Third-Order Pseudo-Hyperbolic Model Equation with Mixed Boundary Constraints: Solving the Inverse Problem

This research explores an inverse boundary value problem associated with a third-order pseudo-hyperbolic differential equation, in which the temporal component of the equation’s right-hand side remains unidentified. The problem is subject to both periodic and integral boundary conditions. To facilitate analysis, the original formulation is transformed into an equivalent auxiliary problem. The Fourier approach is employed to rigorously demonstrate both the existence and uniqueness of solutions to the associated auxiliary problem. Additionally, the third-order pseudo-hyperbolic equation undergoes discretization using the cubic B-spline (CB-spline) collocation procedure, and is subsequently transformed into a nonlinear least-squares optimization framework incorporating the Tikhonov regularization functional. The obtained system is numerically solved using MATLAB’s lsqnonlin subroutine. Numerical findings from a test case are provided and interpreted to validate the approach.