References
- X. T. Xiong, X. H. Liu, Y. M. Yan, and H. B. Guo, “A numerical method for identifying heat transfer coefficient,” Appl. Math. Model., vol. 34, no. 7, pp. 1930–1938, 2010. DOI: https://doi.org/10.1016/j.apm.2009.10.010.
- J. Cheng, Y. Ke, and T. Wei, “The backward problem of parabolic equations with the measurements on a discrete set,” J. Inverse III-Posed Probl., vol. 28, no. 1, pp. 137–144, 2020. DOI: https://doi.org/10.1515/jiip-2019-0079.
- N. Mera, L. Elliott, D. Ingham, and D. Lesnic, “An iterative boundary element method for solving the one dimensional backward heat conduction problem,” Int. J. Heat Mass Transf., vol. 44, no. 10, pp. 1937–1946, 2001. DOI: https://doi.org/10.1016/S0017-9310(00)00235-0.
- X. T. Xiong, C. L. Fu, and Z. Qian, “Two numerical methods for solving a backward heat conduction problem,” Appl. Math. Comput., vol. 179, no. 1, pp. 370–377, 2006. DOI: https://doi.org/10.1016/j.amc.2005.11.114.
- L. Eldén, F. Berntsson, and T. Regińska, “Wavelet and Fourier methods for solving the sideways heat equation,” SIAM J. Sci. Comput., vol. 21, no. 6, pp. 2187–2205, 2000. DOI: https://doi.org/10.1137/S1064827597331394.
- X. T. Xiong, J. X. Wang, and M. Li, “An optimal method for fractional heat conduction problem backward in time,” Appl. Anal., vol. 91, no. 4, pp. 823–840, 2012. DOI: https://doi.org/10.1080/00036811.2011.601455.
- Y. Yu, D. Xu, and Y. C. Hon, “Reconstruction of inaccessible boundary value in a sideways parabolic problem with variable coefficients—forward collocation with finite integration method,” Eng. Anal. Bound. Elem., vol. 61, pp. 78–90, 2015. DOI: https://doi.org/10.1016/j.enganabound.2015.07.007.
- L. Eldén and F. Berntsson, “A stability estimate for a Cauchy problem for an elliptic partial differential equation,” Inverse Probl., vol. 21, no. 5, pp. 1643–1653, 2005. DOI: https://doi.org/10.1088/0266-5611/21/5/008.
- H. W. Engl and J. Zou, “A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction,” Inverse Probl., vol. 16, no. 6, pp. 1907–1923, 2000. DOI: https://doi.org/10.1088/0266-5611/16/6/319.
- Y. C. Hon and M. Li, “A computational method for inverse free boundary determination problem,” Int. J. Numer. Meth. Eng., vol. 73, no. 9, pp. 1291–1309, 2008. DOI: https://doi.org/10.1002/nme.2122.
- Y. S. Li and T. Wei, “Identification of a moving boundary for a heat conduction problem in a multilayer medium,” Heat Mass Transf., vol. 46, no. 7, pp. 779–789, 2010. DOI: https://doi.org/10.1007/s00231-010-0621-7.
- T. Wei and Y. S. Li, “An inverse boundary problem for one-dimensional heat equation with a multilayer domain,” Eng. Anal. Bound. Elem., vol. 33, no. 2, pp. 225–232, 2009. DOI: https://doi.org/10.1016/j.enganabound.2008.04.006.
- A. Farcas and D. Lesnic, “The boundary-element method for the determination of a heat source dependent on one variable,” J. Eng. Math., vol. 54, no. 4, pp. 375–388, 2006. DOI: https://doi.org/10.1007/s10665-005-9023-0.
- B. T. Johansson and D. Lesnic, “A variational method for identifying a spacewise-dependent heat source,” IMA J. Appl. Math., vol. 72, no. 6, pp. 748–760, 2007. DOI: https://doi.org/10.1093/imamat/hxm024.
- J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations. New York: Dover Publications, 1953.
- C. L. Fu, X. T. Xiong, and Z. Qian, “Fourier regularization for a backward heat equation,” J. Math. Anal. Appl., vol. 331, no. 1, pp. 472–480, 2007. DOI: https://doi.org/10.1016/j.jmaa.2006.08.040.
- F. Yang and C. L. Fu, “Two regularization methods to identify time-dependent heat source through an internal measurement of temperature,” Math. Comput. Model., vol. 53, no. 5–6, pp. 793–804, 2011. DOI: https://doi.org/10.1016/j.mcm.2010.10.016.
- F. Yang and C. L. Fu, “A mollification regularization method for the inverse spatial-dependent heat source problem,” J. Comput. Appl. Math., vol. 255, pp. 555–567, 2014. DOI: https://doi.org/10.1016/j.cam.2013.06.012.
- T. Wei and J. C. Wang, “Simultaneous determination for a space-dependent heat source and the initial data by the MFS,” Eng. Anal. Bound. Elem., vol. 36, no. 12, pp. 1848–1855, 2012. DOI: https://doi.org/10.1016/j.enganabound.2012.07.006.
- J. Wen, “A meshless method for reconstructing the heat source and partial initial temperature in heat conduction,” Inverse Probl. Sci. Eng., vol. 19, no. 7, pp. 1007–1022, 2011. DOI: https://doi.org/10.1080/17415977.2011.569711.
- J. Wen, M. Yamamoto, and T. Wei, “Simultaneous determination of a time-dependent heat source and the initial temperature in an inverse heat conduction problem,” Inverse Probl. Sci. Eng., vol. 21, no. 3, pp. 485–499, 2013. DOI: https://doi.org/10.1080/17415977.2012.701626.
- L. Yang, Z. C. Deng, and Y. C. Hon, “Simultaneous identification of unknown initial temperature and heat source,” Dynam. Syst. Appl., vol. 25, pp. 583–602, 2016.
- F. F. Dou, C. L. Fu, and F. L. Yang, “Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation,” J. Comput. Appl. Math., vol. 230, no. 2, pp. 728–737, 2009. DOI: https://doi.org/10.1016/j.cam.2009.01.008.
- F. Yang and C. L. Fu, “The method of simplified Tikhonov regularization for dealing with the inverse time-dependent heat source problem,” Comput. Math. Appl., vol. 60, no. 5, pp. 1228–1236, 2010. DOI: https://doi.org/10.1016/j.camwa.2010.06.004.
- F. Yang and C. L. Fu, “A simplified Tikhonov regularization method for determining the heat source,” Appl. Math. Model., vol. 34, no. 11, pp. 3286–3299, 2010. DOI: https://doi.org/10.1016/j.apm.2010.02.020.
- F. F. Dou and C. L. Fu, “Determining an unknown source in the heat equation by a wavelet dual least squares method,” Appl. Math. Lett., vol. 22, no. 5, pp. 661–667, 2009. DOI: https://doi.org/10.1016/j.aml.2008.08.003.
- Z. Qian, “Optimal modified method for a fractional-diffusion inverse heat conduction problem,” Inverse Probl. Sci. Eng., vol. 18, no. 4, pp. 521–533, 2010. DOI: https://doi.org/10.1080/17415971003624348.
- X. T. Xiong, X. M. Xue, and Z. Qian, “A modified iterative regularization method for ill-posed problems,” Appl. Numer. Math., vol. 122, pp. 108–128, 2017. DOI: https://doi.org/10.1016/j.apnum.2017.08.004.
- F. Yang and C. L. Fu, “Two regularization methods for identification of the heat source depending only on spatial variable for the heat equation,” J. Inverse III-Posed Probl., vol. 17, pp. 815–830, 2009. DOI: https://doi.org/10.1515/JIIP.2009.048.
- W. Cheng and Q. Zhao, “A modified quasi-boundary value method for a two-dimensional inverse heat conduction problem,” Comput. Math. Appl., vol. 79, no. 2, pp. 293–302, 2020. DOI: https://doi.org/10.1016/j.camwa.2019.06.031.
- Z. Ruan, S. Zhang, and S. Xiong, “Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method,” Evol. Equ. Control Theory, vol. 7, no. 4, pp. 669–682, 2018. DOI: https://doi.org/10.3934/eect.2018032.
- A. L. Qian, X. T. Xiong, and Y. J. Wu, “On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation,” J. Comput. Appl. Math., vol. 233, no. 8, pp. 1969–1979, 2010. DOI: https://doi.org/10.1016/j.cam.2009.09.031.
- L. E. Payne, “Improperly posed problems in partial differential equations,” Society for Industrial and Applied Mathematics, Philadelphia, PA, 1975, Regional Conference Series in Applied Mathematics, No. 22.
- M. A. Iglesias, K. J. H. Law, and A. M. Stuart, “Ensemble Kalman methods for inverse problems,” Inverse Probl., vol. 29, no. 4, p. 045001, 2013. DOI: https://doi.org/10.1088/0266-5611/29/4/045001.
- T. Y. Xiao, Y. Zhao, and G. Z. Su, Extrapolation Techniques of Tikhonov Regularization, in Optimization and Regularization for Computational Inverse Problems and Applications. Heidelberg: Springer, 2010, pp. 107–126. DOI: https://doi.org/10.1007/978-3-642-13742-6\_5.
- J. Wen, L. M. Huang, and Z. X. Liu, “A modified quasi-reversibility method for inverse source problem of Poisson equation,” Inverse Probl. Sci. Eng., vol. 29, no. 12, pp. 2098–2109, 2021. DOI: https://doi.org/10.1080/17415977.2021.1902516.
- J. G. Wang and T. Wei, “Quasi-reversibility method to identify a space-dependent source for the time-fractional diffusion equation,” Appl. Math. Model., vol. 39, no. 20, pp. 6139–6149, 2015. DOI: https://doi.org/10.1016/j.apm.2015.01.019.
- Y. Wang and B. Wu, “On the convergence rate of an improved quasi-reversibility method for an inverse source problem of a nonlinear parabolic equation with nonlocal diffusion coefficient,” Appl. Math. Lett., vol. 121, pp. 107491–8, 2021. DOI: https://doi.org/10.1016/j.aml.2021.107491.
- F. Yang, Y. P. Ren, and X. X. Li, “The quasi-reversibility method for a final value problem of the time-fractional diffusion equation with inhomogeneous source,” Math. Methods Appl. Sci., vol. 41, no. 5, pp. 1774–1795, 2018. DOI: https://doi.org/10.1002/mma.4705.
- T. T. Le, L. H. Nguyen, T. P. Nguyen, and W. Powell, “The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations,” J. Sci. Comput., vol. 87, no. 90, pp. 1–23, 2021. DOI: https://doi.org/10.1007/s10915-021-01501-3.
- L. H. Nguyen, “An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method,” Inverse Probl., vol. 35, no. 3, p. 035007, 2019. DOI: https://doi.org/10.1088/1361-6420/aafe8f.
- Z. Qian, C. L. Fu, and R. Shi, “A modified method for a backward heat conduction problem,” Appl. Math. Comput., vol. 185, no. 1, pp. 564–573, 2007. DOI: https://doi.org/10.1016/j.amc.2006.07.055.
- L. Eldén, “Approximations for a Cauchy problem for the heat equation,” Inverse Probl., vol. 3, pp. 263–273, 1987. http://stacks.iop.org/0266-5611/3/263.