Solution of Oxygen Diffusion Moving Boundary Value Problem Based on Variational Iteration Least Square Methods
DOI:
https://doi.org/10.22401/Keywords:
Variational iteration method, Modified variation iteration method, Least Square Method, One-phase Stefan problem, Oxygen diffusion problemAbstract
This article aims to address and solve the oxygen diffusion problem which involves oxygen diffraction into a medium that absorbs and immobilizes the oxygen concentration at a constant rate. Such types of problems are difficult to solve analytically since it is a type of moving boundary value problem or one-phase Stefan problem, and such problems require us to determine the domain boundary as a part of the solution that is unknown or may vary with respect to time. The main governing equation, a partial differential equation used to govern the oxygen diffusion in an absorbing medium, is the heat equation, which has a moving boundary that varies with respect to time. Also, the presence of a moving boundary that indicates the farthest point where oxygen enters the medium, as well as, the initial distribution of oxygen across the medium will raise fundamental mathematical challenges in the solution. The approach followed for solving this problem is a semi-analytical method, which is a hybrid approach between the variational iteration method and the least squares method. These two methods are used to derive an efficient iterative approximate solution of the considered problem of this paper, because of their simplicity, efficiency and reliability in computational applications.
References
[1] Crank G. and Makin M. I., "Oxidations of aromatic amines by superoxide ion". Aust. J. Chem, 37(4): 845-855, 1984.
[2] Kim K., Mueller C. and Sowers R. B. "A stochastic moving boundary value problem". Ill. J. Math., 54(3): 927-962, 2010.
[3] Galib Jr T. A., Bruch Jr J. C. and Sloss J. M. "Solution of an oxygen diffusion-absorption problem". Int. J. Bio-Med. Comput., 12(2) :157-180, 1981.
[4] Crank J. and Gupta R. S. "A moving boundary problem arising from the diffusion of oxygen in absorbing tissue". IMA. J. Appl. Math., 10(1): 19-33, 1972.
[5] Crank J., and Gupta R. S. "A method for solving moving boundary problems in heat flow using cubic splines or polynomials". J. Inst. Math. Its Appl., 10(3): 296-304, 1972.
[6] Gupta R. S. and Kumar D. "Complete numerical solution of the oxygen diffusion problem involving a moving boundary". Comput. Methods Exp. Appl. Mech. Eng., 29(2): 233-239, 1981.
[7] Liapis A. I., Lipscomb G. G., Crosser O. K. and Tsiroyianni-Liapis E. "A model of oxygen diffusion in absorbing tissue". Math. Modell., 3(1): 83-92, 1982.
[8] Ahmed S. G. "An approximate method for oxygen diffusion in a sphere with simultaneous absorption". Int. J. Numer. Methods Heat Fluid Flow, 9(6): 631-643, 1999.
[9] Ahmed S. G. "A numerical method for oxygen diffu5z2sion and absorption in a sike cell". Appl. math. comput., 173(1): 668-682, 2006.
[10] Boureghda A. "Numerical solution of the oxygen diffusion in absorbing tissue with a moving boundary". Commun. Numer. Methods Eng., 22(9): 933-942, 2006.
[11] Bougoffa L. "The Adomian decomposition method for solving a moving boundary problem arising from the diffusion of oxygen in absorbing tissue". Sci. World. J., 2014(1), 579628, 2014.
[12] González A. M., Reginato J. C. and Tarzia D. A. "A free boundary model for oxygen diffusion in a spherical medium". J. Biol. Syst., 23(supp01): S67-S76, 2015.
[13] Gülkaç V. "The new approximate analytic solution for oxygen diffusion problem with time-fractional derivative". Math. Probl. Eng., 2016(1), 8409839, 2016.
[14] Nama A. Y. and Fadhel F. S. "Fractional variational iteration method for solving two-dimensional Stefan problem with fractional order derivative". Int. J. of Nonlinear Anal. Appl., 13(2): 2167-2174, 2022.
[15] Miklavčič M. "Analytic and numeric solutions of moving boundary problems". J. Comput. Appl. Math., 431, 115270, 2023.
[16] Momani S., Abuasad S. and Odibat Z. "Variational iteration method for solving nonlinear boundary value problems". Appl. Math. Comput., 183(2): 1351-1358, 2006.
[17] Jawad U. M. and Fadhel F. S. "Modified variational iteration method for solving sedimentary ocean basin moving boundary value problem". Al-Nahrain. J. Sci., 25(3): 33-39, 2022.
[18] Tari H. "Modified variational iteration method". Phys. Lett. A, 369(4): 290-293, 2007.
[19] Noor M. A. and Mohyud-Din S. T. "Modified variational iteration method for Goursat and Laplace problems". World Appl. Sci. J., 4(4): 487-498, 2008.
[20] Embree M. "Numerical analysis I". Lect. Notes, Rice Univ., 1-207, 2010.
[21] Loach P. D. and Wathen A. J. "On the best least squares approximation of continuous functions using linear splines with free knots". IMA. J. Appl. Math., 11(3): 393-409, 1991.
[22] Słota D. "Direct and inverse one-phase Stefan problem solved by the variational iteration method". Comput. Math. Appl., 54(7-8): 1139-1146, 2007.
[23] Słota D. and Zielonka A. "A new application of He’s variational iteration method for the solution of the one-phase Stefan problem". Comput. Math. Appl., 58(11-12): 2489-2494, 2009.
[24] Abdulhussein, A. I. and Fadhel F. S. "Solution of shoreline sedimentary ocean basin moving boundary value problem based on variational iteration-least square methods", 3rd International Conference on Mathematics, AI, Information And Communication Technologies (ICMAICT-2024), Bayan University, Erbil, Iraq in 27-28 April, 2024
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Asmahan I. Abdulhussein, Fadhel S. Fadhel
This work is licensed under a Creative Commons Attribution 4.0 International License.
.jpg)
