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Numerical Heat Transfer, Part B: Fundamentals
An International Journal of Computation and Methodology
Volume 69, 2016 - Issue 4
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Original Articles

To recover heat source G(x) + H(t) by using the homogenized function and solving rectangular differencing equations

Pages 351-363 | Received 24 Jun 2015, Accepted 10 Sep 2015, Published online: 22 Mar 2016

References

  • A. Karageorghis, D. Lesnic, and L. Marin, A Survey of Applications of the MFS to Inverse Problems, Inv. Prob. Sci. Eng., vol. 19, pp. 309–336, 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.
  • P. C. Rosenbloom and D. V. Widder, Expansions in Terms of Heat Polynomials and Associated Functions, Trans. Am. Math. Soc., vol. 92, pp. 220–266, 1956.
  • D. V. Widder, Series Expansions of Solutions of the Heat Equation in n-Dimensions, Ann. Math. Pure Appl., vol. 55, pp. 389–410, 1961.
  • D. V. Widder, Expansions in Series of Homogeneous Temperature Functions of the First and Second Kinds, Duke Math. J., vol. 36, pp. 495–510, 1969.
  • D. V. Widder, The Heat Equation, Academic Press, New York, 1975.
  • G. N. Hile and A. Stanoyevitch, Heat Polynomial Analogs for Higher Order Evolution Equations, Electr. J. Diff. Eqs., vol. 2001, pp. 1–19, 2001.
  • J. R. Cannon and P. Duchateau, Structural Identification of an Unknown Source Term in a Heat Equation, Inv. Prob., vol. 14, pp. 535–551, 1998.
  • E. G. Savateev and P. Duchateau, On Problems of Determining the Source Function in a Parabolic Equation, J. Inv. Ill-Posed Prob., vol. 3, pp. 83–102, 1995.
  • V. T. Borukhov and P. N. Vabishchevich, Numerical Solution of the Inverse Problem of Reconstructing a Distributed Right-Hand Side of a Parabolic Equation, Comput. Phys. Commun., vol. 126, pp. 32–36, 2000.
  • 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, pp. 375–388, 2006.
  • L. Ling, M. Yamamoto, and Y. C. Hon, Identification of Source Locations in Two-Dimensional Heat Equations, Inv. Prob., vol. 22, pp. 1289–1305, 2006.
  • L. Yan, C. L. Fu, and F. L. Yang, The Method of Fundamental Solutions for the Inverse Heat Source Problem, Eng. Anal. Bound. Elem., vol. 32, pp. 216–222, 2008.
  • 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, pp. 1228–1236, 2010.
  • L. Yang, M. Dehghan, J. N. Yu, and G. W. Luo, Inverse Problem of Time-Dependent Heat Sources Numerical Reconstruction, Math. Comput. Simul., vol. 81, pp. 1656–1672, 2011.
  • A. Hasanov, Identification of Spacewise and Time Dependent Source Terms in 1D Heat Conduction Equation from Temperature Measurement at a Final Time, Int. J. Heat Mass Transfer, vol. 55, pp. 2069–2080, 2012.
  • L. Yang, J. N. Yu, G. W. Luo, and Z. C. Deng, Reconstruction of a Space and Time Dependent Heat Source from Finite Measurement Data, Int. J. Heat Mass Transfer, vol. 55, pp. 6573–6581, 2012.
  • C.-S. Liu, A Self-adaptive LGSM to Recover Initial Condition or Heat Source of One-Dimensional Heat Conduction Equation by Using Only Minimal Boundary Thermal Data, Int. J. Heat Mass Transfer, vol. 54, pp. 1305–1312, 2011.
  • C.-S. Liu, An Iterative Algorithm for Identifying Heat Source by Using a DQ and a Lie-Group Method, Inv. Prob. Sci. Eng., vol. 23, pp. 67–92, 2015.
  • C.-S. Liu, Finding Unknown Heat Source in a Nonlinear Cauchy Problem by the Lie-Group Differential Algebraic Equations Method, Eng. Anal. Bound. Elem., vol. 50, pp. 148–156, 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. Dehghan, Identification of a Time Dependent Coefficient in a Partial Differential Equation Subject to an Extra Measurement, Numer. Meth. Partial Diff. Eq., vol. 21, pp. 611–622, 2005.
  • C.-S. Liu, A Two-Stage LGSM to Identify Time-Dependent Heat Source Through an Internal Measurement of Temperature, Int. J. Heat Mass Transfer, vol. 52, pp. 1635–1642, 2009.
  • W. Yeih and C.-S. Liu, A Three-Point BVP of Time-Dependent Inverse Heat Source Problems and Solving by a TSLGSM, Comput. Model. Eng. Sci., vol. 46, pp. 107–127, 2009.
  • C. L. Kuo, J. R. Chang, and C.-S. Liu, The Modified Polynomial Expansion Method for Solving the Inverse Heat Source Problems, Numer. Heat Transfer B, vol. 63, pp. 357–370, 2013.
  • 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 Lie-Group Shooting Method for Reconstructing a Past Time-Dependent Heat Source, Int. J. Heat Mass Transfer, vol. 55, pp. 1773–1781, 2012.
  • C.-S. Liu, On-line Detecting Heat Source of a Nonlinear Heat Conduction Equation by a Differential Algebraic Equation Method, Int. J. Heat Mass Transfer, vol. 76, pp. 153–161, 2014.
  • C.-S. Liu and S. N. Atluri, A Highly Accurate Technique for Interpolations Using Very High-Order Polynomials, and Its Applications to Some Ill-posed Linear Problems, Comput. Model. Eng. Sci., vol. 43, pp. 253–276, 2009.
  • C.-S. Liu, A Highly Accurate Multi-scale Full/Half-Order Polynomial Interpolation, Comput. Mater. Contin., vol. 25, pp. 239–263, 2011.
  • C.-S. Liu, A Two-Side Equilibration Method to Reduce the Condition Number of an Ill-posed Linear System, Comput. Model. Eng. Sci., vol. 91, pp. 17–42, 2013.
  • C.-S. Liu, An Equilibrated Method of Fundamental Solutions to Choose the Best Source Points for the Laplace Equation, Eng. Anal. Bound. Elem., vol. 36, pp. 1235–1245, 2012.
  • C.-S. Liu and S. N. Atluri, Numerical Solution of the Laplacian Cauchy Problem by Using a Better Postconditioning Collocation Trefftz Method, Eng. Anal. Bound. Elem., vol. 37, pp. 74–83, 2013.
  • C.-S. Liu, Optimally Scaled Vector Regularization Method to Solve Ill-posed Linear Problems, Appl. Math. Comput., vol. 218, pp. 10602–10616, 2012.

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