The use of positive matrices for the analysis of the large time behavior of the numerical solution of reactiondiffusion systems
Author:
Luciano Galeone
Journal:
Math. Comp. 41 (1983), 461472
MSC:
Primary 65M10; Secondary 15A51, 65C20
MathSciNet review:
717696
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Abstract: In this paper we study the numerical solution of nonlinear reactiondiffusion systems with homogeneous Neumann boundary conditions, via the known method. We show that if conditions for the positivity of solutions are imposed, then the study of the asymptotic behavior of the numerical solution can be done by means of the theory of stochastic matrices. In this way it is shown that the numerical solution reproduces the asymptotic behavior of the corresponding theoretical one. In particular, we obtain the decay of the solution to its mean value. An analysis of the asymptotic stability of the equilibrium points and the convergence of the numerical scheme is given based on the use of Mmatrices. Finally we consider the case in which the nonlinear term satisfies a condition of quasimonotonicity.
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 V. Capasso & S. L. PaveriFontana, "Some results on linear stochastic multicompartmental systems," Math. Biosci., v. 55, 1981, pp. 726. MR 625265 (82i:92003)
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 S. L. Campbell & C. D. Mayer, Generalized Inverses of Linear Transformations, Pitman, London, 1979.
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 J. P. Lasalle, "Stability theory for difference equations," Studies in Ordinary Differential Equations, Math. Assoc. Amer., 1978. MR 0481689 (58:1789)
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 R. H. Martin, "Asymptotic stability and critical points for nonlinear quasimonotone parabolic systems," J. Differential Equations, v. 30, 1978, pp. 301423. MR 521861 (80g:35011)
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 R. S. Varga, Matrix Iterative Analysis, PrenticeHall, Englewood Cliffs, N. J., 1962. MR 0158502 (28:1725)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198307176965
PII:
S 00255718(1983)07176965
Keywords:
Nonlinear reactiondiffusion systems,
methods,
Mmatrices,
stochastic matrices,
Astability,
quasimonotone functions
Article copyright:
© Copyright 1983
American Mathematical Society
