Unicity of solution for a semi-infinite inverse heat source problem

Zui-Cha Deng; Liu Yang; Zui-Cha Deng; Liu Yang

doi:10.3934/math.2022391

AIMS Mathematics

2022, Volume 7, Issue 4: 7026-7039. doi: 10.3934/math.2022391

Previous Article Next Article

Research article

Unicity of solution for a semi-infinite inverse heat source problem

Zui-Cha Deng ,
Liu Yang ^,

Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China

Received: 23 October 2021 Revised: 03 January 2022 Accepted: 06 January 2022 Published: 07 February 2022
MSC : 35R30, 49J20

A semi-infinite inverse source problem in heat conduction equations is considered, where the source term is assumed to be compactly supported in the region. After introducing a suitable artificial boundary, the semi-infinite problem is transformed into a bounded one and the corresponding exact expression of the boundary condition is derived. Then we rigorously prove the uniqueness of the solution of original problem, together with the stability of the corresponding optimal control solution.
- semi-infinite domain,
- heat conduction equation,
- inverse source problem,
- uniqueness,
- stability
Citation: Zui-Cha Deng, Liu Yang. Unicity of solution for a semi-infinite inverse heat source problem[J]. AIMS Mathematics, 2022, 7(4): 7026-7039. doi: 10.3934/math.2022391

Related Papers:

Abstract

A semi-infinite inverse source problem in heat conduction equations is considered, where the source term is assumed to be compactly supported in the region. After introducing a suitable artificial boundary, the semi-infinite problem is transformed into a bounded one and the corresponding exact expression of the boundary condition is derived. Then we rigorously prove the uniqueness of the solution of original problem, together with the stability of the corresponding optimal control solution.

References

[1]	I. Bushuyev, Global uniqueness for inverse parabolic problems with final observation, Inverse Probl., 11 (1995), L11.
[2]	P. Cannarsa, J. Tort, M. Yamamoto, Determination of source terms in a degenerate parabolic equation, Inverse Probl., 26 (2010), 105003.
[3]	J. R. Cannon, Y. Lin, An inverse problem of finding a parameter in a semilinear heat equation, J. Math. Anal. Appl., 145 (1990), 470–484. https://doi.org/10.1016/0022-247X(90)90414-B doi: 10.1016/0022-247X(90)90414-B
[4]	M. Dehghan, An inverse problems of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model., 25 (2001), 743–754. https://doi.org/10.1016/S0307-904X(01)00010-5 doi: 10.1016/S0307-904X(01)00010-5
[5]	Z. C. Deng, K. Qian, X. B. Rao, L. Yang, G. W. Luo, An inverse problem of identifying the source coefficient in a degenerate heat equation, Inverse Probl. Sci. Eng., 23 (2015), 498–517. https://doi.org/10.1080/17415977.2014.922079 doi: 10.1080/17415977.2014.922079
[6]	P. Duchateau, W. Rundell, Unicity in an inverse problem for an unknown reaction term in a reaction-diffusion equation, J. Differ. Equations, 59 (1985), 155–164. https://doi.org/10.1016/0022-0396(85)90152-4 doi: 10.1016/0022-0396(85)90152-4
[7]	H. W. Engl, M. Hanke, A. Neubauer, Regularization of inverse problems, Dordrecht: Kluwer Academic Publishers, 1996.
[8]	H. D. Han, X. N. Wu, Artificial boundary method–Numerical solutions of partial differential equations on unbounded domains, Beijing: Tsinghua University Press, 2009.
[9]	A. Hasanov, Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach, J. Math. Anal. Appl., 330 (2007), 766–779. https://doi.org/10.1016/j.jmaa.2006.08.018 doi: 10.1016/j.jmaa.2006.08.018
[10]	V. Isakov, Inverse problems for partial differential equations, New York: Springer, 1998. https://doi.org/10.1007/978-1-4899-0030-2
[11]	V. Isakov, Inverse source problems, American Mathematical Society, 1990.
[12]	V. Isakov, Inverse parabolic problems with the final overdetermination, Commun. Pure Appl. Math., 44 (1991), 185–209. https://doi.org/10.1002/cpa.3160440203 doi: 10.1002/cpa.3160440203
[13]	T. Johansson, D. Lesnic, Determination of a spacewise dependent heat source, J. Comput. Appl. Math., 209 (2007), 66–80. https://doi.org/10.1016/j.cam.2006.10.026 doi: 10.1016/j.cam.2006.10.026
[14]	A. Kirsch, An introduction to the mathematical theory of inverse problem, New York: Springer, 1999.
[15]	C. S. Liu, L. Qiu, J. Lin, Simulating thin plate bending problems by a family of two-parameter homogenization functions, Appl. Math. Model., 79 (2020), 284–299. https://doi.org/10.1016/j.apm.2019.10.036 doi: 10.1016/j.apm.2019.10.036
[16]	M. S. Pilant, W. Rundell, An inverse problem for a nonlinear parabolic equation, Commun. Partial Differ. Equations, 11 (1986), 445–457. https://doi.org/10.1080/03605308608820430 doi: 10.1080/03605308608820430
[17]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
[18]	L. Qiu, J. Lin, F. Wang, Q. Qin, C. Liu, A homogenization function method for inverse heat source problems in 3D functionally graded materials, Appl. Math. Model., 91 (2021), 923–933. https://doi.org/10.1016/j.apm.2020.10.012 doi: 10.1016/j.apm.2020.10.012
[19]	X. B. Rao, Y. X. Wang, K. Qian, Z. C. Deng, L. Yang, Numerical simulation for an inverse source problem in a degenerate parabolic equation, Appl. Math. Model., 39 (2015), 7537–7553. https://doi.org/10.1016/j.apm.2015.03.016 doi: 10.1016/j.apm.2015.03.016
[20]	W. Rundell, The determination of a parabolic equation from initial and final data, Proc. Amer. Math. Soc., 99 (1987), 637–642. https://doi.org/10.1090/S0002-9939-1987-0877031-4 doi: 10.1090/S0002-9939-1987-0877031-4
[21]	A. A. Samarskii, P. N. Vabishchevich, Numerical methods for solving inverse problems of mathematical physics, De Gruyter, 2007. https://doi.org/10.1515/9783110205794
[22]	A. Tikhonov, V. Arsenin, Solutions of ill-posed problems, Beijing: Geology Press, 1979.
[23]	A. Tikhonov, A. Samarsky, Equations of mathematical physics, Beijing: Higher Education Press, 1956.
[24]	L. Yang, Z. C. Deng, Uniqueness for an inverse source problem in degenerate parabolic equations, J. Math. Anal. Appl., 488 (2020), 124095. https://doi.org/10.1016/j.jmaa.2020.124095 doi: 10.1016/j.jmaa.2020.124095
[25]	L. Yang, M. Dehghan, J. N. Yu, G. W. Luo, Inverse problem of time-dependent heat sources numerical reconstruction, Math. Comput. Simul., 81 (2011), 1656–1672. https://doi.org/10.1016/j.matcom.2011.01.001 doi: 10.1016/j.matcom.2011.01.001
[26]	L. Yang, Z. C. Deng, J. N. Yu, G. W. Luo, Two regularization strategies for an evolutional type inverse heat source problem, J. Phys. A: Math. Theor., 42 (2009), 365203.
[27]	L. Yang, J. N. Yu, G. W. Luo, Z. C. Deng, Reconstruction of a space and time dependent heat source from finite measurement data, Int. J. Heat Mass Transfer, 55 (2012), 6573–6581. https://doi.org/10.1016/j.ijheatmasstransfer.2012.06.064 doi: 10.1016/j.ijheatmasstransfer.2012.06.064

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)