## THERMAL SCIENCE

International Scientific Journal

### VARIATIONAL ITERATION METHOD TO SOLVE MOVING BOUNDARY PROBLEM WITH TEMPERATURE DEPENDENT PHYSICAL PROPERTIES

**ABSTRACT**

In this paper, variational iteration method is used to solve a moving boundary problem arising during melting or freezing of a semi infinite egion when physical properties (thermal conductivity and specific heat) of the two regions are temperature dependent. The Result is compared with result obtained by exact method (when thermal conductivity and specific heat in two regions are temperature independent) and semi analytical method (When thermal conductivity and specific heat are temperature dependent) and are in good agreement. We obtain the solution in the form of continuous functions. The method performs extremely well in terms of efficiency and simplicity and effective for solving the moving boundary problem.

**KEYWORDS**

PAPER SUBMITTED: 2010-02-26

PAPER REVISED: 2010-08-27

PAPER ACCEPTED: 2011-03-15

**THERMAL SCIENCE** YEAR

**2011**, VOLUME

**15**, ISSUE

**Supplement 2**, PAGES [S229 - S239]

- Alexiades, V., Solomon, A.D., Mathematical modeling of melting and freezing processes, in Hemisphere, Taylor and Francis, Washington, 1983
- Crank, J., Free and Moving Boundary Problems, Oxford University Press, Oxford, UK, 1984
- Lunardini, V. J., Heat transfer with freezing and thawing, Elsevier, Amsterdam, 1991
- Barber, J. R., An asymptotic solution for short time transient heat conduction between two similar contacting bodies, Int. J. Heat Mass Transfer, 32 (1989), pp. 943‐949
- Tarzia, D. A., An inequality for the coefficient σof the free boundary ttsσ2)(=of the Neumann solution for the two‐phase Stefan problem, Quart. Appl .Math.39 (1981), pp. 491‐497
- Grzymkowski, R., Slota, D., Moving Boundary Problem, Solved by Adomian decomposition Method, in: S. K. Chakrabarti et al. (Eds.), Fluid Structure Interaction and Moving Boundary Problems, Wit Press, Southampton, 2005, pp. 653‐660
- Ang, D. D., Pham, A., Dinh, N., Thanh, D. N., Regularization of an inverse two phase Stefan problem, Nonlinear Anal. 34 (1998), pp. 719‐731
- Carslaw, H. S., Jaeger, J. C., Conduction of Heat in Solids, second ed., Oxford University, Press, London, England, 1956
- Ozisik, M. N., Heat Conduction, Wiley, New York, 1980
- Wrobel, L. C., Brebbia, C. A., (Eds.), Computational Methods for Free and Moving Boundary Problems in Heat and Fluid Flow, Computational Mechanics Publications, Southampton, 1993
- Fredrick, D., Greif, R., A method for the solution of heat transfer problems with a change of Phase, ASME J. Heat transfer 107 (1985), pp. 520‐526
- Natale, M. F., Tarzia, D. A., Explicit solutions to the one phase Stefan problem with temperature‐dependent thermal conductivity and a convective term, Int. J. Engi. Sci. 41 (2003), pp. 1685‐1698
- Gasiorski, A. K., Application of the finite element method for the analysis of nonsteady‐state induction device problems with rotational symmetry, Compt. Elect. Eng.13 (1987), 2, pp. 117‐128
- Chen, H.T., Liu, J.Y., Effect of the potential field on non‐Fickian diffusion problems in a sphere, Int .J. Heat Mass Transfer 46 (2003), 15, pp.2809‐2818
- Goodman, T.R., Shea, J. J., The Melting of Finite Slabs, ASME, J. Appl. Mech. 27 (1960), pp.1624‐1632
- Goodman, T.R., The Heat‐Balance Integral Method and its Application to Problems Involving a Change of Phase, Trans. ASME, 80 (1958), 2, pp. 335‐342
- Savovic, S., Caldwell, J., Numerical solution of Stefan problem with time‐dependent boundary conditions by variable space grid method, Thermal Science 13 (2009), 4, pp.165‐174
- Rai, K.N., and Rai, S., Approximate closed form analytical solution of the desublimation problem in a porous medium, Int . J. Energy Research,19 (1995), pp 279‐288
- Annamalai, K., Lau S.C., and Kondepudi, S.N., A non‐integral technique for the approximate solution of transport problems Int.Comm.Heat Mass Transfer,13 (1986), 5, pp.523‐ 534
- Rai, K.N., Rai, S., An analytical study of the solidification in a semi‐infinite porous medium International J.Engg.Sci.,30 (1992), 2 , pp. 247‐256
- Nayfeh, A.H., Perturbation methods, Wiley, New York,1973
- Liu, X. L., Oliveira, C. S., Iterative modal perturbation and reanalysis of eigen value problem, comm. in Num. Meth in Engg. 19 (2003), 4 , pp. 263‐274
- Prud homme, M., Hung Nguyen T., Long Nguyen, D., A heat transfer analysis for solidification of slabs, cylinders, and spheres Journal of heat Transfer, Vol.111, (1989), pp.699‐705
- He, J. H., Variational iteration method -a kind of non linear analytical technique: some Example, Int. J. Nonlin. Mech.34 (1999), pp.699‐708
- He, J. H., Variational iteration method for autonomous ordinary differential systems, Appl. Math.Comput.114 (2000), pp.115‐123
- Yao, X., A model for initial spherical growth during equiaxed solidification, Int. J. Nonlin. Sci. Num. Sim., 9 (2008), pp. 283‐288
- Chun, C., Variational Iteration Method for a Reliable Treatment of Heat Equations with Ill‐defined Initial Data, Int. J. Nonlin. Sci. Num. Sim., 9 (2008), 4, pp. 435‐440
- Cao, L., Han, B., Wang, W., Homotopy Perturbation Method for Nonlinear Ill‐posed Operator Equations, Int. J. Nonlin. Sci. Num., 9 (2009), pp. 1319‐1322
- He, J. H., Wu, G. C., Austin, F., The Variational iteration method which should be followed, Nonlinear Science Letters A,1 (2010), pp. 1‐30
- Ganji, D. D., Sadighi, A., Application of homotopy‐perturbation and variational iteration Methods to nonlinear heat transfer and porous media equations, J. of computational and Applied Mathematics, 207 (2007), pp. 24 ‐34
- Momani, S., Abuasad, S., Application of He's Variational iteration method to Helmholtz equation, Chaos, Solitons Fractals 27 (2006), pp. 1119‐1123
- Abdou, M.A., Soliman, A.A., New applications of variational iteration method, Physica D 211 (2005), pp. 1‐8
- Abdou, M. A., Soliman, A. A., Variational iteration method for solving Burgers and coupled Burgers equations, J. Comput. Appl. Math. 181(2005), pp.245‐251
- Momani, S., Abuasad, S., Application of He's variational iteration method to Helmholtz equation, Chaos Solution Fractals 27 (2006), pp. 1119‐1123
- Slota, D., Direct and inverse one‐phase Stefan problem solved by the variational Iteration method, Comp. and Math. with Applications 54 (2007), pp. 1139‐1146
- Oliver, D. L. R., Sunderland, J. E., A Phase change problem with temperature dependent thermal conductivity and specific heat, Int. J. Heat Mass Transfer.30 (1987), pp. 2657‐2661