Abstract
In this paper, we consider an inverse heat conduction problem (IHCP). First, the existence, uniqueness and unstability solution of the inverse problem will be studied. Due to the ill-posedness of the inverse problem, we propose a meshless numerical approach based on basis function to solve this problem in the presence of noisy data. The Tikhonov regularization method with generalized discrepancy principle is applied to obtain a stable numerical approximation to the solution. The effectiveness of the algorithm is illustrated by some numerical examples.
Acknowledgements
The authors are very grateful to the reviewer for carefully reading the paper and for his/her comments and suggestions, which have improved the paper.
References
[1] Alifanov O. M., Inverse Heat Transfer Problems, Springer, New York, 1994. 10.1007/978-3-642-76436-3Search in Google Scholar
[2] Alves C. J. S., Chen C. S. and Saler B., The method of fundamental solutions for solving Poisson problems, Boundary Elements XXIV: Incorporating Meshless Solutions. 24th World Conference (Sintra 2002), WIT Press, Southampton (2002), 67–76. Search in Google Scholar
[3] Baumeister J., Stable Solution of Inverse Problems, Vieweg, Wiesbaden, 1987. 10.1007/978-3-322-83967-1Search in Google Scholar
[4] Beck J. V., Blackwell B. and Chair S. R., Inverse Heat Conduction: Ill-Posed Problems, John Wiley & Sons, New York, 1985. Search in Google Scholar
[5] Beck J. V., Blackwell B. and Haji-sheikh A., Comparison of some inverse heat conduction methods using experimental data, Int. J. Heat Mass Transfer 3 (1996), 3649–3657. 10.1016/0017-9310(96)00034-8Search in Google Scholar
[6] Beck J. V. and Murio D. C., Combined function specification-regularization procedure for solution of inverse heat condition problem, AIAA J. 24 (1986), 180–185. 10.2514/3.9240Search in Google Scholar
[7] Cabeza J. M. G., Garcia J. A. M. and Rodriguez A. C., A sequential algorithm of inverse heat conduction problems using singular value decomposition, Int. J. Thermal Sci. 44 (2005), 235–244. 10.1016/j.ijthermalsci.2004.06.009Search in Google Scholar
[8] Cannon J. R., The One-Dimensional Heat Equation, Addison Wesley, Reading, 1984. 10.1017/CBO9781139086967Search in Google Scholar
[9] Engl H. W., Hanke M. and Neubauer A., Regularization of Inverse Problems, Kluwer Academic, Dordrecht, 2000. 10.1007/978-94-009-1740-8_3Search in Google Scholar
[10] Hon Y. and Wei T., A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem. 28 (2004), 489–495. 10.1016/S0955-7997(03)00102-4Search in Google Scholar
[11] Huang C.-H. and Tsai Y. L., A transient 3-D inverse problem in imaging the time-dependent local heat transfer coefficients for plate fin, Appl. Therm. Eng. 25 (2005), 2478–2495. 10.1016/j.applthermaleng.2004.12.003Search in Google Scholar
[12] Huang C.-H., Yeh C.-Y. and Orlande H. R. B., A nonlinear inverse problem in simultaneously estimating the heat and mass production rates for a chemically reacting fluid, Chem. Eng. Sci. 58 (2003), no. 16, 3741–3752. 10.1016/S0009-2509(03)00270-7Search in Google Scholar
[13] Kim S. K., Jung B. S. and Lee W. L., An inverse estimation of surface temperature using the maximum entropy method, Int. Commun. Heat Mass Transf. 34 (2007), 37–44. 10.1615/IHTC12.180Search in Google Scholar
[14] Li W., Liu X. and Yao G., A local meshless collocation method for solving certain inverse problems, Eng. Anal. Bound. Elem. 57 (2015), 9–15. 10.1016/j.enganabound.2014.11.034Search in Google Scholar
[15] Lu S., Pereverzyev S., Shao Y. and Tautenhahn U., On the generalized discrepancy principle for Tikhonov regularization in Hilbert scales, preprint 2009, http://www.ricam.oeaw.ac.at/publications/reports/09/rep09-19.pdf. 10.1216/JIE-2010-22-3-483Search in Google Scholar
[16] Marin L., A meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations, Eng. Anal. Bound. Elem. 34 (2010), 274–288. 10.1016/j.enganabound.2009.03.009Search in Google Scholar
[17] Murio D. C. and Paloschi J. R., Combined mollification-future temperature procedure for solution of inverse heat conduction problem, J. Comput. Appl. Math. 23 (1988), 235–244. 10.1016/0377-0427(88)90282-8Search in Google Scholar
[18] Pourgholi R., Dana H. and Tabasi S. H., Solving an inverse heat conduction problem using genetic algorithm: Sequential and multi-core parallelization approach, Appl. Math. Model. 38 (2014), 1948–1958. 10.1016/j.apm.2013.10.019Search in Google Scholar
[19] Pourgholi R. and Rostamian M., A numerical technique for solving IHCPs using Tikhonov regularization method, Appl. Math. Model. 34 (2010), 2102–2110. 10.1016/j.apm.2009.10.022Search in Google Scholar
[20] Reemtsen R. and Kirsch A., A method for the numerical solution of the one-dimensional inverse Stefan problem, Numer. Math. 45 (1984), 253–273. 10.1007/BF01389470Search in Google Scholar
[21] Rieder A., Keine Probleme mit Inversen Problemen, Vieweg, Wiesbaden, 2003. 10.1007/978-3-322-80234-7Search in Google Scholar
[22] Sarvari S. M. H., Howell J. R. and Mansouri S. H., Inverse boundary design conduction-radiation problem in irregular two-dimensional domains, Numer. Heat Transfer B Fundam. 44 (2003), no. 3, 209–224. 10.1080/713836377Search in Google Scholar
[23] Siegel R. and Howell J. R., Thermal Radiation Heat Transfer, 4th ed., Taylor & Francis, New York, 1992. Search in Google Scholar
[24] Su J. and Silva Neto A. J., Two-dimensional inverse heat conduction problems of source strength estimation in cylindrical rods, Appl. Math. Model. 25 (2001), 861–872. 10.1016/S0307-904X(01)00018-XSearch in Google Scholar
[25] Taigbenu A. E., Inverse solutions of temperature, heat flux and heat source by the Green element method, Appl. Math. Model. 39 (2015), 667–681. 10.1016/j.apm.2014.06.020Search in Google Scholar
[26] Tautenhahn U. and Hämarik U., The use of monotonicity for choosing the regularization parameter in ill-posed problems, Inverse Problems 15 (1999), 1487–1505. 10.1088/0266-5611/15/6/307Search in Google Scholar
[27] Tikhonov A. N. and Arsenin V. Y., Methods for Solving Ill-Posed Problems, Nauka, Moscow, 1986. Search in Google Scholar
[28] Woodbury K. A., Beck J. V. and Najafi H., Filter solution of inverse heat conduction problem using measured temperature history as remote boundary condition, Int. J. Heat Mass Transfer 72 (2014), 139–147. 10.1016/j.ijheatmasstransfer.2013.12.073Search in Google Scholar
[29] Zhou J., Zhang Y., Chen J. K. and Feng Z. C., Inverse heat conduction in a composite slab with pyrolysis effect and temperature-dependent thermophysical properties, J. Heat Transfer 132 (2010), no. 3, 471–481. 10.1115/IMECE2009-10134Search in Google Scholar
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Articles in the same Issue
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Articles in the same Issue
- Frontmatter
- A finite element method for the inverse problem of boundary data recovery in an oxygen balance model
- On regularization and error estimates for the Cauchy problem of the modified inhomogeneous Helmholtz equation
- Application of the factorization method to retrieve a crack from near field data
- Solution to a class of inverse problems for a system of loaded ordinary differential equations with integral conditions
- The variational formulation of an inverse problem for multidimensional nonlinear time-dependent Schrödinger equation
- Integral identity for a class of ill-posed problems generated by a parabolic equation
- A meshless method to the solution of an ill-posed problem
- About an inverse problem for a free boundary compressible problem in hydrodynamic lubrication
- Error analysis for the operator marching method applied to range dependent waveguides