Publication Cover
Numerical Heat Transfer, Part B: Fundamentals
An International Journal of Computation and Methodology
Volume 73, 2018 - Issue 1
163
Views
3
CrossRef citations to date
0
Altmetric
Original Articles

Numerically solving twofold ill-posed inverse problems of heat equation by the adjoint Trefftz method

ORCID Icon, &
Pages 48-61 | Received 25 Oct 2017, Accepted 12 Dec 2017, Published online: 16 Jan 2018

References

  • H. Han, D. B. Ingham, and Y. Yuan, The Boundary Element Method for the Solution of the Backward Heat Conduction Equation, J. Comput. Phys., vol. 116, pp. 292–299, 1995.
  • N. S. Mera, L. Elliott, D. B. Ingham, and D. Lesnic, An Iterative Boundary Element Method for Solving the One-Dimensional Backward Heat Conduction Problem, Int. J. Heat Mass Transfer, vol. 44, pp. 1937–1946, 2001.
  • N. S. Mera, L. Elliott, and D. B. Ingham, An Inversion Method with Decreasing Regularization for the Backward Heat Conduction Problem, Numer. Heat Transfer B, vol. 42, pp. 215–230, 2002.
  • W. B. Muniz and H. F. de Campos Velho, and F. M. Ramos, A Comparison of Some Inverse Methods for Estimating the Initial Condition of the Heat Equation, J. Comput. Appl. Math., vol. 103, pp. 145–163, 1999.
  • W. B. Muniz, F. M. Ramos, and H. F. de, Campos Velho, Entropy- and Tikhonov-Based Regularization Techniques Applied to the Backward Heat Equation, Int. J. Comput. Math., vol. 40, pp. 1071–1084, 2000.
  • S. M. Kirkup and M. Wadsworth, Solution of Inverse Diffusion Problems by Operator-Splitting Methods, Appl. Math. Model., vol. 26, pp. 1003–1018, 2002.
  • K. Iijima, Numerical Solution of Backward Heat Conduction Problems by a High Order Lattice-Free Finite Difference Method, J. Chin. Inst. Eng., vol. 27, pp. 611–620, 2004.
  • Y. C. Hon and M. Li, A Discrepancy Principle for the Source Points Location in Using the MFS for Solving the BHCP, Int. J. Comput. Methods, vol. 6, pp. 181–197, 2009.
  • C. H. Tsai, D. L. Young, and J. Kolibal, An Analysis of Backward Heat Conduction Problems Using the Time Evolution Method of Fundamental Solutions, Comput. Model. Eng. Sci., vol. 66, pp. 53–71, 2010.
  • Z. Qian, C. L. Fu, and R. Shi, A Modified Method for a Backward Heat Conduction Problem, Appl. Math. Comput., vol. 185, pp. 564–573, 2007.
  • C. L. Fu, X. T. Xiong, and Z. Qian, Fourier Regularization for a Backward Heat Equation, J. Math. Anal. Appl., vol. 331, pp. 472–480, 2007.
  • X. T. Xiong, C. L. Fu, and Z. Qian, On Three Spectral Regularization Methods for a Backward Heat Conduction Problem, J. Korean Math. Soc., vol. 44, pp. 1281–1290, 2007.
  • M. Li, T. Jiang, and Y. C. Hon, A Meshless Method Based on RBFs Method for Nonhomogeneous Backward Heat Conduction Problem, Eng. Anal. Boundary Elem., vol. 34, pp. 785–792, 2010.
  • G. W. Clark and S. F. Oppenheimer, Quasireversibility Methods for Non-Well-Posed Problems, Electron. J. Differ. Eqs., vol. 1994, pp. 1–9, 1994.
  • K. A. Ames, G. W. Clark, J. F. Epperson, and S. F. Oppenheimer, A Comparison of Regularizations for an Ill-Posed Problem, Math. Comput., vol. 67, pp. 1451–1471, 1998.
  • C.-S. Liu, Group Preserving Scheme for Backward Heat Conduction Problems, Int. J. Heat Mass Transfer, vol. 47, pp. 2567–2576, 2004.
  • C.-S. Liu, C. W. Chang, and J. R. Chang, Past Cone Dynamics and Backward Group Preserving Schemes for Backward Heat Conduction Problems, Comput. Model. Eng. Sci., vol. 12, pp. 67–81, 2006.
  • J. R. Chang, C.-S. Liu, and C. W. Chang, A New Shooting Method for Quasi-Boundary Regularization of Backward Heat Conduction Problems, Int. J. Heat Mass Transfer, vol. 50, pp. 2325–2332, 2007.
  • C. W. Chang, C.-S. Liu, and J. R. Chang, ”A Quasi-Boundary Semi-Analytical Method for Backward Heat Conduction Problems, J. Chin. Inst. Eng., vol. 33, pp. 163–175, 2010.
  • C.-S. Liu, A New Method for Fredholm Integral Equations of 1D Backward Heat Conduction Problems, Comput. Model. Eng. Sci., vol. 47, pp. 1–21, 2009.
  • C. W. Chang, C.-S. Liu, and J. R. Chang, A New Shooting Method for Quasi-Boundary Regularization of Multi-Dimensional Backward Heat Conduction Problems, J. Chin. Inst. Eng., vol. 32, pp. 307–318, 2009.
  • C.-S. Liu, A Highly Accurate LGSM for Severely Ill-Posed BHCP Under a Large Noise on the Final Time Data, Int. J. Heat Mass Transfer, vol. 53, pp. 4132–4140, 2010.
  • C. W. Chang and C.-S. Liu, A New Algorithm for Direct and Backward Problems of Heat Conduction Equation, Int. J. Heat Mass Transfer, vol. 52, pp. 5552–5569, 2010.
  • C.-S. Liu, A Self-Adaptive LGSM to Recover Initial Condition or Heat Source of One-Dimensional Heat Conduction by Using Only Minimal Boundary Data, Int. J. Heat Mass Transfer, vol. 54, pp. 1305–1312, 2011.
  • C.-S. Liu, The Method of Fundamental Solutions for Solving the Backward Heat Conduction Problem with Conditioning by a New Post-Conditioner, Numer. Heat Transfer, B, vol. 60, pp. 57–72, 2011.
  • C.-S. Liu and C. W. Chang, A Simple Algorithm for Solving Cauchy Problem of Nonlinear Heat Equation without Initial Value, Int. J. Heat Mass Transfer, vol. 80, pp. 562–569, 2015.
  • C.-S. Liu and C. W. Chang, Nonlinear Problems with Unknown Initial Temperature and without Final Temperature, Solved by the GL(N,R) Shooting Method, Int. J. Heat Mass Transfer, vol. 83, pp. 655–678, 2015.
  • C.-S. Liu, C. L. Kuo, and J. R. Chang, Recovering a Heat Source and Initial Value by a Lie-Group Differential Algebraic Equations Method, Numer. Heat Transfer, B, vol. 67, pp. 231–254, 2015.
  • M. Tadi, An Iterative Method for the Solution of Ill-Posed Parabolic Systems, Appl. Math. Comput., vol. 201, pp. 843–851, 2008.
  • C.-S. Liu, Lie-Group Differential Algebraic Equations Method to Recover Heat Source in a Cauchy Problem with Analytic Continuation Data, Int. J. Heat Mass Transfer, vol. 78, pp. 538–547, 2014.
  • C.-S. Liu, A Multiple/Scale/Direction Polynomial Trefftz Method for Solving the BHCP in High-Dimensional Arbitrary Simply-Connected Domains, Int. J. Heat Mass Transfer, vol. 92, pp. 970–978, 2016.
  • C.-S. Liu and C. W. Chang, A Global Boundary Integral Equation Method for Recovering Space-Time Dependent Heat Source, Int. J. Heat Mass Transfer, vol. 92, pp. 1034–1040, 2016.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.