Timewise–dependent Coefficients Identification Problems for Third-Order Pseudo-Parabolic Equations from Nonlocal Extra Conditions

Mohammed  S. Hussein; Sayl Gani; Taysir E.  Dyhoum

doi:10.30526/38.1.3671

Authors

Mohammed S. Hussein Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq. https://orcid.org/0000-0002-9456-4303
Sayl Gani Department of Computer Techniques Engineering, Imam Al-Kadhum College, Baghdad, Iraq https://orcid.org/0000-0003-4595-2251
Taysir E. Dyhoum Department of Computing and Mathematics, Faculty of Science and Engineering, Manchester Metropolitan University, Manchester, UK https://orcid.org/0000-0002-1761-5904

DOI:

https://doi.org/10.30526/38.1.3671

Keywords:

Pseudoparabolic inverse problem, Tikhonov regularization method, Finite difference method, Von Neumann stability analysis

Abstract

This study aims to find the time-dependent potential terms in the two inverse problems of the third-order pseudo-parabolic with initial and various boundary conditions supplemented by the overdetermination data. The nonlinear inverse problems have significant applications in physics and engineering fields. We proved the existence and uniqueness of the solution of the two problems are being proved, but they still need to be proposed (since tiny perturbations in input data cause considerable errors in the output potential term). Consequently, the regularized methods should be employed. A finite difference schema is used for solving direct problems. In contrast, the inverse problems were reformulated as nonlinear least-square minimization and solved efficiently by optimizing MATLAB routine lsqnonlin. Tikhonov's regularization method was applied to get stable results. The numerical results were explained by presenting a test example for each problem. In addition, the stability was discussed by utilizing the Von Neumann stability analysis. The results showed that the time-dependent potential terms were reconstructed successfully and were stable and accurate.

Author Biographies

Mohammed S. Hussein, Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq.

.
Sayl Gani, Department of Computer Techniques Engineering, Imam Al-Kadhum College, Baghdad, Iraq

.

References

Hussein M.S.; Lesnic D.; Ivanchov M.I.; SnitkoH A. Multiple time-dependent coefficient identification thermal problems with a free boundary. Applied numerical mathematics. 2016; 99: 24-50.‏ https://doi.org/10.1016/j.apnum.2015.09.001

Hussein M.S.; Lesnic D. Identification of the time-dependent conductivity of an inhomogeneous diffusive material. Applied Mathematics and Computation. 2015; 269: 35-58.‏ https://doi.org/10.1016/j.amc.2015.07.039

Hussein M. S.; Lesnic D. Simultaneous determination of time and space-dependent coefficients in a parabolic equation. Communications in Nonlinear Science and Numerical Simulation. 2016; 33: 194-217. https://doi.org/10.1016/j.cnsns.2015.09.008 ‏

Huntul M.J.; Hussein M. S. Simultaneous Identification of Thermal Conductivity and Heat Source in the Heat Equation. Iraqi Journal of Science. 2021; 1968-1978.‏ DOI: 10.24996/ijs.2021.62.6.22

Hazanee A.; Lesnic D.; Ismailov M.I.; Kerimov N. B. Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions. Applied Mathematics and Computation. 2019; 346, 800-815.‏ https://doi.org/10.1016/j.amc.2018.10.059

Hussein M.S.; Adil Z. Numerical solution for two-sided Stefan problem. Iraqi Journal of Science. 2020; 61.2: 444-452.‏ https://doi.org/ 10.24996/ijs.2020.61.2.24

Hazanee A.; Lesnic D. Reconstruction of multiplicative space-and time-dependent sources. Inverse Problems in Science and Engineering. 2016; 24(9), 1528-1549.‏ https://doi.org/10.1080/17415977.2015.1130041

Qassim M.; Hussein M.S. Numerical Solution to Recover Time-dependent Coefficient and Free Boundary from Nonlocal and Stefan Type Overdetermination Conditions in Heat Equation. Iraqi Journal of Science. 2021; 950-960.‏ https://doi.org/10.24996/ijs.2021.62.3.25

Asanov A.; Atamanov E;. Nonclassical and inverse problems for pseudo-parabolic equations,1997, Vol. 7. VSP.‏

Colton D. Pseudoparabolic equations in one space variable. Journal of Differential Equations. 1972; 12(3):559–565. https://core.ac.uk/reader/82773480

Khompysh, K. Inverse problem for 1D pseudo-parabolic equation, Springer New York LLC. 2017; 216, 382–387. https://doi.org/10.100

Sobolev, S. L. On a new problem of mathematical physics. In Selected Works of SL Sobolev. 2006; 279–332. Springer. https://doi.org/10.1007/978-0-387-34149-1

Barenblatt G.I.; Zheltov I. P.; Kochina I. N. Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata]. Journal of applied mathematics and mechanics. 1960; 24 (5):1286–1303. https://blasingame.engr.tamu.edu/z_zCourse_Archive/P620_14C/P620_14C_zReference/(Barenblatt_Zheltov_Kochina)_Basic_Concepts_in_the_Theory_of_Seepage_of_Homogeneous_Liquids_in_Fissured_Rocks.pdf

Lyubanova A.S. and Tani A. An inverse problem for the pseudo-parabolic equation of filtration: the stabilization. Applicable Analysis. 2013; 92(3):573–585. https://doi.org/10.1080/00036811.2011.630667

Mehraliyev Y. T.; Shafiyeva G.K. Inverse boundary value problem for the pseudoparabolic equation of the third order with periodic and integral conditions. Applied Mathematical Sciences. 2014; 8(23):1145–1155. http://dx.doi.org/10.12988/ams.2014.4167

Abylkairov U. U.; Khompysh K. H. An inverse problem of identifying the coefficient in kelvin-voight equations. Appl Math Sci. 2015; 9(101-104):5079–5088. http://dx.doi.org/10.12988/ams.2015.57464

Antontsev S. N.; Aitzhanov S. E.; Ashurova G. R. An inverse problem for the pseudo-parabolic equation with p-laplacian. Evolution Equations and Control Theory. 2022; 11(2):399. http://dx.doi.org/10.3934/eect.2021005

Lyubanova A. S. Inverse problem for a pseudoparabolic equation with integral overdetermination conditions. Differential Equations. 2014; 50.4: 502-512.‏ https://doi.org/10.1134/S0012266114040089

Lyubanova A. Sh.; and Tani A. An inverse problem for the pseudo-parabolic equation of filtration: the existence, uniqueness and regularity. Applicable Analysis. 2011; 90.10: 1557-1571.‏ https://doi.org/10.1080/00036811.2010.530258

Yaman M.; Özukizil G.; Faruk Ö. Asymptotic behaviour of the solutions of inverse problems for pseudo-parabolic equations. Applied mathematics and computation. 2004; 154.1: 69-74.‏ https://doi.org/10.1016/S0096-3003(03)00691-X

Barenblat G. I.; Zheltov Y. P.; Kochina I. N. Basic concept in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 1960; 24, 1286–1303. https://blasingame.engr.tamu.edu/z_zCourse_Archive/P620_14C/P620_14C_zReference/(Barenblatt_Zheltov_Kochina)_Basic_Concepts_in_the_Theory_of_Seepage_of_Homogeneous_Liquids_in_Fissured_Rocks.pdf

Rubinshtein L. I. On heat propagation in heterogeneous media. Izvestiya Rossiiskoi Akademii Nauk. Seriya geograficheskaya. 1948; 12, 27–45.

Ting T.W. A cooling process according to the two-temperature theory of heat conduction. J. Math. Anal. Appl. 1974; 45, 23–31. https://doi.org/10.1016/0022-247X(74)90116-4

Huntul M. J.; Dhiman N.; Tamsir, M. Reconstructing an unknown potential term in the third-order pseudo-parabolic problem. Comput. Appl. Math. 2021; 40: 140.

Huntul M. J.; Tamsir M.; Dhiman N. An inverse problem of identifying the time dependent potential in a fourth-order pseudo-parabolic equation from additional condition. Num. Meth. Part. Differential Equations. 2021. https://doi.org/10.1002/num.22778.

Huntul M. J. Determination of a time-dependent potential in the higher-order pseudo-hyperbolic problem. Inverse Prob. Sci. Eng. 2021. https://doi.org/10.1080/17415977.2021.1964496.

Ramazanova A. T.; Mehraliyev Y. T.; Allahverdieva S. I. 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 modelling. 2019; 101–103. https://conf.svmo.ru/files/2019/papers/paper32.

Mehraliyev Y.; Shafiyeva T.; Gulshan Kh. Determination of an unknown coefficient in the third-order pseudo-parabolic equation with non-self-adjoint boundary conditions. Journal of Applied Mathematics. 2014.‏ https://doi.org/10.1155/2014/358696

Mehraliyev Y. T.; Shafiyeva G. Inverse boundary value problem for the pseudo-parabolic equation of the third order with periodic and integral conditions. Applied Mathematical Sciences. 2014; 8(23), 1145-1155.‏ http://dx.doi.org/10.12988/ams.2014.4167

Dhiman N.; Tamsir, M. A collocation technique based on a modified form of trigonometric cubic b-spline basis functions for Fisher's reaction-diffusion equation. Multidiscipline Modeling in Materials and Structures. 2018; 14:923–939, 10. https://doi.org/10.1108/MMMS-12-2017-0150

Sousa E. On the edge of stability analysis. Applied Numerical Mathematics. 2009; 59(6): 1322-1336.‏ https://doi.org/10.1016/j.apnum.2008.08.001

Strikwerda J. C. Finite Difference Schemes and Partial Differential Equations. Pacific Grove, CA: Wadsworth and Brooks, 1989.

Alosaimi M.; Lesnic D.; Niesen J. Identification of the thermo-physical properties of a stratified tissue. Adiabatic hypodermic wall. International Communications in Heat and Mass Transfer. 2021;126, 105376.‏ https://doi.org/10.1016/j.icheatmasstransfer.2021.105376

Cao Kai.; Lesnic D.; Ismailov M. I. Determination of the time-dependent thermal grooving coefficient. Journal of Applied Mathematics and Computing. 2021; 65: 199-221.‏ https://doi.org/10.1007/s12190-020-01388-7

Hussein M. S.; Lesnic D.; Kamynin V. L.; Kostin, A. B. Direct and inverse source problems for degenerate parabolic equations. Journal of Inverse and Ill-Posed Problems. 2020; 28.3: 425-448.‏ https://doi.org/10.1515/jiip-2019-0046

Alosaimi M.; Lesnic D.; B. T. Johansson B. T. Solution of the Cauchy problem for the wave equation using iterative regularization. Inverse Problems in Science and Engineering. 2021; 29(13): 2757-2771.‏ https://doi.org/10.1080/17415977.2021.1949590

Anwer F.; Hussein M. S. Retrieval of timewise coefficients in the heat equation from nonlocal overdetermination conditions. Iraqi Journal of Science. 2022; 1184–1199.https://doi.org/ 10.24996/ijs.2022.63.3.24

Hussein M. S.; Lesnic D.; Johansson B. T.; Hazanee A. A. Identification of a multidimensional space-dependent heat source from boundary data. Applied Mathematical Modelling. 2018; 54:202–220. https://doi.org/10.1016/j.apm.2017.09.029

Hussein M. S.; Kinash N. S. E. P.; Lesnic D.; Ivanchov, M. Retrieving the time-dependent thermal conductivity of an orthotropic rectangular conductor. Applicable Analysis. 2017; 96.15: 2604-2618.‏ https://doi.org/10.1080/00036811.2016.1232401

Ibraheem Q. W.; Hussein M. S. Determination of time-dependent coefficient in time-fractional heat equation. Partial Differential Equations in Applied Mathematics. 2023; 100492.‏ https://doi.org/10.1016/j.padiff.2023.100492

Hansen, P. C. Analysis of discrete ill-posed problems by means of the l-curve. SIAM review. 1992; 34(4):561–580. https://doi.org/10.1137/1034115

Morozov V. A. On the solution of functional equations by the method of regularization. In Doklady Akademii Nauk. Russian Academy of Sciences. 1966; 167: 510–512. http://www.mathnet.ru/eng/agreement

Hussein S. O. Determination force/source function dependent on space under the non-classical condition data. Journal of University of Babylon for Pure and Applied Sciences. 2020; 28(3): 68-75.‏ https://www.journalofbabylon.com/index.php/JUBPAS/article/view/3333/2546

Hussein S. O. Splitting the one-dimensional wave equation. Part I: Solving by finite-difference method and separation variables. Baghdad Science Journal. 2020; 17(2 (SI)): 0675-0675.‏ http://dx.doi.org/10.21123/bsj.2020.17.2(SI).0675

Alosaimi M. A.; Lesnic D. Determination of the space-dependent blood perfusion coefficient in the thermal-wave model of bio-heat transfer. Engineering Computations. 2023; 129: 34-49. https://doi.org/10.1108/EC-07-2022-0467

Alosaimi M.; Lesnic D. Determination of a space-dependent source in the thermal-wave model of bio-heat transfer. Computers & Mathematics with Applications. 2023; 129: 34-49.‏ https://doi.org/10.1016/j.camwa.2022.10.026

Hussein S. O, Placement Inverse Source Problem under Partition Hyperbolic Equation. Iraqi Journal of Science. 2021; 1994-1999.‏ https://doi.org/10.24996/ijs.2021.62.6.25

Hussein S. O; T. E. Dyhoum. Solutions for non-homogeneous wave equations subject to unusual and Neumann boundary conditions. Applied Mathematics and Computation. 2022; 430, 127285.‏ https://doi.org/10.1016/j.amc.2022.127285

Karami A.; Abbasbandy S.; Shivanian E. Inverse problem of identifying a time-dependent coefficient and free boundary in heat conduction equation by using the meshless local Petrov-Galerkin (MLPG) method via moving least squares approximation. Filomat. 2020; 34(10): 3319-3337.‏ https://doi.org/10.2298/FIL2010319K

Karami A.; Abbasbandy S.; Shivanian E. Meshless local Petrov–Galerkin formulation of inverse Stefan problem via moving least squares approximation. Mathematical and Computational Applications. 2019; 24(4): 101.Doi:‏ 10.3390/mca24040101

Huntul M. J.; Tamsir M. Reconstruction of a potential coefficient in the Rayleigh–Love equation with non-classical boundary condition. Engineering Computations. 2022; 39(10): 3442-3458.‏ https://doi.org/10.1108/EC-01-2022-0010