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An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation

Year 2023, Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications, 1578 - 1599, 03.11.2023

Mousa J. Huntul , İbrahim Tekin

https://doi.org/10.15672/hujms.1118138
Cited By: 2

Abstract

In this article, simultaneous identification of the time-dependent lowest and source terms in a two-dimensional (2D) parabolic equation from knowledge of additional measurements is studied. Existence and uniqueness of the solution is proved by means of the contraction mapping on a small time interval. Since the governing equation is yet ill-posed (very slight errors in the time-average temperature input may cause relatively significant errors in the output potential and source terms), we need to regularize the solution. Therefore, regularization is needed for the retrieval of unknown terms. The 2D problem is discretized using the alternating direction explicit (ADE) method and reshaped as non-linear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine $lsqnonlin$ tool. Finally, we present a numerical example to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the ADE is an efficient and unconditionally stable scheme for reconstructing the potential and source coefficients from minimal data which makes the solution of the inverse problem (IP) unique.

Keywords

Existence and uniqueness Contraction mapping 2D parabolic equation Inverse identification problem Nonlinear optimization Tikhonov regularization

References

  • [1] E.I. Azizbayov and Y.T. Mehraliyev, Nonlocal inverse boundary-value problem for a 2D parabolic equation with integral overdetermination condition, Carpathian Math. Publ. 12, 23–33, 2020.
  • [2] I. Baglan and F. Kanca, Two-dimensional inverse quasilinear parabolic problem with periodic boundary condition, Appl. Anal. 98, 1549–1565, 2019.
  • [3] H.Z. Barakat and A.J. Clark, On the solution of the diffusion equations by numerical methods, Journal of Heat Transfer 88, 421–427, 1996.
  • [4] F.S.V. Bazán, M.I. Ismailov and L. Bedin Time-dependent lowest term estimation in a 2D bioheat transfer problem with nonlocal and convective boundary conditions, Inverse Probl. Sci. Eng. 29, 1282–1307, 2021.
  • [5] F.S.V. Bazán, L. Bedin and L.S. Borges, Space-dependent perfusion coefficient estimation in a 2D bioheat transfer problem, Comput. Phys. Commun. 214, 18–30, 2017.
  • [6] L. Bedin and F.S.V. Bazán, On the 2D bioheat equation with convective boundary conditions and its numerical realization via a highly accurate approach, Appl. Math. Comput. 236, 422–436, 2014.
  • [7] H. Berestycki, J. Busca and I. Florent, An inverse parabolic problem arising in finance, C. R. Acad. Sci. Paris Sér. I Math. 331, 965–969, 2000.
  • [8] L.A. Caffarelli and A.Friedman Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. Soc. 252, 99–113, 1979.
  • [9] L.J. Campbell and B. Yin, On the stability of alternating-direction explicit methods for advection-diffusion equations, Numer. Methods Partial Differential Equations 23, 1429–1444, 2007.
  • [10] J.R. Cannon and J. van der Hoek, Diffusion subject to the specification of mass, J. Math. Anal. Appl. 115, 517–529, 1986.
  • [11] J.R. Cannon, The One-dimensional Heat Equation, Addison-Wesley, Menlo Park, California, 1984.
  • [12] J.A. Carrillo and J.L. Vázquez, Some free boundary problems involving non-local diffusion and aggregation, Philos. Trans. Roy. Soc. A 373, 20140275, 2015.
  • [13] J. Cen, A.A. Khan, D. Motreanu and S. Zeng, Inverse problems for generalized quasi-variational inequalities with application to elliptic mixed boundary value systems, In-verse Problems 38, 065006, 2022.
  • [14] T.F. Coleman and Y. Li, An interior trust region approach for nonlinear minimization subject to bounds, SIAM Journal on Optimization, 6, 418–445, 1996.
  • [15] M. Dehghan, Determination of a control parameter in the two-dimensional diffusion equation, Appl. Numer. Math. 37, 489-502, 2001.
  • [16] B.H. Dennis, G.S. Dulikravich and S. Yoshimura, A finite element formulation for the determination of unknown boundary conditions for three-dimensional steady ther- moelastic problems, Journal of Heat Transfer 126, 110–118, 2004.
  • [17] C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, 5th Edition, Addison- Wesley, Reading, MA, 1994.
  • [18] V. Grebenev, On a system of degenerate parabolic equations that arises in fluid dynamics, Sib. Mat. J. 35, 753–767, 1994.
  • [19] P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review 34, 561–580, 1992.
  • [20] M.J. Huntul and D. Lesnic, Determination of time-dependent coefficients and multiple free boundaries, Eurasian J. Math. Comput. Appl. 5, 15–43, 2017.
  • [21] M.J. Huntul and D. Lesnic, Time-dependent reaction coefficient identification problems with a free boundary, Int. J. Comput. Methods Eng. Sci. Mech. 20, 99–114, 2019.
  • [22] M.J. Huntul and D. Lesnic, Determination of a time-dependent free boundary in a two-dimensional parabolic problem, Int. J. Appl. Comput. Math. 5, (4), 1–15, 2019.
  • [23] M.J. Huntul, Recovering the timewise reaction coefficient for a two-dimensional free boundary problem, Eurasian J. Math. Comput. Appl. 7, 66–85, 2019.
  • [24] M.J. Huntul Identification of the timewise thermal conductivity in a 2D heat equation from local heat flux conditions, Inverse Probl. Sci. Eng. 29, 903–919, 2021.
  • [25] M.J. Huntul and D. Lesnic, Determination of the time-dependent convection coefficient in two-dimensional free boundary problems, Engineering Computations 38, 3694–3709, 2021.
  • [26] M.J. Huntul, Reconstructing the time-dependent thermal coefficient in 2D free boundary problems, CMC-Computers, Materials & Continua 67, 3681–3699, 2021.
  • [27] M.J. Huntul, Finding the time-dependent term in 2D heat equation from nonlocal integral conditions, Comput. Syst. Sci. Eng. 39, 415–429, 2021.
  • [28] M.J. Huntul, N. Dhiman and M. Tamsir, Reconstructing an unknown potential term in the third-order pseudo-parabolic problem, Comput. Appl. Math. 40, 140, 2021.
  • [29] M.J. Huntul, Identifying an unknown heat source term in the third-order pseudo-parabolic equation from nonlocal integral observation, Int. Commun. Heat Mass Transf. 128, 105550, 2021.
  • [30] M.J. Huntul, M. Tamsir and N. Dhiman, Identification of time-dependent potential in a fourth-order pseudo-hyperbolic equation from additional measurement, Math. Meth- ods Appl. Sci. 45(9), 5249–5266, 2022.
  • [31] M.J. Huntul, Recovering a source term in the higher-order pseudo-parabolic equation via cubic spline functions, Physica Scripta 97, 035004, 2022.
  • [32] M.J. Huntul, M. Abbas and M.K. Iqbal, An inverse problem for investigating the time-dependent coefficient in a higher-order equation, Comput. Appl. Math. 41, 1– 21, 2022.
  • [33] M.J. Huntul and I. Tekin, On an inverse problem for a nonlinear third order in time partial differential equation, Results Appl. Math. 15, 100314, 2022.
  • [34] M.J. Huntul and M. Abbas,An inverse problem of fourth-order partial differential equation with nonlocal integral condition, Adv. Contin. Discrete Models 2022, 1–27, 2022.
  • [35] M.I. Ismailov, S. Erkovan and A.A. Huseynova, Fourier series analysis of a time-dependent perfusion coefficient determination in a 2D bioheat transfer process, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 38, 70–78, 2018.
  • [36] M.I. Ismailov and S. Erkovan, Inverse problem of finding the coefficient of the lowest term in two-dimensional heat equation with Ionkin-type boundary condition, Comput. Math. Math. Phys. 59, 791–808, 2019.
  • [37] M. Ivanchov and V. Vlasov, Inverse problem for a two-dimensional strongly degenerate heat equation, Visnyk of the Lviv Univ. Series Mech. Math. 2018, 1–17, 2018.
  • [38] V.L. Kamynin, The inverse problem of determining the lower-order coefficient in parabolic equations with integral observation, Math Notes 94, 205213, 2013.
  • [39] M. Karazym, T. Ozawa and D. Suragan, Multidimensional inverse Cauchy problems for evolution equations, Inverse Probl. Sci. Eng. 28 (11), 1582-1590, 2020.
  • [40] N.Ye. Kinash, An inverse problem for a 2D parabolic equation with nonlocal overde- termination condition, Carpathian Math. Publ. 8, 107–117, 2016.
  • [41] Mathworks, Documentation Optimization Toolbox-Least Squares (Model Fitting) Algorithms, available at www.mathworks.com, 2016.
  • [42] L. Marin, L. Elliott, P.J. Heggs, D.B. Ingham and D. Lesnic and X. Wen, Analysis of polygonal fins using the boundary element method, Appl. Therm. Eng. 24, 1321–1339, 2004.
  • [43] Y.T. Mehraliyev, A.T. Ramazanova and M.J. Huntul, An inverse boundary value problem for a two-dimensional pseudo-parabolic equation of third order, Results Appl. Math. 14, 100274, 2022.
  • [44] V.A. Morozov, On the solution of functional equations by the method of regularization, Soviet Mathematics Doklady 7, 414-417, 1996.
  • [45] M.N. Ozisik, Finite Difference Methods in Heat Transfer, Boca Raton, FL: CRC Press, 1994.
  • [46] S.G. Pyatkov, Solvability of some inverse problems for parabolic equations, Journal of Inverse and Ill-posed Problems 12(4), 397–412, 2004.

Year 2023, Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications, 1578 - 1599, 03.11.2023

Mousa J. Huntul , İbrahim Tekin

https://doi.org/10.15672/hujms.1118138
Cited By: 2

Abstract

References

  • [1] E.I. Azizbayov and Y.T. Mehraliyev, Nonlocal inverse boundary-value problem for a 2D parabolic equation with integral overdetermination condition, Carpathian Math. Publ. 12, 23–33, 2020.
  • [2] I. Baglan and F. Kanca, Two-dimensional inverse quasilinear parabolic problem with periodic boundary condition, Appl. Anal. 98, 1549–1565, 2019.
  • [3] H.Z. Barakat and A.J. Clark, On the solution of the diffusion equations by numerical methods, Journal of Heat Transfer 88, 421–427, 1996.
  • [4] F.S.V. Bazán, M.I. Ismailov and L. Bedin Time-dependent lowest term estimation in a 2D bioheat transfer problem with nonlocal and convective boundary conditions, Inverse Probl. Sci. Eng. 29, 1282–1307, 2021.
  • [5] F.S.V. Bazán, L. Bedin and L.S. Borges, Space-dependent perfusion coefficient estimation in a 2D bioheat transfer problem, Comput. Phys. Commun. 214, 18–30, 2017.
  • [6] L. Bedin and F.S.V. Bazán, On the 2D bioheat equation with convective boundary conditions and its numerical realization via a highly accurate approach, Appl. Math. Comput. 236, 422–436, 2014.
  • [7] H. Berestycki, J. Busca and I. Florent, An inverse parabolic problem arising in finance, C. R. Acad. Sci. Paris Sér. I Math. 331, 965–969, 2000.
  • [8] L.A. Caffarelli and A.Friedman Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. Soc. 252, 99–113, 1979.
  • [9] L.J. Campbell and B. Yin, On the stability of alternating-direction explicit methods for advection-diffusion equations, Numer. Methods Partial Differential Equations 23, 1429–1444, 2007.
  • [10] J.R. Cannon and J. van der Hoek, Diffusion subject to the specification of mass, J. Math. Anal. Appl. 115, 517–529, 1986.
  • [11] J.R. Cannon, The One-dimensional Heat Equation, Addison-Wesley, Menlo Park, California, 1984.
  • [12] J.A. Carrillo and J.L. Vázquez, Some free boundary problems involving non-local diffusion and aggregation, Philos. Trans. Roy. Soc. A 373, 20140275, 2015.
  • [13] J. Cen, A.A. Khan, D. Motreanu and S. Zeng, Inverse problems for generalized quasi-variational inequalities with application to elliptic mixed boundary value systems, In-verse Problems 38, 065006, 2022.
  • [14] T.F. Coleman and Y. Li, An interior trust region approach for nonlinear minimization subject to bounds, SIAM Journal on Optimization, 6, 418–445, 1996.
  • [15] M. Dehghan, Determination of a control parameter in the two-dimensional diffusion equation, Appl. Numer. Math. 37, 489-502, 2001.
  • [16] B.H. Dennis, G.S. Dulikravich and S. Yoshimura, A finite element formulation for the determination of unknown boundary conditions for three-dimensional steady ther- moelastic problems, Journal of Heat Transfer 126, 110–118, 2004.
  • [17] C.F. Gerald and P.O. Wheatley, Applied Numerical Analysis, 5th Edition, Addison- Wesley, Reading, MA, 1994.
  • [18] V. Grebenev, On a system of degenerate parabolic equations that arises in fluid dynamics, Sib. Mat. J. 35, 753–767, 1994.
  • [19] P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review 34, 561–580, 1992.
  • [20] M.J. Huntul and D. Lesnic, Determination of time-dependent coefficients and multiple free boundaries, Eurasian J. Math. Comput. Appl. 5, 15–43, 2017.
  • [21] M.J. Huntul and D. Lesnic, Time-dependent reaction coefficient identification problems with a free boundary, Int. J. Comput. Methods Eng. Sci. Mech. 20, 99–114, 2019.
  • [22] M.J. Huntul and D. Lesnic, Determination of a time-dependent free boundary in a two-dimensional parabolic problem, Int. J. Appl. Comput. Math. 5, (4), 1–15, 2019.
  • [23] M.J. Huntul, Recovering the timewise reaction coefficient for a two-dimensional free boundary problem, Eurasian J. Math. Comput. Appl. 7, 66–85, 2019.
  • [24] M.J. Huntul Identification of the timewise thermal conductivity in a 2D heat equation from local heat flux conditions, Inverse Probl. Sci. Eng. 29, 903–919, 2021.
  • [25] M.J. Huntul and D. Lesnic, Determination of the time-dependent convection coefficient in two-dimensional free boundary problems, Engineering Computations 38, 3694–3709, 2021.
  • [26] M.J. Huntul, Reconstructing the time-dependent thermal coefficient in 2D free boundary problems, CMC-Computers, Materials & Continua 67, 3681–3699, 2021.
  • [27] M.J. Huntul, Finding the time-dependent term in 2D heat equation from nonlocal integral conditions, Comput. Syst. Sci. Eng. 39, 415–429, 2021.
  • [28] M.J. Huntul, N. Dhiman and M. Tamsir, Reconstructing an unknown potential term in the third-order pseudo-parabolic problem, Comput. Appl. Math. 40, 140, 2021.
  • [29] M.J. Huntul, Identifying an unknown heat source term in the third-order pseudo-parabolic equation from nonlocal integral observation, Int. Commun. Heat Mass Transf. 128, 105550, 2021.
  • [30] M.J. Huntul, M. Tamsir and N. Dhiman, Identification of time-dependent potential in a fourth-order pseudo-hyperbolic equation from additional measurement, Math. Meth- ods Appl. Sci. 45(9), 5249–5266, 2022.
  • [31] M.J. Huntul, Recovering a source term in the higher-order pseudo-parabolic equation via cubic spline functions, Physica Scripta 97, 035004, 2022.
  • [32] M.J. Huntul, M. Abbas and M.K. Iqbal, An inverse problem for investigating the time-dependent coefficient in a higher-order equation, Comput. Appl. Math. 41, 1– 21, 2022.
  • [33] M.J. Huntul and I. Tekin, On an inverse problem for a nonlinear third order in time partial differential equation, Results Appl. Math. 15, 100314, 2022.
  • [34] M.J. Huntul and M. Abbas,An inverse problem of fourth-order partial differential equation with nonlocal integral condition, Adv. Contin. Discrete Models 2022, 1–27, 2022.
  • [35] M.I. Ismailov, S. Erkovan and A.A. Huseynova, Fourier series analysis of a time-dependent perfusion coefficient determination in a 2D bioheat transfer process, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 38, 70–78, 2018.
  • [36] M.I. Ismailov and S. Erkovan, Inverse problem of finding the coefficient of the lowest term in two-dimensional heat equation with Ionkin-type boundary condition, Comput. Math. Math. Phys. 59, 791–808, 2019.
  • [37] M. Ivanchov and V. Vlasov, Inverse problem for a two-dimensional strongly degenerate heat equation, Visnyk of the Lviv Univ. Series Mech. Math. 2018, 1–17, 2018.
  • [38] V.L. Kamynin, The inverse problem of determining the lower-order coefficient in parabolic equations with integral observation, Math Notes 94, 205213, 2013.
  • [39] M. Karazym, T. Ozawa and D. Suragan, Multidimensional inverse Cauchy problems for evolution equations, Inverse Probl. Sci. Eng. 28 (11), 1582-1590, 2020.
  • [40] N.Ye. Kinash, An inverse problem for a 2D parabolic equation with nonlocal overde- termination condition, Carpathian Math. Publ. 8, 107–117, 2016.
  • [41] Mathworks, Documentation Optimization Toolbox-Least Squares (Model Fitting) Algorithms, available at www.mathworks.com, 2016.
  • [42] L. Marin, L. Elliott, P.J. Heggs, D.B. Ingham and D. Lesnic and X. Wen, Analysis of polygonal fins using the boundary element method, Appl. Therm. Eng. 24, 1321–1339, 2004.
  • [43] Y.T. Mehraliyev, A.T. Ramazanova and M.J. Huntul, An inverse boundary value problem for a two-dimensional pseudo-parabolic equation of third order, Results Appl. Math. 14, 100274, 2022.
  • [44] V.A. Morozov, On the solution of functional equations by the method of regularization, Soviet Mathematics Doklady 7, 414-417, 1996.
  • [45] M.N. Ozisik, Finite Difference Methods in Heat Transfer, Boca Raton, FL: CRC Press, 1994.
  • [46] S.G. Pyatkov, Solvability of some inverse problems for parabolic equations, Journal of Inverse and Ill-posed Problems 12(4), 397–412, 2004.
There are 46 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Mousa J. Huntul 0000-0001-5247-2913

İbrahim Tekin 0000-0001-6725-5663

Publication Date November 3, 2023
Published in Issue Year 2023 Volume: 52 Issue: 6 - Special Issue: Nonlinear Evolution Problems with Applications

Cite

APA Huntul, M. J., & Tekin, İ. (2023). An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation. Hacettepe Journal of Mathematics and Statistics, 52(6), 1578-1599. https://doi.org/10.15672/hujms.1118138
AMA Huntul MJ, Tekin İ. An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation. Hacettepe Journal of Mathematics and Statistics. November 2023;52(6):1578-1599. doi:10.15672/hujms.1118138
Chicago Huntul, Mousa J., and İbrahim Tekin. “An Inverse Problem of Identifying the Time-Dependent Potential and Source Terms in a Two-Dimensional Parabolic Equation”. Hacettepe Journal of Mathematics and Statistics 52, no. 6 (November 2023): 1578-99. https://doi.org/10.15672/hujms.1118138.
EndNote Huntul MJ, Tekin İ (November 1, 2023) An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation. Hacettepe Journal of Mathematics and Statistics 52 6 1578–1599.
IEEE M. J. Huntul and İ. Tekin, “An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 6, pp. 1578–1599, 2023, doi: 10.15672/hujms.1118138.
ISNAD Huntul, Mousa J. - Tekin, İbrahim. “An Inverse Problem of Identifying the Time-Dependent Potential and Source Terms in a Two-Dimensional Parabolic Equation”. Hacettepe Journal of Mathematics and Statistics 52/6 (November 2023), 1578-1599. https://doi.org/10.15672/hujms.1118138.
JAMA Huntul MJ, Tekin İ. An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation. Hacettepe Journal of Mathematics and Statistics. 2023;52:1578–1599.
MLA Huntul, Mousa J. and İbrahim Tekin. “An Inverse Problem of Identifying the Time-Dependent Potential and Source Terms in a Two-Dimensional Parabolic Equation”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 6, 2023, pp. 1578-99, doi:10.15672/hujms.1118138.
Vancouver Huntul MJ, Tekin İ. An inverse problem of identifying the time-dependent potential and source terms in a two-dimensional parabolic equation. Hacettepe Journal of Mathematics and Statistics. 2023;52(6):1578-99.

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