Variational iteration method for solving two phases continuous casting moving boundary value problem

Zaboon, Radhi A.; Al-Shimmary, Abbas

doi:10.47974/JIM-2284

Open Access Research Article

Variational iteration method for solving two phases continuous casting moving boundary value problem

^*Radhi A. ZaboonCorresponding authorr.zaboon@uomustansiriyah.edu.iqDepartment of Mathematics College of Science Mustansiriyah UniversityBaghdad, IraqView full profile → , Abbas Al-Shimmaryabbs.fadhil62@uomustansiriyah.edu.iqDepartment of Electrical Engineering College of Engineering Mustansiriyah UniversityBaghdad, IraqView full profile →

* Corresponding author · click or hover a name for details

pp. 1713–1721Vol. 29Issue 5-BMay 2026DOI: 10.47974/JIM-2284XML

Received:: 01 May 2025
Published Online:: 27 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2284
Pages:: 1713–1721

Abstract

This paper’s primary goal is to find an approximate solution for the nonlinear continuous casting of liquids and solids in two phases. It is a type of moving boundary value problem, which is nonlinear, and the unknown moving interface position makes it hard to solve using analytical methods. The He’s Variational Iteration Method (HVIM) procedure begins by initializing a solution with an initial guess in order to construct a correction functional based on optimal values of the Lagrange multiplier parameters. The approximate solution is then built up as a series of approximations, which quickly come together to the exact solution. As an operator equation, we have used the Stefan condition, along with the approximate solutions of the liquid and the solid phases, to build up a correction functional to the unknown moving boundary operator equation, which is obtained concurrently. It provides very accurate solutions to the unknown two-phase moving boundary values problem with an unknown moving interface, as indicated in the numerical values and comparisons (of the various initial guesses).

Keywords

Variational iteration method Continuous casting Stefan problem Moving boundary problem

Subject Classifications

35R3535K20

References

[1] Z. Chen, J. Bentsman, and B. G. Thomas, “Optimal control of free boundary of a stefan problem for metallurgical length maintenance in continuous steel casting,” in 2019 American Control Conference (ACC), IEEE, pp. 3206-3211 (2019).
[2] Z. Chen, J. Bentsman, B. G. Thomas, and A. Matsui, “Study of spray cooling control to maintain metallurgical length during speed drop in steel continuous casting,” Iron Steel Technol, vol. 14, no. 10, pp. 92-103 (2017).
[3] R. Grzymkowski, M. Pleszczyński, and E. Hetmaniok, “Problem of the moving boundary in continuous casting solved by the analytic-numerical method,” Archives of Foundry Engineering, vol. 13, no. 1, pp. 33-38 (2013).
[4] G. Abd and R. Ali, “Parametrization approach for solving index-4 linear differential-algebraic control systems,” International Journal of Mathematics and Computer Science, vol. 17, no. 2, pp. 815-825 (2022).
[5] A. Hemeda, “Variational iteration method for solving non-linear partial differential equations,” Chaos, Solitons & Fractals, vol. 39, no. 3, pp. 1297-1303 (2009).
[6] G. F. Abd and R. A. Zaboon, “Parameterization technique for solving variational formulation of index three linear time-varying singular control systems,” Journal of Interdisciplinary Mathematics, vol. 24, no. 7, pp. 1935–1945 (2021), doi: 10.1080/09720502.2021.1966949.
[7] H. Ahmad, T. A. Khan, P. S. Stanimirović, Y.-M. Chu, and I. Ahmad, “Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models,” Complexity, vol. 2020, no. 1, p. 8841718 (2020).
[8] J.-H. He and H. Latifizadeh, “A general numerical algorithm for nonlinear differential equations by the variational iteration method,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30, no. 11, pp. 4797-4810 (2020).
[9] X. Wang, Q. Xu, and S. N. Atluri, “Combination of the variational iteration method and numerical algorithms for nonlinear problems,” Applied Mathematical Modelling, vol. 79, pp. 243-259 (2020).
[10] Z. M. Odibat, “A study on the convergence of variational iteration method,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1181-1192 (2010).
[11] D. Słota, “Homotopy perturbation method for solving the two-phase inverse Stefan problem,” Numerical Heat Transfer, Part A: Applications, vol. 59, no. 10, pp. 755-768 (2011).
[12] L. Sowa and A. Bokota, “Numerical model of thermal and flow phenomena in the process of growing the CC slab,” Archives of metallurgy and materials, vol. 56, pp. 359-366 (2011).
[13] J. Singh, K. P. Gupta, and N. K. Rai, “Variational iteration method to solve moving boundary problem with temperature dependent physical properties,” Thermal Science, vol. 15, no. suppl. 2, pp. 229-239 (2011).

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