Analysis of a class of nonlinear integro-differential equations arising in a forestry application
Authors:
Michael A. Kraemer and Leonid V. Kalachev
Journal:
Quart. Appl. Math. 61 (2003), 513-535
MSC:
Primary 45J05; Secondary 34E10, 34K26
DOI:
https://doi.org/10.1090/qam/1999835
MathSciNet review:
MR1999835
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Abstract: Certain models describing the age dynamics of a natural forest give rise to nonlinear integro-differential equations for the seedlings density as a function of time. The special feature of the problem is that corresponding solutions have non-smooth second derivatives. Since the biological model contains a small parameter, a perturbation method can be used to find an asymptotic solution. Banach’s fixed point theorem is used to prove existence and uniqueness of the solution, the convergence of a numerical scheme, and the validity of the asymptotic approximation. In an example numerical and asymptotic approximations are compared for various choices of time steps.
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M. A. Kraemer, L. V. Kalachev, and D. W. Coble, A Class of Models Describing Age Structure Dynamics in a Natural Forest, Natural Resource Modeling, to appear.
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R. E. O’Malley, Singular Perturbations Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.
C. D. Oliver and B. C. Larson, Forest Stand Dynamics, McGraw-Hill, New York, 1990.
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- Adelaida B. Vasil′eva, Valentin F. Butuzov, and Leonid V. Kalachev, The boundary function method for singular perturbation problems, SIAM Studies in Applied Mathematics, vol. 14, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. With a foreword by Robert E. O’Malley, Jr. MR 1316892
J. A. Cochran, The analysis of linear integral equations, McGraw-Hill, New York, 1972.
L. V. Kalachev, Asymptotic methods: application to reduction of models, Natural Resource Modeling 13 (2000), 305–338.
J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics, Springer-Verlag, New York, 1981.
M. A. Kraemer, L. V. Kalachev, and D. W. Coble, A Class of Models Describing Age Structure Dynamics in a Natural Forest, Natural Resource Modeling, to appear.
G. McFadden and C. D. Oliver, Three-dimensional forest growth model relating tree size, tree number, and stand age: Relation to previous growth models and to self-thinning, Forest Science 34 (1988), 662–676.
R. E. O’Malley, Singular Perturbations Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.
C. D. Oliver and B. C. Larson, Forest Stand Dynamics, McGraw-Hill, New York, 1990.
D. R. Smith, Singular-Perturbation Theory, Cambridge University Press, Cambridge, 1985.
A. B. Vasil’eva and V. F. Butuzov, Asymptotic behavior of the solution of an integro-differential equation with a small parameter multiplying the derivative (in Russian), Zhurnal Vychislitel’noi’ Matematiki i Matematicheskoi’ Fiziki 4 (1964), 183–191.
A. B. Vasil’eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems, SIAM, Philadelphia, 1995.
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© Copyright 2003
American Mathematical Society
