Solution of the solidification problem by using the variational iteration method

• Autorzy: Hetmaniok E. , Słota D. , Zielonka A.
• Języki publikacji: EN

Abstrakty

EN

The paper presents the approximated solution of the solidification problem, modelled with the aid of the one-phase Stefan problem with the boundary condition of the second kind, by using the variational iteration method. For solving this problem one needs to determine the distribution of temperature in the given domain and the position of the moving interface. The proposed method of solution consists of describing the considered problem with a system of differential equations in a domain with known boundary, and solving the received system with the aid of VIM method. The accuracy of the obtained approximated solution is verified by comparing it with the analytical solution.

Słowa kluczowe

EN

solidification process; information technology; foundry industry; Stefan problem; moving boundary problem; variational iteration method

PL

proces krzepnięcia; technologia informacyjna; odlewnictwo; problem Stefana; problem brzegowy; metoda iteracyjna

• Wydawca: Komisja Odlewnictwa Polskiej Akademii Nauk Oddział w Katowicach
• Czasopismo: Archives of Foundry Engineering
• Rocznik: 2009
• Tom: Vol. 9, iss. 4
• Strony: 63--68
• Opis fizyczny: Bibliogr. 30 poz., rys., tab., wykr.

Twórcy

autor

Hetmaniok E.

autor

Słota D.

autor

Zielonka A.

• Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland,

Bibliografia

• [1] A. M. Meirmanov, The Stefan Problem, Walter de Gruyter, Berlin, 1992.
• [2] J. Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford, 1984.
• [3] S. C. Gupta, The Classical Stefan Problem. Basic Concepts, Modelling and Analysis, Elsevier, Amsterdam, 2003.
• [4] L. I.Rubinstein, The Stefan Problem, AMS, Providence, 1971.
• [5] M. Zerroukat, C. R. Chatwin, Computational Moving Boundary Problems, Research Studies Press, Taunton, 1994.
• [6] R. Grzymkowski, D.Słota, Stefan problem solved by Adomian decomposition method, Inter. J. Computer Math., Vol. 82 (2005), 851-856.
• [7] R. Grzymkowski, M. Pleszczyński, D. Słota, Application of the Adomian decomposition method for solving the Stefan problem, in: A. J. Nowak, R. A. Białecki, G. Węcel (Eds.), Numerical Heat Transfer 2005, EUROTHERM Seminar 82, ITT, Silesian Univ. of Technology, Gliwice-Kraków, 2005, 249-258.
• [8] R. Grzymkowski, M. Pleszczyński, D. Słota, Comparing the Adomian decomposition method and Runge-Kutta method for the solutions of the Stefan problem, Inter. J.Computer Math., Vol.83 (2006), 409-417.
• [9] D. Słota, Direct and inverse one-phase Stefan problem solved by variational iteration method, Comput. Math. Appl., Vol. 54 (2007), 1139-1146.
• [10] D. Słota, Application of the variational iteration method for inverse Stefan problem with Neumann's boundary condition, in: M. Bubak, G. D. van Albada, J. Dongarra, P. M. Sloot (Eds.), Computational Science ICCS 2008, Vol. 5101 of LNCS, Springer, 2008, 1005-1012.
• [11] D. Słota, A. Zielonka, A new application of he's variational iteration method for the solution of the one-phase Stefan problem, Comput. Math. Appl. doi: 10.1016 j.camwa. 2009.03.070, (2009) (w druku).
• [12] J. H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg., Vol.167 (1998), 57-68.
• [13] J. H. He, Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. Methods Appl. Mech. Engrg., Vol.167 (1998), 69-73.
• [14] J. H. He, Variational iteration method - a kind of non-linear analytical technique: some examples, Int. J. Non-Linear Mech., Vol.34 (1999), 699-708.
• [15] A. M. Wazwaz, The variational iteration method for exact solutions of Laplace equation, Physics Letters A, Vol. 363, (2007) 260-262.
• [16] M. M. D. D. Ganji, M. Miansari, Application of He's variational iteration method to nonlinear heat transfer equations, Physics Letters A, Vol.372 (2008), 779-785.
• [17] H. Khaleghi, D. D. Ganji, A. Sadighi, Application of variational iteration and homotopy-perturbation methods to nonlinear heat transfer equations with variable coefficients, Numer. Heat Transfer A, Vol. 52 (2007), 25-42.
• [18] B. Batiha, M. S. M. Noorani, I. Hashim, Application of variational iteration method to heat- and wave-like equations, Physics Letters A, Vol.369 (2007), 55-61.
• [19] J. Biazar, H. Ghazvini, An analytical approximation to the solution of a wave equation by a variational iteration method, Appl. Math. Lett., Vol.21 (2008), 780-785.
• [20] M. Dehghan, F. Shakeri, Application of He's variational iteration method for solving the Cauchy reaction-diffusion problem, J. Comput. Appl. Math., Vol.214 (2008), 435-446.
• [21] J. Biazar, H. Ghazvini, He's variational iteration method for solving hyperbolic differential equations, Int. J. Nonlin. Sci Numer. Simulat., Vol. 8 (2007), 311-314.
• [22] A. M. Wazwaz, The variational iteration method for solving linear and nonlinear systems of PDEs, Comput. Math. Appl., Vol. 54 (2007), 895-902.
• [23] M. Dehghan, M. Tatari, Identifying an unknown function in a parabolic equation with overspecified data via He's variational iteration method, Chaos, Solitions and Fractals, Vol. 36 (2008), 157-166.
• [24] M. Tatari, M. Dehghan, He's variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation, Chaos, Solitions and Fractals, Vol.33 (2007), 671-677.
• [25] S. M. Varedi, M. J. Hosseini, M. Rahimiand, D.D. Ganji, He's variational iteration method for solving a semi-linear inverse parabolic equation, Physics Letters A, Vol.370 (2007), 275-280.
• [26] F. Shakeri, M. Dehghan, Solution of a model describing biological species living together using the variational iteration method, Math. Comput. Modelling, Vol. 48 (2008) 685-699.
• [27] M. Dehghan, F. Shakeri, Approximate solution of a differential equation arising in astrophysics using the variational iteration method, New Astronomy, Vol.13 (2008), 53-59.
• [28] M. Tatari, M. Dehghan, On the convergence of He's variational iteration method, J.Comput. Appl. Math., Vol.207 (2007), 121-128.
• [29] M. Inokuti, H. Sekine, T. Mura, General use Lagrange multiplier in non-linear mathematical physics, in: S.Nemat-Nasser (Ed.), Variational Method in the Mechanics of Solids, Pergamon Press, Oxford, 1978, 156-162.
• [30] B. A. Finlayson, The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972.