An inverse source problem for a pseudoparabolic equation with memory

M. J. Huntul; Kh. Khompysh; M. K. Shazyndayeva; M. K. Iqbal; M. J. Huntul; Kh. Khompysh; M. K. Shazyndayeva; M. K. Iqbal

doi:10.3934/math.2024689

AIMS Mathematics

2024, Volume 9, Issue 6: 14186-14212. doi: 10.3934/math.2024689

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An inverse source problem for a pseudoparabolic equation with memory

1.
Department of Mathematics, College of Science, Jazan University, P.O. Box 114, Jazan 45142, Saudi Arabia
2.
Al-Farabi Kazakh National University, Almaty, Kazakhstan
3.
Department of Mathematics, Government College University, Faisalabad, Pakistan

Received: 28 January 2024 Revised: 10 April 2024 Accepted: 11 April 2024 Published: 18 April 2024
MSC : 35R30, 35K10, 35A09, 35A01, 35A02

This paper is devoted to investigating the well-posedness, as well as performing the numerical analysis, of an inverse source problem for linear pseudoparabolic equations with a memory term. The investigated inverse problem involves determining a right-hand side that depends on the spatial variable under the given observation at a final time along with the solution function. Under suitable assumptions on the problem data, the existence, uniqueness and stability of a strong generalized solution of the studied inverse problem are obtained. In addition, the pseudoparabolic problem is discretized using extended cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this problem is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for a benchmark test example are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.
- inverse problem,
- pseudoparabolic equation,
- memory term,
- Tikhonov regularization,
- stability analysis,
- nonlinear optimization
Citation: M. J. Huntul, Kh. Khompysh, M. K. Shazyndayeva, M. K. Iqbal. An inverse source problem for a pseudoparabolic equation with memory[J]. AIMS Mathematics, 2024, 9(6): 14186-14212. doi: 10.3934/math.2024689

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Abstract

This paper is devoted to investigating the well-posedness, as well as performing the numerical analysis, of an inverse source problem for linear pseudoparabolic equations with a memory term. The investigated inverse problem involves determining a right-hand side that depends on the spatial variable under the given observation at a final time along with the solution function. Under suitable assumptions on the problem data, the existence, uniqueness and stability of a strong generalized solution of the studied inverse problem are obtained. In addition, the pseudoparabolic problem is discretized using extended cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this problem is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for a benchmark test example are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.

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