Abstract
In this chapter we provide a gentle introduction of the Chapman–Enskog-type asymptotic expansion and of the basic techniques of proving its convergence. To make the presentation not too technical, it is illustrated on systems of linear ordinary differential equations. The chapter begins with a survey of necessary results from linear algebra and theory of finite-dimensional dynamical systems and it is concluded with a detailed analysis of linear population models with geographical structure in which the migration between geographical patches is much faster than the demographic processes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Banasiak, J.: Mathematical Modelling in One Dimension. An Introduction via Difference and Differential Equations. Cambridge University Press, Cambridge (2013)
Braun, M.: Differential Equations and Their Applications. Springer, New York (1993)
Cercignani, C., Illner R., Pulvirenti M.: The Mathematical Theory of Dilute Gases. Springer, New York (1994)
Hirsch, M., Smale, S.: Differential Equations, Dynamical Systems, and Linear Algebra. Academic, San Diego (1974)
Hoppensteadt, F.C.: Singular perturbations on the infinite interval. Trans. Am. Math. Soc. 123, 521–535 (1966)
Kato, T.: Perturbation Theory for Linear Operators. 2nd edn. Springer, Berlin (1984)
Mika, J.R.: New asymptotic expansion algorithm for singularly perturbed evolution equations. Math. Methods Appl. Sci. 3, 172–188 (1981)
Mika, J., Banasiak, J.: Singularly Perturbed Evolution Equations with Applications to Kinetic Theory. World Scientific, River Edge (1995)
O’Malley, R.E., Jr.: Singular Perturbation Methods for Ordinary Differential Equations. Springer, New York (1991)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Banasiak, J., Lachowicz, M. (2014). Introduction to the Chapman–Enskog Method: Linear Models with Migrations. In: Methods of Small Parameter in Mathematical Biology. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-05140-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-05140-6_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-05139-0
Online ISBN: 978-3-319-05140-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)