The Ill-posedness and Fourier regularization for the backward heat conduction equation with uncertainty

Hong Yang; Yiliang He; Hong Yang; Yiliang He

doi:10.3934/era.2025089

Electronic Research Archive

2025, Volume 33, Issue 4: 1998-2031. doi: 10.3934/era.2025089

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The Ill-posedness and Fourier regularization for the backward heat conduction equation with uncertainty

Hong Yang ^,,
Yiliang He

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received: 16 December 2024 Revised: 26 March 2025 Accepted: 29 March 2025 Published: 09 April 2025

The backward heat conduction problem (BHCP) is an important branch of inverse problems in mathematical physics. It is widely used and is one of the hot research fields nowadays. However, due to the ill-posedness of the inverse problem, the solution of the problem is difficult to obtain. Therefore, it is necessary to study regularization methods to solve such problems. This paper introduces the backward heat conduction equation (BHCE) with uncertainty and investigates its numerical solution. The uncertainty of parameters in the system is modeled by a fuzzy number. In the solution process, it is first proved that the equation is seriously ill-posed under the concept of granular differentiability. Second, a Fourier regularization method based on the fuzzy Fourier transform is proposed to solve the BHCE with uncertainty, and the granular representation of the approximate solution is given. Finally, some error estimates between the approximate solution and exact solution are provided under the condition of the prior bound assumptions and suitable choices of regularization parameters. Numerical examples also demonstrate the effectiveness and practicability of the proposed method.
- fuzzy numbers,
- ill-posedness,
- granular differentiability,
- Fourier regularization,
- fuzzy Fourier transform,
- error estimate
Citation: Hong Yang, Yiliang He. The Ill-posedness and Fourier regularization for the backward heat conduction equation with uncertainty[J]. Electronic Research Archive, 2025, 33(4): 1998-2031. doi: 10.3934/era.2025089

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Abstract

The backward heat conduction problem (BHCP) is an important branch of inverse problems in mathematical physics. It is widely used and is one of the hot research fields nowadays. However, due to the ill-posedness of the inverse problem, the solution of the problem is difficult to obtain. Therefore, it is necessary to study regularization methods to solve such problems. This paper introduces the backward heat conduction equation (BHCE) with uncertainty and investigates its numerical solution. The uncertainty of parameters in the system is modeled by a fuzzy number. In the solution process, it is first proved that the equation is seriously ill-posed under the concept of granular differentiability. Second, a Fourier regularization method based on the fuzzy Fourier transform is proposed to solve the BHCE with uncertainty, and the granular representation of the approximate solution is given. Finally, some error estimates between the approximate solution and exact solution are provided under the condition of the prior bound assumptions and suitable choices of regularization parameters. Numerical examples also demonstrate the effectiveness and practicability of the proposed method.

References

[1]	J. B. Keller, Inverse problems, Am. Math. Mon., 2 (1976), 107–118. https://doi.org/10.2307/2976988
[2]	J. Hadamard, Lecture on Cauchy Problems in Linear Partial Differential Equations, Yale University Press, New Haven, 1923.
[3]	A. E. Abouelregal, R. A. Alharb, M. Yaylaci, B. O. Mohamed, S. F. Megahid, Analysis of temperature changes in living tissue using the modified fractional thermal conduction model under laser heat flux on the skin surface, Continuum Mech. Thermodyn., 37 (2025), 1011–1014. https://doi.org/10.1007/s00161-024-01343-y doi: 10.1007/s00161-024-01343-y
[4]	E. A. N. Al-Lehaibi, H. M. Youssef, The heat transfer in skin tissues under the general two-temperature three-phase-lag model of heat conduction with a comparative study, Heliyon, 10 (2024), 40257. https://doi.org/10.1016/j.heliyon.2024.e40257 doi: 10.1016/j.heliyon.2024.e40257
[5]	L. Liu, C. Liu, Q. M. Zhu, Y. H. Li, Inversion of spatio-temporal distribution heat flux and reconstruction of transient temperature field of three-layered skin tissue during hyperthermia, J. Therm. Biol., 114 (2023), 103515. https://doi.org/10.1016/j.jtherbio.2023.103515 doi: 10.1016/j.jtherbio.2023.103515
[6]	A. Carasso, Error bounds in the final value problem for the heat equation, SIAM J. Math. Anal., 7 (1976), 195–199. https://doi.org/10.1137/0507015 doi: 10.1137/0507015
[7]	V. Isakov, Inverse Problems for Partial Differenticl Equations, Springer-Verlag, New York, 1998. https://doi.org/10.1007/978-3-319-51658-5
[8]	W. B. Muniz, H. F. de Campos Veiho, F. M. Ramos, A comparison of some inverse methods for estimating the initial condition of the heat equation, J. Comput. Appl. Math., 103 (1999), 145–163. https://doi.org/10.1016/S0377-0427(98)00249-0 doi: 10.1016/S0377-0427(98)00249-0
[9]	W. B. Muniz, F. M. Ramos, H. F. de Campos Veiho, Entropy- and Tihonov-based regularization techniques applied to the backward heat equation, Int. J. Comput. Math., 40 (2000), 1071–1084. https://doi.org/10.1016/S0898-1221(00)85017-8 doi: 10.1016/S0898-1221(00)85017-8
[10]	W. Cheng, C. L. Fu, A modified Tikhonov regularization method for an axisymmetric backward heat equation, Acta Math. Sin. Engl. Ser., 26 (2010), 2157–2164. https://doi.org/10.1007/s10114-010-8509-5 doi: 10.1007/s10114-010-8509-5
[11]	J. R. Chang, C. S. Liu, C. W. Chang, A new shooting method for quasi-boundary regularization of backward heat conduction problems, Int. J. Heat Mass Transfer, 50 (2007), 2325–2332. https://doi.org/10.1016/j.ijheatmasstransfer.2006.10.050 doi: 10.1016/j.ijheatmasstransfer.2006.10.050
[12]	S. M. Kirkup, M. Wadsworth, Solution of inverse diffusion problems by operator-splitting methods, Appl. Math. Model., 26 (2002), 1003–1018. https://doi.org/10.1016/S0307-904X(02)00053-7 doi: 10.1016/S0307-904X(02)00053-7
[13]	G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (1988), 501–544. https://doi.org/10.1016/0022-247X(88)90170-9 doi: 10.1016/0022-247X(88)90170-9
[14]	R. Grzymkowski, M. Pleszczynski, D. Slota, Comparing the Adomian decomposition method and Runge-Kutta method for the solutions of the Stefan problem, Int. J. Comput. Math., 83 (2006), 409–417. http://doi.org/10.1080/00207160600961729 doi: 10.1080/00207160600961729
[15]	M. J. Li, Z. J. Fu, W. Z. Xu, C. M. Fan, A novel spatial-temporal radial Trefftz collocation method for the backward heat conduction analysis with time-dependent source term, Int. J. Heat Mass Transfer, 201 (2023), 123627. https://doi.org/10.1016/j.ijheatmasstransfer.2022.123627 doi: 10.1016/j.ijheatmasstransfer.2022.123627
[16]	C. L. Fu, X. T. Xiong, P. Fu, Fourier regularization method for solving the surface heat flux from interior observations, Math. Comput. Modell., 42 (2005), 489–498. https://doi.org/10.1016/j.mcm.2005.08.003 doi: 10.1016/j.mcm.2005.08.003
[17]	C. L. Fu, Simplified Tikhonov and Fourier regularization methods on a general equation, J. Comput. Appl. Math., 167 (2004), 449–463. https://doi.org/10.1016/j.cam.2003.10.011 doi: 10.1016/j.cam.2003.10.011
[18]	Z. Qian, C. L. Fu, X. T. Xiong, T. Wei, Fourier truncation method for high order numerical derivatives, Appl. Math. Comput., 181 (2006), 940–948. https://doi.org/10.1016/j.amc.2006.01.057 doi: 10.1016/j.amc.2006.01.057
[19]	C. L. Fu, Y. X. Fu, H. Cheng, Y. J. Ma, The a posteriori Fourier method for solving ill-posed problem, Inverse Probl., 28 (2012), 095002. https://doi.org/10.1088/0266-5611/28/9/095002 doi: 10.1088/0266-5611/28/9/095002
[20]	C. L. Fu, X. L. Feng, Z. Qian, The Fourier regularization for solving Cauchy problem for the Helmholtz equation, Appl. Numer. Math., 59 (2009), 2625–2640. https://doi.org/10.1016/j.apnum.2009.05.014 doi: 10.1016/j.apnum.2009.05.014
[21]	Y. X. Zhang, C. L. Fu, Z. L. Deng, An a posteriori truncation method for some Cauchy problems associated with Helmholtz-type equations, Inverse Probl. Sci. Eng., 21 (2013), 1151–1168. https://doi.org/10.1080/17415977.2012.743538 doi: 10.1080/17415977.2012.743538
[22]	F. Yang, P. Zhang, X. X. Li, The truncation method for the Cauchy problem of the inhomogeneous Helmholtz equation, Appl. Anal., 4 (2017), 991–1004. https://doi.org/10.1080/00036811.2017.1408080 doi: 10.1080/00036811.2017.1408080
[23]	J. Kokila, M. T. Nair, Fourier truncation method for the non-homogeneous time fractional backward heat conduction problem, Inverse Probl. Sci. Eng., 28 (2020), 402–426. https://doi.org/10.1080/17415977.2019.1580707 doi: 10.1080/17415977.2019.1580707
[24]	S. Y. Duan, B. T. Yang, Determination of singular value truncation threshold for regularization in ill-posed problems, Inverse Probl. Sci. Eng., 8 (2020), 1127–1157. https://doi.org/10.1080/17415977.2020.1832090 doi: 10.1080/17415977.2020.1832090
[25]	V. N. Doan, H. T. Nguyen, V. A. Khoa, V. A. Vo, A note on the derivation of filter regularization operators for nonlinear evolution equations, Appl. Anal., 97 (2016), 3–12. https://doi.org/10.1080/00036811.2016.1276176 doi: 10.1080/00036811.2016.1276176
[26]	F. Andrzej, W. Agnieszka, C. Michal, Trefftz numerical functions for solving inverse heat conduction problems, Int. J. Therm. Sci., 177 (2022), 107566. https://doi.org/10.1016/j.ijthermalsci.2022.107566 doi: 10.1016/j.ijthermalsci.2022.107566
[27]	M. Ahsan, W. Lei, M. Ahmad, M. S. Hussein, Z. Uddin, A wavelet-based collocation technique to find the discontinuous heat source in inverse heat conduction problems, Phys. Scr., 97 (2022), 125208. https://doi.org/10.1088/1402-4896/ac9dc6 doi: 10.1088/1402-4896/ac9dc6
[28]	D. N. Hao, T. T. Le, L. H. Nguyen, The Fourier-based dimensional reduction method for solving a nonlinear inverse heat conduction problem with limited boundary data, Commun. Nonlinear Sci. Numer. Simul., 128 (2024), 107679. https://doi.org/10.1016/j.cnsns.2023.107679 doi: 10.1016/j.cnsns.2023.107679
[29]	Y. Wang, Z. Qian, Regularizing a two-dimensional time-fractional inverse heat conduction problem by a fractional Landweber iteration method, Comput. Math. Appl., 164 (2024), 104–115. https://doi.org/10.1016/j.camwa.2024.04.001 doi: 10.1016/j.camwa.2024.04.001
[30]	Y. Wang, Z. Qian, A quasi-reversibility method for solving a two-dimensional time-fractional inverse heat conduction problem, Math. Comput. Simul., 212 (2023), 423–440. https://doi.org/10.1016/j.matcom.2023.05.012 doi: 10.1016/j.matcom.2023.05.012
[31]	L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
[32]	S. S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE Trans. Syst. Man Cybern., 2 (1972), 30–34. https://doi.org/10.1109/TSMC.1972.5408553
[33]	D. Dubois, H. Prade, Towards fuzzy differential calculus part 1: Integration of fuzzy mappings, Fuzzy Sets Syst., 8 (1982), 1–17. https://doi.org/10.1016/0165-0114(82)90025-2 doi: 10.1016/0165-0114(82)90025-2
[34]	O. Kaleva, The cauchy problem for fuzzy differential equations, Fuzzy Sets Syst., 35 (1990), 389–396. https://doi.org/10.1016/0165-0114(90)90010-4 doi: 10.1016/0165-0114(90)90010-4
[35]	S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets Syst., 24 (1987), 319–330. https://doi.org/10.1016/0165-0114(87)90030-3 doi: 10.1016/0165-0114(87)90030-3
[36]	B. Bede, Note on "Numerical solutions of fuzzy differential equations by predictor-corrector method", Inf. Sci., 178 (2008), 1917–1922. https://doi.org/10.1016/j.ins.2007.11.016 doi: 10.1016/j.ins.2007.11.016
[37]	N. Mikaeilvand, S. Khakrangin, Solving fuzzy partial differential equations by fuzzy two-dimensional differential transform method, Neural Comput. Appl., 21 (2012), 307–312. https://doi.org/10.1007/s00521-012-0901-x doi: 10.1007/s00521-012-0901-x
[38]	M. S. Cecconello, R. C. Bassanezi, A. J. V. Brandao, J. Leite, On the stability of fuzzy dynamical systems, Fuzzy Sets Syst., 248 (2014), 106–121. https://doi.org/10.1016/j.fss.2013.12.009 doi: 10.1016/j.fss.2013.12.009
[39]	M. S. Cecconello, J. Leite, R. C. Bassanezi, A. J. V. Brandao, Invariant and attractor sets for fuzzy dynamical systems, Fuzzy Sets Syst., 265 (2015), 99–109. https://doi.org/10.1016/j.fss.2014.07.017 doi: 10.1016/j.fss.2014.07.017
[40]	S. Abbasbandy, J. Nieto, M. Alavi, Tuning of reachable set in one dimensional fuzzy differential inclusions, Chaos Solitons Fractals, 26 (2005), 1337–1341. https://doi.org/10.1016/j.chaos.2005.03.018 doi: 10.1016/j.chaos.2005.03.018
[41]	M. Chen, Y. Fu, X. Xue, C. Wu, Two-point boundary value problems of undamped uncertain dynamical systems, Fuzzy Sets Syst., 159 (2008), 2077–2089. https://doi.org/10.1016/j.fss.2008.03.006 doi: 10.1016/j.fss.2008.03.006
[42]	R. Agarwal, V. Lakshmikantham, J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal., 72 (2010), 2859–2862. https://doi.org/10.1016/j.na.2009.11.029 doi: 10.1016/j.na.2009.11.029
[43]	L. A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic, Fuzzy Sets Syst., 90 (1997), 111–127. https://doi.org/10.1016/S0165-0114(97)00077-8 doi: 10.1016/S0165-0114(97)00077-8
[44]	M. Mazandarani, N. Pariz, A. V. Kamyad, Granular differentiability of fuzzy-number-valued functions, IEEE Trans Fuzzy Syst., 26 (2018), 310–323. https://doi.org/10.1109/TFUZZ.2017.2659731 doi: 10.1109/TFUZZ.2017.2659731
[45]	Z. T. Gong, H. Yang, Ill-posed fuzzy initial-boundary value problems based on generalized differentiability and regularization, Fuzzy Sets Syst., 295 (2016), 99–113. https://doi.org/10.1016/j.fss.2015.04.016 doi: 10.1016/j.fss.2015.04.016
[46]	H. Yang, Z. T. Gong, Ill-posedness for fuzzy Fredholm integral equation of the first kind and regularization methods, Fuzzy Sets Syst., 358 (2019), 132–149. https://doi.org/10.1016/j.fss.2018.05.010 doi: 10.1016/j.fss.2018.05.010
[47]	H. Yang, Z. T. Gong, Numerical solutions for fuzzy Fredholm inregral equations of the first kind using Landweber iterative method, J. Intell. Fuzzy Syst., 38 (2020), 3059–3074. https://doi.org/10.3233/JIFS-190972 doi: 10.3233/JIFS-190972
[48]	O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24 (1987), 301–317. https://doi.org/10.1016/0165-0114(87)90029-7
[49]	H. Yang, F. Wang, L. N. Wang, Solving the homogeneous BVP of second order linear FDEs with fuzzy parameters under granular differentiability concept, J. Intell. Fuzzy Syst., 22 (2023), 6327–6340. https://doi.org/10.3233/JIFS-223003 doi: 10.3233/JIFS-223003
[50]	Z. Gouyandeh, T. Allahviranloo, S. Abbasbandy, A. Armand, A fuzzy solution of heat equation under generalized Hukuhara differentiability by fuzzy Fourier transform, Fuzzy Sets Syst., 309 (2017), 81–97. https://doi.org/10.1016/j.fss.2016.04.010 doi: 10.1016/j.fss.2016.04.010
[51]	H. Yang, Y. Chen, Lyapunov stability of fuzzy dynamical systems based on fuzzy-number-valued function granular differentiability, Commun. Nonlinear Sci. Numer. Simul., 133 (2024), 107984. https://doi.org/10.1016/j.cnsns.2024.107984 doi: 10.1016/j.cnsns.2024.107984

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