# Levenberg-marquardt Method for Solving the Inverse Heat Transfer Problems

Volume 13, Issue 4, pp 300-310
• 6304 Views

### Authors

Nasibeh Asa Golsorkhi - Department of mathematics , Shahrood University , Shahrood , Iran. Hojat Ahsani Tehrani - Department of mathematics , Shahrood University , Shahrood , Iran.

### Abstract

In this paper, The Levenberg-Marquardt method is used in order to solve the inverse heat conduction problem. One-dimensional formulation of heat conduction problem was used. The direct problem was solved with finite-volumes by using an implicit discretization in time. Simulated measurements are obtained from the solution of the direct Problem at the sensor location. Results obtained in this inverse problem will be justified based on the numerical experiments. The results show that the speed of convergence is considerably fast and The Levenberg-Marquardt method is an accurate and stable method to determine the strength of the heat source in the inverse heat conduction problems.

### Share and Cite

##### ISRP Style

Nasibeh Asa Golsorkhi, Hojat Ahsani Tehrani, Levenberg-marquardt Method for Solving the Inverse Heat Transfer Problems, Journal of Mathematics and Computer Science, 13 (2014), no. 4, 300-310

##### AMA Style

Golsorkhi Nasibeh Asa, Tehrani Hojat Ahsani, Levenberg-marquardt Method for Solving the Inverse Heat Transfer Problems. J Math Comput SCI-JM. (2014); 13(4):300-310

##### Chicago/Turabian Style

Golsorkhi, Nasibeh Asa, Tehrani, Hojat Ahsani. "Levenberg-marquardt Method for Solving the Inverse Heat Transfer Problems." Journal of Mathematics and Computer Science, 13, no. 4 (2014): 300-310

### Keywords

• Levenberg-Marquardt method
• inverse problem
• heat conduction .

•  80A23
•  80A20
•  65J15
•  65N21

### References

• [1] K . Levenberg, A Method for the Solution of Certain Non-linear Problems in Least Squares, Quart . Appl . Math ., 2 (1944), 164 -168.

• [2] D . W . Marquardt , An Algorithm for Least Squares Estimation of Nonlinear Parameters, J . Soc . Ind . Appl . Math ., 11 (1963), 431-441.

• [3] Y . B . Bard , Nonlinear Parameter Estimation, Acad . Press, New York (1974.)

• [4] J . V . Beck, K . J . Arnold , Parameter Estimation in Engineering and Science, Wiley , New York (1977)

• [5] J . Dennis, R . Schnabel , Numerical Methodr for Unconstrained Optimization and Nonlinear Equations, Prentice Hall , (1983)

• [6] W . H . Press, B . F . Flannery, S . A . Teukolsky, W . T . Wetterling, Numerical Recipes , Cambridge University Press, New York (1989)

• [7] M. N . Ozisik , Heat Conduction, 2nd ed . John Wiley, New York (1993)

• [8] S. R . Arridge, M . Schweiger, A General framework for iterative reconstruction algorithms in optical tomography , using a finite element method, Appl Opt , 42 (1998), 9683-9706.

• [9] E. Ghasemi, A. Ranjbar, A. Ramiar , Three-dimensional Numerical Analysis of Heat Transfer Characteristics of Solar Parabolic Collector With Two Segmental Rings, Journal of mathematics and computer Science, 7 (2013), 89 - 100.

• [10] H. Heidarzadeh, M. Mashinchi joubari, R. Asghari , Application of Adomian Decomposition Method to Nonlinear Heat Transfer Equation, Journal of mathematics and computer Science, 4.3 (2012), 436 – 447.

• [11] A. O . Kuye , C. O. C . Oko, S. N. Nnamchi, Determination of the thermal conductivity and specific heat capacity of neem seeds by inverse problem method, Journal of Engineering Science and Technology Review , 3.1 (2010), 1-6.

• [12] J. Pujo, The solution of nonlinear inverse problems and the Levenberg-Marquardt method, GEOPHYSICS, 72 (2007), 1-16.

• [13] A. N. Bondarenko, D. S. Ivaschenko , Levenberg–Marquardt Method for Restoration of Layered Medium Key Parameters, Russian-Korean International Symposium on Science and Technology, (2005), 43- 46.

• [14] R. K. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, (2002)

• [15] H. Versteeg, W. Malalasekera , An Introduction to Computational Fluid Dynamics: The Finite Volume Method , Prentice Hall, (2007)

• [16] M. N . Ozisik, H. R. B. Orlande, Inverse Heat Transfer Fundamentals and Applications , Taylor and Francis, New York (2000)