A Banach space solution of an abstract Cauchy problem which has Robin boundary data
Author:
Dennis W. Quinn
Journal:
Quart. Appl. Math. 49 (1991), 179-200
MSC:
Primary 92C05; Secondary 65J15, 76Z05, 93C20
DOI:
https://doi.org/10.1090/qam/1096240
MathSciNet review:
MR1096240
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Abstract: In this paper, a problem which involves the diffusion equation coupled by way of Robin boundary data with a nonlinear system of ordinary differential equations is considered. This problem has been proposed as a model to describe the absorption through the skin, the distribution throughout the body, and the metabolism of a substance in a mammal. The problem is set as an abstract Cauchy problem in a Banach space and is shown to have a unique solution. Finite dimensional approximations of the problem are obtained by replacing the spatial partial derivatives with finite differences. The approximate solutions are shown to converge to the exact solution of the original problem. Comparisons of numerical solutions with experimental data are presented.
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J. N. McDougal, G. W. Jepson, H. J. Clewell III, M. G. MacNaughton, and M. E. Andersen, A physiological pharmacokinetic model for dermal absorption of vapors in the rat, Toxicology and Applied Pharmacology 85, 286–294 (1986)
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D. W. Quinn, A distributed parameter identification problem which arises in pharmacokinetics, Proceedings of the University of Arkansas Eighth Annual Lecture Series in the Mathematical Sciences, 1984
J. N. McDougal, G. W. Jepson, H. J. Clewell III, and M. E. Andersen, Dermal absorption of dihalomethane vapors, Toxicology and Applied Pharmacology 79, 150–158 (1985)
J. N. McDougal, G. W. Jepson, H. J. Clewell III, M. G. MacNaughton, and M. E. Andersen, A physiological pharmacokinetic model for dermal absorption of vapors in the rat, Toxicology and Applied Pharmacology 85, 286–294 (1986)
H. T. Banks and K. Kunisch, An Approximation theory for nonlinear partial differential equations with applications to identification and control, SIAM J. Control Optim. 20, 815–849 (1982)
A. Belleni-Morante, Applied Semigroups and Evolution Equations, Clarendon Press, Oxford, 1980
H. O. Fattorini, The Cauchy Problem, Addison-Wesley, Reading, Massachusetts, 1983
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966
R, H. Martin, Jr., Nonlinear Operators and Differential Equations in Banach Spaces, John Wiley & Sons, New York, 1976
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York, 1983
A. Pazy, A class of semi-linear equations of evolution, Israel J. Math. 20, 23–36 (1975)
H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8, 887–919 (1959)
D. W. Quinn, A distributed parameter identification problem which arises in pharmacokinetics, Proceedings of the University of Arkansas Eighth Annual Lecture Series in the Mathematical Sciences, 1984
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Article copyright:
© Copyright 1991
American Mathematical Society
