Inverse Problem Analysis for the 2D Sixth-Order Boussinesq Equation Subject to Extra Conditions

The current research work studies an inverse boundary value problem (InBVP) for the two-dimensional sixth-order Boussinesq model equation under additional constraints. First, we introduce an auxiliary InBVP and establish its equivalence (in a specified sense) to the original problem. To analyze the auxiliary problem, we employ the method of separation of variables. Using this approach, the solution of the direct boundary value problem (for a given unknown function) is reduced to solving a problem with undetermined coefficients. This leads to a countable system of integrodifferential equations (IDEs) governing these coefficients. The system is then reformulated as a single IDE for the desired solution. Next, by incorporating the additional conditions of the auxiliary inverse problem, we derive a system of two nonlinear integral equations to determine the unknown functions. Consequently, resolving the auxiliary InBVP essentially reduces to examining a system of three coupled nonlinear IDEs for the unknown functions. To address this system, we construct a specialized Banach space. Within a ball in this space, we apply contraction mapping principles to demonstrate both the existence and uniqueness of solutions to the nonlinear integrodifferential system, thereby obtaining the solution to the auxiliary inverse problem. Finally, leveraging the equivalence between the auxiliary and original problems, we showcase that the original InBVP possesses exactly one classical solution in the specified function space.MSC2020 Classification : 35R30, 35D30, 35A01