×

An inverse boundary value problem for a two-dimensional pseudo-parabolic equation of third order. (English) Zbl 07534467

Summary: In the present work, we consider an inverse boundary value problem for a two-dimensional pseudo-parabolic equation of the third-order. Using analytical and operator-theoretic methods, as well as the Fourier method, the existence and uniqueness of the classical solution of this problem is proved. This inverse problem is formulated as an auxiliary inverse problem which, in turn, is reduced to the operator equation in a specified Banach space using the method of spectral analysis. In addition, the two-dimensional pseudo-parabolic problem is discretized using the FDM 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 benchmark test example is reported and discussed.

MSC:

65-XX Numerical analysis
35-XX Partial differential equations
34-XX Ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Tikhonov, A. N., On stability of inverse problems, Proc USSR Acad Sci, 39, 195-198 (1943)
[2] Lavrent’ev, M. M., On an inverse problem for the wave equation, Dokl AN USSR, 157, 520-521 (1964) · Zbl 0138.34901
[3] Lavrent_ev, M. M.; Romanov, V. G.; Shishatski, S. P., Ill-posed problems of mathematical physics and analysis, Vol. 64 (1986), American Mathematical Soc. · Zbl 0593.35003
[4] Ivanov, V. K.; Vasin, V. V.; Tanana, V. P., Theory of linear Ill-posed problems and its applications (2013), De Gruyter
[5] Bukhgeim, A. L., Introduction to the theory of inverse problems, AL Bukhgeim (2000), VSP: VSP Utrecht
[6] Mehraliyev, Y. T.; Shafiyeva, G. K., 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 (2015) · Zbl 1327.35443
[7] Mehraliyev, Y. T.; Shafiyeva, G. K., Determination of an unknown coefficient in the third order pseudoparabolic equation with non-self-adjoint boundary conditions, J Appl Math, 2014 (2014) · Zbl 1442.35549
[8] Mehraliyev, Y. T.; Shafiyeva, G. K., Inverse boundary value problem for the pseudoparabolic equation of the third order with periodic and integral conditions, Appl Math Sci, 8, 1145-1155 (2014)
[9] Khompysh, K., Inverse problem for 1D pseudo-parabolic equation, Funct Anal Interdiscip Appl, 216, 382-387 (2017) · Zbl 1402.35313
[10] Ramazanova AT, Mehraliyev YT, Allahverdieva SI. On an inverse boundary value problem with non-local integral terms condition for the pseudoparabolic equation of the fourth order. In: Differential equations and their applications in mathematical modeling. Saransk; 2019, p. 101-3, July 9-12.
[11] Rudenko, O. V.; Soluyan, S. I.; Beyer, R. T., Theoretical foundations of nonlinear acoustics, Vol. 274 (1977), Consultants Bureau: Consultants Bureau New York · Zbl 0413.76059
[12] Beshtokov, M. K., A numerical method for solving one nonlocal boundary value problem for a third-order hyperbolic equation, Comput Math Math Phys, 54, 1441-1458 (2014) · Zbl 1327.65157
[13] Gekkieva, S. K., Nonlocal boundary-value problem for the generalized ller-Lykov moisture transport equation, Vestnik KRAUNC Fiz-Mat Nauki, 4, 19-28 (2018) · Zbl 1406.35463
[14] Huntul, M. J.; Lesnic, D., Determination of time-dependent coefficients and multiple free boundaries, Eurasian J Math Comput Appl, 5, 15-43 (2017)
[15] Huntul, M. J.; Lesnic, D., Time-dependent reaction coefficient identification problems with a free boundary, Int J Comput Methods Eng Sci Mech, 20, 99-114 (2019)
[16] Huntul, M. J.; Lesnic, D., Determination of a time-dependent free boundary in a two-dimensional parabolic problem, Int J Appl Comput Math, 5, 4, Article 118 pp. (2019), 15 pages · Zbl 1427.35357
[17] Huntul, M. J., Identification of the timewise thermal conductivity in a 2D heat equation from local heat flux conditions, Inverse Probl Sci Eng, 29, 903-919 (2021) · Zbl 1473.65166
[18] Huntul, M. J.; Lesnic, D., Determination of the time-dependent convection coefficient in two-dimensional free boundary problems, Eng Comput, 38, 3694-3709 (2021)
[19] Huntul, M. J., Reconstructing the time-dependent thermal coefficient in 2D free boundary problems, CMC-Comput, Mater Continua, 67, 3681-3699 (2021)
[20] Huntul, M. J., Finding the time-dependent term in 2D heat equation from nonlocal integral conditions, Comput Syst Sci Eng, 39, 415-429 (2021)
[21] Carrillo, J. A.; Vázquez, J. L., Some free boundary problems involving non-local diffusion and aggregation, Phil Trans R Soc A, 373, Article 20140275 pp. (2015) · Zbl 1353.35320
[22] Snitko, H. A., Inverse problem for a parabolic equation with unknown minor coefficient in a free boundary domain, (Visnyk of the Lviv University series mechanics and mathematics, vol. 77 (2012)), 218-230 · Zbl 1289.35340
[23] Huntul, M. J., Recovering the timewise reaction coefficient for a two-dimensional free boundary problem, Eurasian J Math Comput Appl, 7, 66-85 (2019)
[24] Huntul, M. J.; Dhiman, N.; Tamsir, M., Reconstructing an unknown potential term in the third-order pseudo-parabolic problem, Comput Appl Math, 40, 140 (2021) · Zbl 1476.65219
[25] Huntul, M. J., Identifying an unknown heat source term in the third-order pseudo-parabolic equation from nonlocal integral observation, Int Commun Heat Mass Transfer, 128, Article 105550 pp. (2021)
[26] Huntul, M. J.; Tamsir, M.; Dhiman, N., Identification of time-dependent potential in a fourth-order pseudo-hyperbolic equation from additional measurement, Math Methods Appl Sci, 1-18 (2022)
[27] Khudaverdiev, K. I.; Veliev, A. A., Investigation of a one-dimensional mixed problem for one class of pseudohyperbolic equations of the third order with nonlinear operator right-hand side, 168 (2010), Chashyoglu: Chashyoglu Baku
[28] Smith, G. D., (Numerical solution of partial differential equations: Finite difference methods. Numerical solution of partial differential equations: Finite difference methods, Oxford applied mathematics and computing science series (1985)) · Zbl 0576.65089
[29] Huntul, M. J., Space-dependent heat source determination problem with nonlocal periodic boundary conditions, Results Appl Math, 12, Article 100223 pp. (2021) · Zbl 1481.35402
[30] Mathworks, Documentation optimization toolbox-least squares (Model Fitting) algorithms (2016), available at https://www.mathworks.com
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.