Nonlinear Analysis: Real World Applications 12 4 , , Transport Theory and Statistical Physics 29 , , Mathematical Models and Methods in Applied Sciences 11 08 , , SIAM journal on mathematical analysis 34 6 , , Mathematical Models and Methods in Applied Sciences 12 07 , , Computational and Mathematical Methods in Medicine 4 1 , , Hence, there has been a growing interest in providing a more focused description of multiscale processes by aggregating variables in a way that is relevant for a particular purpose and that preserves the salient features of the dynamics and many ad hoc methods for this have been devised in the applied sciences.
The aim of this book is to describe some tools which provide a systematic way of deriving the so-called limit equations for such aggregated variables and ensuring that the coefficients of these equations encapsulate the relevant information from the discarded levels of description. Since any approximation is only valid if an estimate of the incurred error is available, the tools we describe allow for proving that the solutions to the original multiscale family of equations converge to the solution of the limit equation if the relevant parameter converges to its critical value.
All problems discussed in this book belong to the class of singularly perturbed problems; that is, problems in which the structure of the limit equation is signifi- cantly different from that of the multiscale model.https://fiarenkare.gq
Mathematical and theoretical biology
Such problems appear in all areas of science and can be approached by many techniques. In this book we present the classical asymptotic analysis based on the expansion of the solution in a series of powers of the parameter and, particularly, for the finite dimensional models, we explore the full power of the Tikhonov—Vasilyeva theory. The applications mostly are drawn from mathematical biology and epidemiology, but we discuss also some classical problems in other applied sciences.
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It is important, however, to realize that the approach to singularly perturbed problems presented in the book is by no means unique. There is a similar comprehensive theory based on the centre manifold theorem, called the geometric singular perturbation theory see, e. In our opinion, however, the asymptotic expansion method, being possibly less elegant, is nevertheless more intuitive and more flexible and requires less theoretical background.
The book is organized as follows.
In Chap. Such models in a natural way contain a small parameter which is the ratio of the slow and the fast rates, thus lending themselves to asymptotic analysis. We discuss, among others, classical models of fluid dynamics and kinetic theory, population problems with fast migrations, epidemiological problems concerning diseases with quick turnover, models of enzyme kinetics and Brownian motion with fast direction changes.
We also discuss initial and boundary layer phenomena using a simplified fluid dynamics equation as an example. In the conclusion of the chapter we discuss a model of enzyme kinetics and show in detail the application of the Hilbert expansion method to derive formally the Michaelis—Menten model; a rigorous derivation is referred to Chap. All problems discussed in this book belong to the class of singularly perturbed problems; that is, problems in which the structure of the limit equation is significantly different from that of the multiscale model.
Such problems appear in all areas of science and can be attacked using many techniques. Methods of Small Parameter in Mathematical Biology will appeal to senior undergraduate and graduate students in applied and biomathematics, as well as researchers specializing in differential equations and asymptotic analysis. Enter your Postcode or Suburb to view availability and delivery times.
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