Uniqueness and Fourier Spectral Recovery of a Time-Varying Coefficient in the Two-Dimensional Sixth-Order Boussinesq Equation with Integral Data

This work studies an inverse boundary value problem for the two-dimensional sixth-order Boussinesq equation. The source term contains an unknown time-dependent coefficient, and an integral observation of the solution supplies the extra condition. The aim is to recover the solution and the coefficient simultaneously. We define a classical solution and introduce an auxiliary inverse formulation. We show that it is equivalent to the original problem. By the Fourier method, the auxiliary problem has a unique solution under explicit conditions on the data. The equivalence then gives existence and uniqueness of the classical solution of the original problem. The numerical method combines a Fourier spectral solver for the direct problem with a regularized Volterra reconstruction of the coefficient. Two estimators of the second derivative of the observation are compared on the same noisy data. They are central differences and a boundary-corrected weak-form estimator with a bandwidth selected from the data. The L-curve rule selects the regularization parameter in the three test problems. These are a one-mode case, a two-mode case with a variable denominator, and a case with an infinite Fourier expansion. A fourth test uses a rapidly oscillating coefficient. Finite-difference and quintic B-spline solvers of the direct problem converge at an observed order of about 2 in space and time. At 0.5% noise, the mean relative error is between 57% and 249% for central differences and between 3.0% and 5.5% for the weak-form estimator. The weak-form error has a bias floor. The bandwidth rule assumes independent noise of constant variance.