Uniqueness and Fourier Spectral Recovery of a Time-Varying Coefficient in the Two-Dimensional Sixth-Order Boussinesq Equation with Integral Data
5 affiliations
- 1Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia
- 2Department of Differential and Control Theory, Baku State University, Baku AZ1148, Azerbaijan
- 3Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 76169-14111, Iran
- 4Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, Tamil Nadu, India
- 5Department of Mathematics, Government College University Faisalabad, Faisalabad 38000, Pakistan
Abstract
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.
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BibTeX
@article{huntul2026uniqueness,
title = {{Uniqueness and Fourier Spectral Recovery of a Time-Varying Coefficient in the Two-Dimensional Sixth-Order Boussinesq Equation with Integral Data}},
author = {Mousa J. Huntul and Yashar T. Mehraliyev and Mohammad Izadi and Muhammad K. Iqbal},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203695},
url = {https://www.mdpi.com/2227-7390/14/20/3695},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Uniqueness and Fourier Spectral Recovery of a Time-Varying Coefficient in the Two-Dimensional Sixth-Order Boussinesq Equation with Integral Data
AU - Mousa J. Huntul
AU - Yashar T. Mehraliyev
AU - Mohammad Izadi
AU - Muhammad K. Iqbal
PY - 2026
DA - 2026/10/12
JO - Mathematics
DO - 10.3390/math14203695
UR - https://www.mdpi.com/2227-7390/14/20/3695
AB - 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.
SN - 2227-7390
ER -