Recovering A Heat Source and Initial Value by a Lie-Group Differential Algebraic Equations Method

REFERENCES

• J. V. Beck and K. J. Arnold , Parameter Estimation in Engineering and Science , Wiley , New York , 1997 .
   Google Scholar
• R. Luce and D. Perez , Parameter Identification for an Elliptic Partial Differential Equation with Distributed Noisy Data , Inverse Prob. , vol. 15 , pp. 291 – 307 , 1995 .
   Google Scholar
• G. Stolz , Numerical Solutions to an Inverse Problem of Heat Conduction for Simple Shapes , J. Heat Transfer , vol. 82 , pp. 20 – 26 , 1960 .
   Google Scholar
• 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 .
   Web of Science ®Google Scholar
• M. Dehghan and M. Tatari , The Radial Basis Functions Method for Identifying an Unknown Parameter in a Parabolic Equation with Overspecified Data , Numer. Meth. Partial Diff. Eq. , vol. 23 , pp. 984 – 997 , 2007 .
   Web of Science ®Google Scholar
• C.-S. Liu , An LGEM to Identify Time-Dependent Heat Conductivity Function by an Extra Measurement of Temperature Gradient , CMC: Comput. Mater. Continua , vol. 7 , pp. 81 – 96 , 2008 .
   Web of Science ®Google Scholar
• C.-S. Liu , An LGSM to Identify Nonhomogeneous Heat Conductivity Functions by an Extra Measurement of Temperature , Int. J. Heat Mass Transfer , vol. 51 , pp. 2603 – 2613 , 2008 .
   Web of Science ®Google Scholar
• B. T. Johansson and D. A. Lesnic , A Procedure for Determining a Spacewise Dependent Heat Source and the Initial Temperature , Appl. Anal. , vol. 87 , pp. 265 – 276 , 2008 .
   Web of Science ®Google Scholar
• 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, pp. 1848–1855, 2012.
   Web of Science ®Google Scholar
• 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 .
   Web of Science ®Google Scholar
• M. N. Ahmadabadi , M. Arab , and F. M. M. Ghaini , The Method of Fundamental Solutions for the Inverse Space-Dependent Heat Source Problem , Eng. Anal. Bound. Elem. , vol. 33 , pp. 1231 – 1235 , 2009 .
   Web of Science ®Google Scholar
• 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 .
   Web of Science ®Google Scholar
• L. Yang , Z. C. Deng , J. N. Yu , and G. W. Guo , Optimization Method for the Inverse Problem of Reconstructing the Source Term in a Parabolic Equation , Math. Comput. Simul. , vol. 80 , pp. 314 – 326 , 2009 .
   Web of Science ®Google Scholar
• J. R. Cannon and P. Duchateau , Structural Identification of an Unknown Source Term in a Heat Equation , Inverse Prob. , vol. 14 , pp. 535 – 551 , 1998 .
   Web of Science ®Google Scholar
• E. G. Savateev and P. Duchateau , On Problems of Determining the Source Function in a Parabolic Equation , J. Inv. Ill-Posed Problem , vol. 3 , pp. 83 – 102 , 1995 .
   Google Scholar
• V. T. Borukhov and P. N. Vabishchevich , Numerical Solution of the Inverse Problem of Reconstructing a Distributed Right-Hand Side of a Parabolic Equation , Comp. Phys. Commun. , vol. 126 , pp. 32 – 36 , 2000 .
   Web of Science ®Google Scholar
• L. Ling , M. Yamamoto , and Y. C. Hon , Identification of Source Locations in Two-Dimensional Heat Equations , Inverse Prob. , vol. 22 , pp. 1289 – 1305 , 2006 .
   Web of Science ®Google Scholar
• 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 .
   Web of Science ®Google Scholar
• W. Yeih and C.-S. Liu , A Three-Point BVP of Time-Dependent Inverse Heat Source Problems and Solving by a TSLGSM , CMES: Comput. Model. Eng. Sci. , vol. 46 , pp. 107 – 127 , 2009 .
   Web of Science ®Google Scholar
• 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 .
   Web of Science ®Google Scholar
• 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 .
   Web of Science ®Google Scholar
• C.-S. Liu , An Iterative Method to Recover Heat Conductivity Function of a Nonlinear Heat Conduction Equation , Numer. Heat Transfer B , vol. 65 , pp. 80 – 101 , 2014 .
   Web of Science ®Google Scholar
• C.-S. Liu , An Iterative Algorithm for Identifying Heat Source by Using a DQ and a Lie-Group Method , Inverse Prob. Sci. Eng. , doi: dx.doi.org/10.1080/17415977.2014.880907 , 2014 .
   Web of Science ®Google Scholar
• Y. Wang , J. Cheng , J. Nakagawa , and M. Yamamoto , A Numerical Method for Solving the Inverse Heat Conduction Problem Without Initial Value , Inverse Prob. Sci. Eng. , vol. 18 , pp. 655 – 671 , 2010 .
   Web of Science ®Google Scholar
• C.-S. Liu , A Method of Lie-Symmetry GL(n,ℝ) for Solving Non-Linear Dynamical Systems , Int. J. Non-Linear Mech. , vol. 52 , pp. 85 – 95 , 2013 .
   Web of Science ®Google Scholar
• C.-S. Liu , A State Feedback Controller Used to Solve an Ill-Posed Linear System by a GL(n,ℝ) Iterative Algorithm , Commun. Numer. Anal. , 2013 ( 2013 ), Article ID cna-00181, 22 pages.
   Google Scholar
• C.-S. Liu , Solving Nonlinear Differential Algebraic Equations by an Implicit GL(n,ℝ) Lie-group Method , J. Appl. Math. , ID 987905, 2013.
   Google Scholar
• C.-S. Liu , A New Sliding Control Strategy for Nonlinear System Solved by the Lie-Group Differential Algebraic Equation Method , Commun. Nonlinear Sci. Numer. Simul. , vol. 19 , pp. 2012 – 2038 , 2014 .
   Web of Science ®Google Scholar
• C.-S. Liu , A Lie-Group Adaptive Differential Quadrature Method to Identify Unknown Force in an Euler-Bernoulli Beam Equation , Acta Mech. , vol. 223 , pp. 2207 – 2223 , 2012 .
   Web of Science ®Google Scholar
• C.-S. Liu , A Manifold-Based Exponentially Convergent Algorithm for Solving Non-Linear Partial Differential Equations , J. Marine Sci. Technol. , vol. 20 , pp. 441 – 449 , 2012 .
   Web of Science ®Google Scholar
• Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/unhb.
   Google Scholar