Inverse problem for time dependent coefficients in the higher order pseudo-parabolic equation

M. J. Huntul; I. Tekin; M. J. Huntul; I. Tekin

doi:10.3934/mmc.2025017

Mathematical Modelling and Control

2025, Volume 5, Issue 3: 236-257. doi: 10.3934/mmc.2025017

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Research article

Inverse problem for time dependent coefficients in the higher order pseudo-parabolic equation

M. J. Huntul ^{1
,
,},
I. Tekin ²

1.
Department of Mathematics, College of Science, Jazan University, P.O. Box. 114, Jazan 45142, Saudi Arabia
2.
Department of Fundamental Sciences, Rafet Kayis Faculty of Engineering, Alanya Alaaddin Keykubat University, 07425 Antalya, Turkey

Received: 31 May 2024 Revised: 27 September 2024 Accepted: 12 October 2024 Published: 18 July 2025

In this paper, we considered an inverse problem of recovering the time dependent potential and force coefficients in the third order pseudoparabolic equation from nonlocal integral observations. Existence and uniqueness of the solution are proved by means of the contraction principle on a small time interval. The stability results for the inverse problem is presented. The unique solvability theorem for this inverse problem is proved. However, since the governing equation is yet ill-posed (very slight errors in the integral input may cause relatively significant errors in the output potential and heat source terms), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown terms. The third order pseudoparabolic problem is discretized using the cubic B-spline (CB-spline) collocation technique and reshaped as nonlinear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine $ lsqnonlin $ tool. Both analytical and perturbed data are inverted. Numerical outcomes for two benchmark test examples are reported and discussed. In addition, the von Neumann stability analysis for the proposed numerical approach has also been discussed.
- third-order pseudo-parabolic equation,
- inverse problem,
- stability analysis,
- Tikhonov regularization,
- nonlinear optimization
Citation: M. J. Huntul, I. Tekin. Inverse problem for time dependent coefficients in the higher order pseudo-parabolic equation[J]. Mathematical Modelling and Control, 2025, 5(3): 236-257. doi: 10.3934/mmc.2025017

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Abstract

In this paper, we considered an inverse problem of recovering the time dependent potential and force coefficients in the third order pseudoparabolic equation from nonlocal integral observations. Existence and uniqueness of the solution are proved by means of the contraction principle on a small time interval. The stability results for the inverse problem is presented. The unique solvability theorem for this inverse problem is proved. However, since the governing equation is yet ill-posed (very slight errors in the integral input may cause relatively significant errors in the output potential and heat source terms), we need to regularize the solution. Therefore, to get a stable solution, a regularized cost function is to be minimized for retrieval of the unknown terms. The third order pseudoparabolic problem is discretized using the cubic B-spline (CB-spline) collocation technique and reshaped as nonlinear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine $ lsqnonlin $ tool. Both analytical and perturbed data are inverted. Numerical outcomes for two benchmark test examples are reported and discussed. In addition, the von Neumann stability analysis for the proposed numerical approach has also been discussed.

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