The solution of a one-phase Stefan problem with a forcing term by homotopy analysis method

 
 
 
  • Abstract
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  • References
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  • Abstract


    In this paper, we use the homotopy analysis method (HAM), to obtain the solutions of the temperature distribution, the position of the moving boundary and the Stefan condition. There are advantages to using HAM, firstly it is independent of small/large physical parameters, there is flexibility on the choice of base function and initial guess of solution and lastly there is great generality. The results obtained from this method shows high accuracy, computational efficiency and a strong rate of convergence.

    Keywords: Dirichlet; Forcing Term; Homotopy Analysis Method; Stefan Problem.


  • References


    1. A.K.Alomari, Modifications of Homotopy Analysis Method for Differential Equations: Modifications of Homotopy Analysis Method, Ordinary, Fractional, Delay and Algebraic Equations, Lambert Academic Publishing, Germany, 2012.
    2. A.Fasano, M. Primicerio,Free Boundary problems for nonlinear parabolic equations with nonlinear free boundary conditions, J.Math.Anal.Appl,72(1979),247 -273.
    3. D. Slota, Homotopy Perturbation Method for solving the two-phase inverse Stefan problem, Numer. Heat Trans, 59(2011), 755-768.
    4. H. Jafari, M.Saeidy and M.A.Frozjaei, Homotopy Analysis Method: a tool for solving a Stefan Problem, J. Adv Res. Sci. Comput, 2(2010), 61-68.
    5. H.Jafari,A.Golbabai,E.Salehpoor and Kh. Sayehvand, Application of Variational Iteration Method, Appl.Math.Sci, 2(2008),3001 -3004.
    6. O.O.Onyejekwe and O.N.Onyejekwe, Numerical solutions of the one-phase classical Stefan Problem using an enthalpy green element method, Adv. Eng. Soft, 42(2011), 743 749.
    7. R. Grzymkowski, M.Pleszczy?ski and D.Slota, Comparing the Adomian Decomposition Method and the Runge-Kutta Method for solutions of the Stefan problem, Intl.J. Comput.83 (2006), 409 -417.
    8. R. Grzymkowski and D.Slota, Stefan Problem solved by Adomian Decomposition Method, Intl. J. of Comput. Math, 82(2005), 851 856.
    9. S. Kutluay, Numerical Schemes for one-dimensional Stefan like problems with a forcing term, Appl. Math. And Comput, 168(2005), 1159 1168.
    10. S. Kutluay, A.R.Bahadir and A.Ozdes, The numerical solution of one-phase classical Stefan problem, J. Comput. And Appl. Math, 81(1997), 135-144.
    11. S.Liao, Homotopy Analysis Method in Nonlinear Equations, Springer, New York, 2012.
    12. S.Liao, Beyond Perturbation: Introduction to homotopy Analysis Method, Chapman & Hall CRC, 2004.
    13. S.Liao, Notes on the homotopy analysis method- Some definitions and theorems, Common, Nonlinear Sci. Numer. Simulat, 14(2009), 983-997.
    14. S.L.Mitchell and T.G.Myers, Application of Heat Balance Integral Methods to one-dimensional phase change problems, Intl. J. Diff. Equa, 2012, 1-22.
    15. S. Savovic and J.Caldwell, Numerical Solution of Stefan Problem with time-dependent boundary conditions by variable space grid method, Ther. Sci, 13(2009) 165-174.
    16. S. Savovic and J.Caldwell, Finite Difference solution of one-dimensional Stefan problem with periodic boundary conditions, Intl.J. Heat. Mass Transr, 46 (2003), 2911-2916.
    17. T.C.Smith, A finite difference method for a Stefan problem, CALCOLO, 18(1981), 131-154.

 

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Article ID: 2300
 
DOI: 10.14419/ijams.v2i2.2300




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