**Author**

School of Mathematics and Computer Science, Damghan University, P.O. Box 36715-364, Damghan, Iran.

**Abstract**

In this paper, a numerical solution of an inverse non-dimensional heat conduction problem by spline method will be considered. The given heat conduction equation, the boundary condition, and the initial condition are presented in a dimensionless form. A set of temperature measurements at a single sensor location inside the heat conduction body is required. The result show that the proposed method can predict the unknown parameters in the current inverse problem with an acceptable error.

**Keywords**

1. O. M. Alifanov, Inverse Heat Transfer Problems, Springer, NewYork, 1994.

2. N. Al-Khalidy, Analysis of bundary inverse heat conduction problems using space marching with Savitzky-Gollay digital filter, Int. Commun. Heat Mass Transfer 26 (2) (1999) 199-208.

3. J. V. Beck, B. Blackwell, C.R. St. Clair, Inverse Heat Conduction Ill-posed Problems, Wiley, New York, 1985, PP. 1-8.

4. 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.

5. Jianhua Zhou, Yuwen Zhang, J. K. Chen, and Z. C. Feng, Inverse Heat Conduction in a Composite Slab With Pyrolysis Effect and Temperature-Dependent Thermophysical Properties, J. Heat Transfer, 132 (3) (2010) 034502 (3 pages) .

6. T.C. Chen, P.C. Tuan, Input estimation method including finite-element scheme for solving inverse heat conduction problems, Num. Heat Transfer, Part B: Fundamentals 47 (3) (2005) 277-290.

7. D. Lesnic, L.Elliott, D.B. Ingham, Application of the boundary element method to inverse heat conduction problems, Int. J. Heat Mass Transfer 39 (7) (1996) 1503-1517.

8. J. Krejsa, K.A. Woodbury, J.D. Ratliff, M. Raudensky, Assessment of strategies and potential for neural networks in the inverse heat conduction problems, Inverse Probl. Eng. 7 (1999) 197-213.

9. M. Raudensky, J. Horsky, J. Krejsa, Usage of neural network for coupled parameter and function spesification inverse heat conduction problem, Int. Commun. Heat Mass Transfer 22 (5) (1995) 661-670.

10. J.M.G Cabeza, J.A.M Garcia, and A.C. Rodriguez, A Sequential Algorithm of Inverse Heat Conduction Problems Using Singular Value Decomposition, International Journal of Thermal Sciences 44 (2005) 235–244.

11. L. Elden, A Note on the Computation of the Generalized Cross-validation Function for Ill-conditioned Least Squares Problems, BIT, 24 (1984) 467–472.

12. G. H. Golub, M. Heath and G.Wahba, Generalized Cross-validation as a Method for Choosing a Good Ridge Parameter, Technometrics, 21 (2) (1979) 215–223.

13. P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev 34 (1992) 561–80.

14. C. L. Lawson and R. J. Hanson , Solving Least Squares Problems, Philadelphia, PA: SIAM, (1995).

15. R. Pourgholi and M. Rostamian, A numerical technique for solving IHCPs using Tikhonov regularization method, Applied Mathematical Modelling 34 (8) (2010) 2102–2110.

16. R. Pourgholi, N. Tavallaie and S. Foadian, Applications of Haar basis method for solving some ill-posed inverse problems, J. Math. Chem., 2012, Volume 50, Number 8, Pages 2317- 2337, DOI 10.1007/s10910-012-0036-4.

17. J. Rashidinia, R. Jalilian, V. Kazemi, Spline methods for the solutions of hyperbolic equations, Appl. Math. Comput. 190 (2007) 882-886 .

18. M. S. Shin, J. W. Lee, Prediction of the inner wall shape of an eroded furnace by the nonlinear inverse heat conduction technique, JSME Int. J. B 43 (4) (2000) 544-545.

19. A.N. Tikhonov and V.Y. Arsenin, On the solution of ill-posed problems, New York, Wiley, 1977.

20. G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 59, SIAM, Philadelphia, 1990.