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Analytical bounding functions for diffusion problems with Michaelis- Menten kinetics. (English) Zbl 0554.92005

Pointwise upper and lower bounds for the solution of a class of nonlinear diffusion problems with Michaelis-Menten kinetics are presented. Simple analytical bounding curves are obtained and for an illustrative case the calculated values bound the recent numerical solution of P. Hiltmann and P. Lory, ibid. 45, 661-664 (1983; Zbl 0512.92008).

MSC:

92Cxx Physiological, cellular and medical topics
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
35K20 Initial-boundary value problems for second-order parabolic equations
34B15 Nonlinear boundary value problems for ordinary differential equations

Citations:

Zbl 0512.92008
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Full Text: DOI

References:

[1] Amann, H. 1971. ”On the Existence of Positive Solutions of Nonlinear Elliptic Boundary Value Problems.”Indiana Univ. Math. J. 21, 125–146. · Zbl 0219.35037 · doi:10.1512/iumj.1972.21.21012
[2] Anderson, N. and A. M. Arthurs. 1980. ”Complementary Variational Principles for Diffusion Problems with Michaelis-Menten Kinetics.”Bull. math. Biol. 42, 131–135. · Zbl 0423.92040 · doi:10.1007/BF02462371
[3] Ghim, Y. S. and H. N. Chang. 1983. ”Diffusional Falsification of Kinetic Constants on Lineweaver-Burk Plots.”J. theor. Biol. 105, 91–102. · doi:10.1016/0022-5193(83)90426-5
[4] Hiltmann, P. and P. Lory. 1983. ”On Oxygen Diffusion in a Spherical Cell with Michaelis-Menten Oxygen Uptake Kinetics.”Bull. math. Biol. 45, 661–664. · Zbl 0512.92008 · doi:10.1007/BF02460043
[5] Lin, S. H. 1976. ”Oxygen Diffusion in a Spherical Cell with Nonlinear Oxygen Uptake Kinetics.”J. theor. Biol. 60, 449–457. · doi:10.1016/0022-5193(76)90071-0
[6] McElwain, D. L. S. 1978. ”A Re-examination of Oxygen Diffusion in a Spherical Cell with Michaelis-Menten Oxygen Uptake Kinetics.”J. theor. Biol. 71, 255–263. · doi:10.1016/0022-5193(78)90270-9
[7] Villadsen, J. and M. L. Michelsen. 1978.Solution of Differetial Equation Models by Polynomial Approximation. Englewood Cliffs, New Jersey: Prentice-Hall. · Zbl 0464.34001
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