A numerical method for oxygen diffusion and absorption in a sike cell. (English) Zbl 1090.65115

Ahmed, S. G.

Appl. Math. Comput. 173, No. 1, 668-682 (2006).

Summary: Oxygen diffusion in a sike cell with simultaneous absorption is an important problem and has a wide range of medical applications. The problem mathematically formulated through two different stages, the present paper, concerns mainly on the second stage, in which, the injected oxygen through the cell starts absorbing. The basic idea of the proposed numerical method is to form a double linear system of equations, each of dimension (\(4 \times 4\)).
The first system is formed through applying first, second, third and fourth moments and using assumed profile for the concentration containing four unknowns functions of time. Numerical solution through a proposed scheme leads to the unknown functions. The second system is formed through applying the boundary conditions given in addition to another assumed condition, that is the concentration at \(x = 0\) is an unknown function of time. The results of the first system becomes an entry data for the second one leading to the concentration at the fixed surface \(x = 0\). The result obtained by the present method are compared with two different methods and the results give a good agreement.

MSC:

65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K05	Heat equation

Keywords:

moving boundary problem; Euler equation; integral equation method; algorithm; numerical examples; oxygen diffusion

References:

[1]	Crank, J.; Gupta, R. S., A moving boundary problem arising from the diffusion of oxygen in absorbing tissue, J. Inst. Math. Appl., 10, 19-33 (1972) · Zbl 0247.65064
[2]	Crank, J.; Gupta, R. S., J. Inst. Math. Appl., 10, 269-304 (1972)
[3]	Hansen, E.; Hougaard, P., On moving problem from biomechanics, J. Inst. Math. Appl., 13, 385-398 (1974) · Zbl 0307.45016
[4]	Berger, A. E.; Ciment, M.; Rogers, J. C.W., Numerical solution of a diffusion consumption problem with a free boundary, SIAM J. Numer. Anal., 12, 646-672 (1975) · Zbl 0317.65032
[5]	Miller, J. V.; Morton, K. W.; Baines, M. J., A finite element moving boundary computation with an adaptive mesh, J. Inst. Math. Appl., 22, 467-477 (1978) · Zbl 0394.65032
[6]	Furzeland, R. M., A comparative study of numerical methods for moving boundary problems, J. Inst. Math. Appl., 26, 411-429 (1980) · Zbl 0468.65063
[7]	(Crank, J., Free and Moving Boundary Problems (1984), Clarendon Press: Clarendon Press Oxford) · Zbl 0547.35001
[8]	Gupta, R. S.; Kumar, D., Variable time step methods for one-dimensional Stefan problems with mixed boundary conditions, Int. J. Heat Transfer, 24, 251-259 (1981) · Zbl 0462.76095
[9]	Zerroukat, M.; Chatwin, R., An explicit unconditionally stable variable time step method for 1D Stefan problems, Int. J. Numer. Methods Eng., 35, 1503-1520 (1992) · Zbl 0788.73079
[10]	Gupta, R. S.; Banik, N. C., Approximate method for the oxygen diffusion problem, Int. J. Heat Mass Transfer, 32, 781 (1989)
[11]	Catal, S., Numerical approximation for oxygen diffusion problem, Appl. Math. Comput., 145, 361-369 (2003) · Zbl 1101.80305
[12]	Ahmed, S. G., An approximate method for oxygen diffusion in a sphere with simultaneous absorption, Int. J. Numer. Methods Heat Fluid Flow, 9, 6, 631-642 (1999) · Zbl 0969.76083
[15]	Murray, W. D.; Landis, F., Numerical and machine solutions of transient heat conduction problems involving melting and freezing, J. Heat Transfer, 81, 106-112 (1959)
[16]	Gupta, R. S.; Banik, N. C., Diffusion of oxygen in a sphere with simultaneous absorption, Appl. Math. Model., 14, 114-121 (1990) · Zbl 0701.65084