Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] James Alan Cochran, The analysis of linear integral equations, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1972. McGraw-Hill Series in Modern Applied Mathematics. MR 0447991
  • [2] Leonid V. Kalachev, Asymptotic methods: application to reduction of models, Natur. Resource Modeling 13 (2000), no. 3, 305–338. Perturbation methods and their applications. MR 1776820, https://doi.org/10.1111/j.1939-7445.2000.tb00038.x
  • [3] J. Kevorkian and Julian D. Cole, Perturbation methods in applied mathematics, Applied Mathematical Sciences, vol. 34, Springer-Verlag, New York-Berlin, 1981. MR 608029
  • [4] 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.
  • [5] 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.
  • [6] R. E. O'Malley, Singular Perturbations Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991.
  • [7] C. D. Oliver and B. C. Larson, Forest Stand Dynamics, McGraw-Hill, New York, 1990.
  • [8] Donald R. Smith, Singular-perturbation theory, Cambridge University Press, Cambridge, 1985. An introduction with applications. MR 812466
  • [9] A. B. Vasil′eva and V. F. Butuzov, Asymptotic behaviour of the solution of an integro-differential equation with a small parameter multiplying the derivative, Ž. Vyčisl. Mat. i Mat. Fiz. 4 (1964), no. 4, suppl., 183–191 (Russian). MR 0172082
  • [10] 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

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 45J05, 34E10, 34K26

Retrieve articles in all journals with MSC: 45J05, 34E10, 34K26

Additional Information

DOI: https://doi.org/10.1090/qam/1999835
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society