Abstract
In this paper, we considered an inverse problem of recovering the time-dependent potential coefficient, for the first time, in the sixth-order Boussinesq-type equation from additional data as an over-specification condition. The unique solvability theorem for this inverse problem is supplied. However, since the governing equation is yet ill-posed (very slight errors in the additional input may cause relatively significant errors in the output potential term), 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 term. The sixth-order Boussinesq-Type problem is discretized using the Septic B-spline (SB-spline) collocation technique and reshaped as non-linear least-squares optimization of the Tikhonov Regularization (TR) function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Both perturbed data and analytical solution are inverted. Numerical outcomes are reported and discussed. In addition, the Von Neumann stability analysis for the proposed numerical approach has also been discussed.
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References
Aliyev ZS, Yashar T, Yusifova EH (2021) On some nonlocal inverse boundary problem for partial differential equations of third order. Turk J Math 45:1871–1886
Antontsev SN, Aitzhanov SE, Ashurova GR (2021) An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evol Equ Control Theory 11:399
Azizbayov EI (2019) The nonlocal inverse problem of the identification of the lowest coefficient and the right-hand side in a second-order parabolic equation with integral conditions. Bound Value Probl 2019:1–19
Abylkairov UU, Khompysh K (2015) An inverse problem of identifying the coefficient in Kelvin-Voight equations. Appl Math Sci 9:5079–5088
Asanov A, Atmanov ER (1994) An inverse problem for a pseudoparabolic operator equation. J Inverse Ill-Posed Probl 2:1–14
Baglan I, Canel T (2020) An inverse coefficient problem for quasilinear pseudo-parabolic of heat conduction of Poly (methyl methacrylate)(PMMA). Turk J Sci 5:199–207
Beshtokov MKh (2017) Differential and difference boundary value problem for loaded third-order pseudo-parabolic differential equations and difference methods for their numerical solution. Comput Math Math Phys 57:1973–1993
Huntul MJ, Tamsir M, Dhiman N (2021) An inverse problem of identifying the time-dependent potential in a fourth-order pseudo-parabolic equation from additional condition. Num Methods Part Differ Equ. https://doi.org/10.1002/num.22778
Huntul MJ, Dhiman N, Tamsir M (2021) Reconstructing an unknown potential term in the third-order pseudo-parabolic problem. Comput Appl Math 40:140
Huntul MJ (2021) Determination of a time-dependent potential in the higher-order pseudo-hyperbolic problem. Inverse Probl Sci Eng 29:3006–3023
Huntul MJ, Tamsir M (2021) Identifying an unknown potential term in the fourth-order Boussinesq-Love equation from mass measurement. Eng Comput 38:3944–3968
Huntul MJ, Tamsir M, Ahmadini A (2021) An inverse problem of determining the time-dependent potential in a higher-order Boussinesq-Love equation from boundary data. Eng Comput 38:3768–3784
Huntul MJ (2021) Identifying an unknown heat source term in the third-order pseudo-parabolic equation from nonlocal integral observation. Int Commun Heat Mass Transfer 128:105550
Huntul MJ, Tamsir M, Dhiman N (2022) Identification of time-dependent potential in a fourth-order pseudo-hyperbolic equation from additional measurement. Math Methods Appl Sci 1–18. https://doi.org/10.1002/mma.8104
Huntul MJ (2022) Recovering a source term in the higher-order pseudo-parabolic equation via cubic spline functions. Phys Scr 97:035004
Lyubanova AS, Velisevich AV (2019) Inverse problems for the stationary and pseudoparabolic equations of diffusion. Appl Anal 98:1997–2010
Iqbal MK, Abbas M, Nazir T, Ali N (2020) Application of new quintic polynomial B-spline approximation for numerical investigation of Kuramoto-Sivashinsky equation. Adv Diff Equ 2020:1–21
Khompysh K (2017) Inverse problem for 1D pseudo-parabolic equation. Funct Anal Interdiscip Appl 216:382–387
Khompysh K, Shakir A (2020) The inverse problem for determining the right part of the pseudo-parabolic equation. J Math Comput Sci 105:87–98
Mathworks (2019) Documentation optimization toolbox-least squares (model fitting) algorithms. Available at www.mathworks
Mehraliyev YT, Shafiyeva GKh (2014) Inverse boundary value problem for the pseudoparabolic equation of the third order with periodic and integral conditions. Appl Math Sci 8:1145–1155
Mehraliyev YT, Shafiyeva GKh (2015) On an inverse boundary-value problem for a pseudoparabolic third-order equation with integral condition of the first kind. J Math Sci 204:343–350
Nazir T, Abbas M, Iqbal MK (2020) New cubic B-spline approximation technique for numerical solutions of coupled viscous Burgers equations. Eng Comput 1:83–106
Ramazanova AT, Mehraliyev YT, Allahverdieva SI (2019) On an inverse boundary value problem with non-local integral terms condition for the pseudo-parabolic equation of the fourth order, Differential Equations and Their Applications in Mathematical Modeling, Saransk, July 9–12, 101–103
Ruzhansky M, Serikbaev D, Tokmagambetov N (2019) An inverse problem for the pseudo-parabolic equation for Laplace operator. Int J Math Phys 10:23–28
Shallal MA, Ali KK, Raslan KR, Taqi AH (2019) Septic B-spline collocation method for numerical solution of the coupled Burgers’ equations. Arab J Basic Appl Sci 26:331–341
Wasim I, Abbas M, Iqbal MK, Hayat AM (2020) Exponential B-spline collocation method for solving the gen-eralized Newell-Whitehead-Segel equation. J Math Comput Sci 20:313–324
Yuldashev TK (2020) Inverse boundary-value problem for an integro-differential Boussinesq-type equation with degenerate kernel. J Math Sci 250:847–858
Yuldashev TK, Kadirkulov BJ (2021) Inverse boundary value problem for a fractional differential equations of mixed type with integral redefinition conditions. Lobachevskii J Math 42:649–662
Yang H (2020) An inverse problem for the sixth-order linear Boussinesq-type equation. UPB Sci Bull Ser A Appl Math Phys 82:27–36
Yaman M, Gözükizil ÖF (2004) Asymptotic behaviour of the solutions of inverse problems for pseudo-parabolic equations. Appl Math Comput 154:69–74
Acknowledgements
We thank Dr Muhammad Amin for his assistance in proofreading of the manuscript. The authors are also grateful to the anonymous referees for their valuable suggestions that significantly improved this manuscript.
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Communicated by Antonio C. G. Leitao.
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Huntul, M.J., Abbas, M. & Iqbal, M.K. An inverse problem for investigating the time-dependent coefficient in a higher-order equation. Comp. Appl. Math. 41, 120 (2022). https://doi.org/10.1007/s40314-022-01829-y
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DOI: https://doi.org/10.1007/s40314-022-01829-y
Keywords
- Inverse problem
- Sixth-order PDE
- Tikhonov regularization
- Nonlinear optimization
- Stability analysis
- Optimization