Simultaneous identification of the right-hand side and time-dependent coefficients in a two-dimensional parabolic equation
Abstract
This paper investigates the simultaneous identification of time-dependent lowest and source terms in a two-dimensional (2D) parabolic equation from the additional measurements. To investigate the solvability of the inverse problem, we first examine an auxiliary inverse boundary value problem and prove its equivalence to the original problem in a certain sense. Then, applying the contraction mappings principle existence and uniqueness of the solution of an equivalent problem is proved. Furthermore, using the equivalency, the existence and uniqueness theorem for the classical solution of the original problem is obtained and some discussions on the numerical solutions for this inverse problem are presented including numerical examples.
Keyword : inverse identification problem, 2D parabolic equation, Fourier method, classical solution, nonlinear optimization, Tikhonov regularization
How to Cite
Mehraliyev, Y. T., Huntul, M. J., & Azizbayov, E. I. (2024). Simultaneous identification of the right-hand side and time-dependent coefficients in a two-dimensional parabolic equation. Mathematical Modelling and Analysis, 29(1), 90–108. https://doi.org/10.3846/mma.2024.17974
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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M. Dehghan. Finite difference schemes for two-dimensional parabolic inverse problem with temperature overspecification. Int. J. Comput. Math., 75(3):339– 349, 2000. https://doi.org/10.1080/00207160008804989
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M.J. Huntul. Recovering the timewise reaction coefficient for a two-dimensional free boundary problem. Eurasian J. Math. Comput. Appl., 7(4):66–85, 2019. https://doi.org/10.32523/2306-6172-2019-7-4-66-85
M.J. Huntul. Finding the time-dependent term in 2d heat equation from nonlocal integral conditions. Comput. Syst. Sci. Eng., 39(3):415–429, 2021. https://doi.org/10.32604/csse.2021.017924
M.J. Huntul. Identification of the timewise thermal conductivity in a 2D heat equation from local heat flux conditions. Inverse Probl. Sci. Eng., 29(7):903–919, 2021. https://doi.org/10.1080/17415977.2020.1814282
M.J. Huntul. Reconstructing the time-dependent thermal coefficient in 2D free boundary problems. CMC- Comput. Mater. Contin., 67(3):3681–3699, 2021. https://doi.org/10.32604/cmc.2021.016036
M.J. Huntul and D. Lesnic. Determination of a time-dependent free boundary in a two-dimensional parabolic problem. Int. J. Appl. Math., 5(4):1–15, 2019. https://doi.org/10.1007/s40819-019-0700-5
M.J. Huntul and D. Lesnic. Determination of the time-dependent convection coefficient in two-dimensional free boundary problems. Eng. Comput., 38(10):3694–3709, 2021. https://doi.org/10.1108/EC-10-2020-0562
M.I. Ismailov, S. Erkovan and A.A. Huseynova. Fourier series analysis of a timedependent perfusion coefficient determination in a 2D bioheat transfer process. Trans. Issue Math. Azerbaijan Natl. Acad. Sci., 38(4):70–78, 2018.
M.I. Ivanchov. Inverse Problem for Equations of Parabolic Type. VNTL, Mathematical Studies, Lviv, 2003.
M.I. Ivanchov and N.E. Kinash. Inverse problem for the heat-conduction equation in a rectangular domain. Ukr. Math. J., 69(12):1865–1876, 2018. https://doi.org/10.1007/s11253-018-1476-1
V.K. Ivanov. On linear ill-posed problems. Dokl. Akad. Nauk SSSR, 145(2):270– 272, 1962.
N.B. Kerimov and M.I. Ismailov. An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions. J. Math. Anal. Appl., 396(2):546–554, 2012. https://doi.org/10.1016/j.jmaa.2012.06.046
K.I. Khudaverdiyev and A.A. Veliyev. Investigation of One-Dimensional Mixed Problem for a Class of Pseudohyperbolic Equations of Third Order with Nonlinear Operator Right Side. Chashyoghly, Baku, 2010.
N.Y. Kinash. An inverse problem for a 2D parabolic equation with nonlocal overdetermination condition. Carpathian J. Math., 8(1):107–117, 2016. https://doi.org/10.15330/cmp.8.1.107-117
A.I. Kozhanov and T.N. Shipina. Inverse problems of finding the lowest coefficient in the elliptic equation. J. Sib. Fed. Univ., 14(4):528–542, 2021. https://doi.org/10.17516/1997-1397-2021-14-4-528-542
M.M. Lavrentiev. On Some Ill-posed Problems of Mathematical Physics. Nauka, Novosibirsk, 1962.
D. Lesnic. Inverse Problems with Applications in Science and Engineering. Chapman and Hall/CRC, 2021. https://doi.org/10.1201/9780429400629
Ya.T. Megraliev. On one nonlocal boundary value problem for the inverse hyperbolic equation of second order. Herald of Tver State University, 30(3):27–38, 2013.
Y.T. Mehraliyev and F. Kanca. An inverse boundary value problem for a second order elliptic equation in a rectangle. Math. Model. Anal., 19(2):241–256, 2014. https://doi.org/10.3846/13926292.2014.910278
M.N. Ozisik. Finite Difference Methods in Heat Transfer. Boca Raton, FL: CRC Press, 1994.
A.I. Prilepko and D.G. Orlovsky. Inverse problems for semilinear evolution equations. Dokl. Akad. Nauk SSSR, 277(4):799–803, 1984.
M.A. Ragusa, A. Razani and F. Safari. Existence of radial solutions for a p(x)laplacian dirichlet problem. Adv Differ Equ., 2021(1):1–14, 2021. https://doi.org/10.1186/s13662-021-03369-x
V.G. Romanov. Inverse Problems of Mathematical Physics. De Gruyter, 1986. https://doi.org/10.1515/9783110926019
K.B. Sabitov and G.R Yunusova. Inverse problem for an equation of parabolichyperbolic type with a nonlocal boundary condition. Differ. Equ., 48(2):246–254, 2012. https://doi.org/10.1134/S0012266112020085
A.N. Tikhonov. On stability of inverse problems. Dokl. Akad. Nauk SSSR, 39(5):195–198, 1943.
A.R. Zaynullov. An inverse problem for two-dimensional equations of finding the thermal conductivity of the initial distribution. Journal of Samara State Technical University: Series Physical and Mathematical Sciences, 19(4):667– 679, 2015.
E.I. Azizbayov and Y.T. Mehraliyev. Nonlocal inverse boundary-value problem for a 2D parabolic equation with integral overdetermination condition. Carpathian J. Math., 12(1):23–33, 2020. https://doi.org/10.15330/cmp.12.1.23-33
I. Baglan and F. Kanca. Two-dimensional inverse quasilinear parabolic problem with periodic boundary condition. Appl. Anal., 98(8):1549–1565, 2019. https://doi.org/10.1080/00036811.2018.1434149
H.Z. Barakat and A.J. Clark. On the solution of the diffusion equations by numerical methods. J. Heat Transfer., 88(4):421–427, 1966. https://doi.org/10.1115/1.3691590
L.J. Campbell and B. Yin. On the stability of alternating-direction explicit methods for advection-diffusion equations. Numer. Methods Partial Differ. Equ., 23(6):1429–1444, 2007. https://doi.org/10.1002/num.20233
J.R. Cannon and Y.P. Lin. An inverse problem of finding a parameter in a semi-linear heat equation. J. Math.Anal. Appl., 145(2):470–484, 1990. https://doi.org/10.1016/0022-247X(90)90414-B
T.F. Coleman and Y. Li. An interior trust region approach for nonlinear minimization subject to bounds. SIAM J. Optim., 6(2):418–445, 1996. https://doi.org/10.1137/0806023
M. Dehghan. Finite difference schemes for two-dimensional parabolic inverse problem with temperature overspecification. Int. J. Comput. Math., 75(3):339– 349, 2000. https://doi.org/10.1080/00207160008804989
M. Dehghan. Identifying a control function in two-dimensional parabolic inverse problems. Appl. Math. Comput., 143(2):375–391, 2003. https://doi.org/10.1016/S0096-3003(02)00369-7
A.A. Dezin. The simplest solvable extensions of ultrahyperbolic and pseudoparabolic operators. Dokl. Akad. Nauk SSSR, 148(5):1013–1016, 1963.
M.J. Huntul. Recovering the timewise reaction coefficient for a two-dimensional free boundary problem. Eurasian J. Math. Comput. Appl., 7(4):66–85, 2019. https://doi.org/10.32523/2306-6172-2019-7-4-66-85
M.J. Huntul. Finding the time-dependent term in 2d heat equation from nonlocal integral conditions. Comput. Syst. Sci. Eng., 39(3):415–429, 2021. https://doi.org/10.32604/csse.2021.017924
M.J. Huntul. Identification of the timewise thermal conductivity in a 2D heat equation from local heat flux conditions. Inverse Probl. Sci. Eng., 29(7):903–919, 2021. https://doi.org/10.1080/17415977.2020.1814282
M.J. Huntul. Reconstructing the time-dependent thermal coefficient in 2D free boundary problems. CMC- Comput. Mater. Contin., 67(3):3681–3699, 2021. https://doi.org/10.32604/cmc.2021.016036
M.J. Huntul and D. Lesnic. Determination of a time-dependent free boundary in a two-dimensional parabolic problem. Int. J. Appl. Math., 5(4):1–15, 2019. https://doi.org/10.1007/s40819-019-0700-5
M.J. Huntul and D. Lesnic. Determination of the time-dependent convection coefficient in two-dimensional free boundary problems. Eng. Comput., 38(10):3694–3709, 2021. https://doi.org/10.1108/EC-10-2020-0562
M.I. Ismailov, S. Erkovan and A.A. Huseynova. Fourier series analysis of a timedependent perfusion coefficient determination in a 2D bioheat transfer process. Trans. Issue Math. Azerbaijan Natl. Acad. Sci., 38(4):70–78, 2018.
M.I. Ivanchov. Inverse Problem for Equations of Parabolic Type. VNTL, Mathematical Studies, Lviv, 2003.
M.I. Ivanchov and N.E. Kinash. Inverse problem for the heat-conduction equation in a rectangular domain. Ukr. Math. J., 69(12):1865–1876, 2018. https://doi.org/10.1007/s11253-018-1476-1
V.K. Ivanov. On linear ill-posed problems. Dokl. Akad. Nauk SSSR, 145(2):270– 272, 1962.
N.B. Kerimov and M.I. Ismailov. An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions. J. Math. Anal. Appl., 396(2):546–554, 2012. https://doi.org/10.1016/j.jmaa.2012.06.046
K.I. Khudaverdiyev and A.A. Veliyev. Investigation of One-Dimensional Mixed Problem for a Class of Pseudohyperbolic Equations of Third Order with Nonlinear Operator Right Side. Chashyoghly, Baku, 2010.
N.Y. Kinash. An inverse problem for a 2D parabolic equation with nonlocal overdetermination condition. Carpathian J. Math., 8(1):107–117, 2016. https://doi.org/10.15330/cmp.8.1.107-117
A.I. Kozhanov and T.N. Shipina. Inverse problems of finding the lowest coefficient in the elliptic equation. J. Sib. Fed. Univ., 14(4):528–542, 2021. https://doi.org/10.17516/1997-1397-2021-14-4-528-542
M.M. Lavrentiev. On Some Ill-posed Problems of Mathematical Physics. Nauka, Novosibirsk, 1962.
D. Lesnic. Inverse Problems with Applications in Science and Engineering. Chapman and Hall/CRC, 2021. https://doi.org/10.1201/9780429400629
Ya.T. Megraliev. On one nonlocal boundary value problem for the inverse hyperbolic equation of second order. Herald of Tver State University, 30(3):27–38, 2013.
Y.T. Mehraliyev and F. Kanca. An inverse boundary value problem for a second order elliptic equation in a rectangle. Math. Model. Anal., 19(2):241–256, 2014. https://doi.org/10.3846/13926292.2014.910278
M.N. Ozisik. Finite Difference Methods in Heat Transfer. Boca Raton, FL: CRC Press, 1994.
A.I. Prilepko and D.G. Orlovsky. Inverse problems for semilinear evolution equations. Dokl. Akad. Nauk SSSR, 277(4):799–803, 1984.
M.A. Ragusa, A. Razani and F. Safari. Existence of radial solutions for a p(x)laplacian dirichlet problem. Adv Differ Equ., 2021(1):1–14, 2021. https://doi.org/10.1186/s13662-021-03369-x
V.G. Romanov. Inverse Problems of Mathematical Physics. De Gruyter, 1986. https://doi.org/10.1515/9783110926019
K.B. Sabitov and G.R Yunusova. Inverse problem for an equation of parabolichyperbolic type with a nonlocal boundary condition. Differ. Equ., 48(2):246–254, 2012. https://doi.org/10.1134/S0012266112020085
A.N. Tikhonov. On stability of inverse problems. Dokl. Akad. Nauk SSSR, 39(5):195–198, 1943.
A.R. Zaynullov. An inverse problem for two-dimensional equations of finding the thermal conductivity of the initial distribution. Journal of Samara State Technical University: Series Physical and Mathematical Sciences, 19(4):667– 679, 2015.