A semi-analytic method to solve the oxygen diffusion problem

Subject Areas : Statistics

Mojtaba  Moradipour ¹ (Assistant Professor of Applied Mathematics, Department of Mathematics, Faculty of Basic Sciences, Lorestan University, 68137-17133, Khorramabad, Iran.)

Received: 2021-01-01 Accepted : 2021-12-09 Published : 2022-06-22

Keywords: معادلات دیفرانسیل جزیی, مسایل مرز-آزاد, مساله پخش اکسیژن,

Abstract :

In this paper, a semi-analytic approach is proposed to solve the oxygen diffusion problem.First, we discretize the partial differential equation of the oxygen diffusion problem in temporal direction using the backward finite difference Euler method. We achieve a sequence of free boundary problems in the form of ordinary differential equations (ODEs) in the spatial direction. The ODEs are then solved analytically and a recursive formula is presented to compute the solutions of the ordinary differential equations. The problems of finding unknown boundaries are reduced to nonlinear algebraic problems. Finally, the nonlinear algebraic problems are solved using the root-finding methods such as the false position method. The method proposed in this paper is easy to implement and a comparison with other numerical methods shows that the proposed approach is very efficient and gives very accurate numerical results.Some tables and figures are included to show the efficiency and effectiveness of the proposed technique.

References:

[1] M. H. Aliabadi and E.L. Ortiz, Numerical treatment of moving and free boundary value problems with the tau method, Computers & Mathematics with Applications, 35, 53 – 61, 1998.

[2] R. W. Cottle, Numerical methods for complementarity problems in engineering and applied science, Proc. Comput. Meth. Appl. Sci. Engng., Springer Lec. Notes, No. 704, pp. 37-52, Springer-Verlag, Berlin, 1979.

[3] J. Crank, Free and moving boundary problems, Oxford science publications, Clarendon Press, 1984.

[4] J. Crank and R.S. Gupta, A moving boundary problem arising from the diffusion of oxygen in absorbing tissue, J. Inst. Math. Appl. 10, 19-33, 1972.

[5] C. Elliott and J.R. Ockendon, Weak and variational methods for moving boundary problems, Research Notes in Mathematics, No. 59, Pitman, London, 1982.

[6] T.A.Galib, J.C. Bruch, and J. M. Sloss, Solution of an oxygen diffusion-absorption problem, International Journal of Bio-Medical Computing, 12(2): 157–180, 2004.

[7] R.S. Gupta, Moving grid methods without interpolations, Comput. Meth. Appl. Mech. Engng. 4, 143-152, 1974.

[8] R.S. Gupta and D. Kumar, A modified variable time step method for the one-dimensional Stefan problem, Comput. Meth. Appl. Mech. Engng. 23, 101-109, 1980.

[9] R.S. Gupta and D. Kumar, Complete numerical solution of the oxygen diffusion problem involving a moving boundary, Comput. Meth. Appl. Mech. Engng. 29, 233-239, 1981.

[10] E.B. Hansen and P. Hougaard, On a moving boundary problem from biomechanics, J. Inst. Math. Appl, 13, 385-398, 1974.

[11] J.V. Miller, K.W. Morton and M.J. Balnes, A finite element moving boundary computation with an adaptive mesh, J. Inst. Math. Appl. 22, 467-477, 1978.

[12] S.L. Mitchell and M. Vynnycky, The oxygen diffusion problem: Analysis and numerical solution, Applied Mathematical Modelling, 2763–2776, 2015.

[13] M. Moradipour, A Variational Inequality Approach for One Dimensional Stefan Problem, Mathematical Analysis and Convex Optimization, 1(2): 35–43, 2020.

[14] M. Moradipour, An effective algorithm to solve option pricing problems, International Journal of Nonlinear Analysis and Applications, 12(1): 261–271, 2021.

[15] W.D. Murray and F. Landis, Numerical and machine solutions of transient heat-condition problems involving melting or freezing, Part 1–Method of analysis and sample solutions, J. Heat Transfer 81 (2), 106-112, 1959.

[16] E.L. Ortiz, The tau method, SIAM J. Numer. Anal., 480-492, 1969.