknown kernel. The problem is considered in a rectangular area. When solving the original inverse boundary value problem, a transition from the original inverse problem to some auxiliary inverse problem is carried out. The existence and uniqueness of a solution to an auxiliary problem are proved with the help of contracted mappings. Then a transition to the original inverse problem is made again; as a result, a conclusion is made about the solvability of the original inverse problem. The proposed methods for finding solutions to the inverse problem can be used in the study of solvability for various problems of mathematical physics. [ABSTRACT FROM AUTHOR]