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Über dieses Buch

This book results from various lectures given in recent years. Early drafts were used for several single semester courses on singular perturbation meth­ ods given at Rensselaer, and a more complete version was used for a one year course at the Technische Universitat Wien. Some portions have been used for short lecture series at Universidad Central de Venezuela, West Vir­ ginia University, the University of Southern California, the University of California at Davis, East China Normal University, the University of Texas at Arlington, Universita di Padova, and the University of New Hampshire, among other places. As a result, I've obtained lots of valuable feedback from students and listeners, for which I am grateful. This writing continues a pattern. Earlier lectures at Bell Laboratories, at the University of Edin­ burgh and New York University, and at the Australian National University led to my earlier works (1968, 1974, and 1978). All seem to have been useful for the study of singular perturbations, and I hope the same will be true of this monograph. I've personally learned much from reading and analyzing the works of others, so I would especially encourage readers to treat this book as an introduction to a diverse and exciting literature. The topic coverage selected is personal and reflects my current opin­ ions. An attempt has been made to encourage a consistent method of ap­ proaching problems, largely through correcting outer limits in regions of rapid change. Formal proofs of correctness are not emphasized.



Chapter 1. Examples Illustrating Regular and Singular Perturbation Concepts

Consider a linear spring-mass system with forcing, but without damping, and with a small spring constant. This yields the differential equation
$$ y + \in y = f\left( x \right) $$
for the displacement y(x) as a function of time x, with the small positive parameter ε being the ratio of the spring constant to the mass of the spring. This should be solved on the semi-infinite interval x ≥ 0, with both the initial displacement y(0) and the initial velocity y’(0) prescribed. The traditional approach to solving such initial value problems [cf. Boyce and DiPrima (1986)] is to note that for ε small the homogeneous equation has the slowly varying solutions \( cos\left( {\sqrt \in x} \right) \) and \( sin\left( {\sqrt \in x} \right) \) and to look for a solution through variation of parameters. Specifically, one sets \( y\left( x \right) = v_1 \left( x \right)\cos \left( {\sqrt \in x} \right) + v_2 \left( x \right)\sin \left( {\sqrt \in x} \right) \), where \( v'_1 \cos \left( {\sqrt \in x} \right) + v'_2 \sin \left( {\sqrt \in x} \right) = 0 \) and \( - \sqrt \in v'_1 \sin \left( {\sqrt \in x} \right) + \sqrt \in v'_2 \cos \left( {\sqrt \in x} \right) = f\left( x \right). \) Since \( y\left( 0 \right) = v_1 \left( 0 \right) \) and \( y'\left( 0 \right) = \sqrt \in v_2 \left( 0 \right) \), solving for v’1 and v’2 and integrating provides the unique solution
$$ \begin{array}{*{20}c} {y(x, \in ) = y(0)\cos \left( {\sqrt \in x} \right) + \frac{1} {{\sqrt \in }}y'(0)\sin (\sqrt \in x)} \\ { - \frac{1} {{\sqrt \in }}\cos (\sqrt { \in x} )\int_0^x {\sin } (\sqrt \in t)f(t)dt} \\ { + \frac{1} {{\sqrt \in }}\sin (\sqrt \in x)\int_0^x {\cos (\sqrt \in t)f(t)dt.} } \\ \end{array} $$
Robert E. O’Malley

Chapter 2. Singularly Perturbed Initial Value Problems

Readers should refer to Murray (1977) and to earlier chemical engineering literature [especially Bowen et al. (1963) and Heinekin et al. (1967)] for experts’ explanations of the significance of the pseudo-steady-state hypothesis in biochemistry. The theory of Michaelis and Menton (1913) and Briggs and Haldane (1925) concerns a substrate S being converted irreversibly by a single enzyme E into a product P. There is also an intermediate substrate-enzyme complex SE. Since the back reaction is negligible, we shall systematically write
$$S + E\begin{array}{*{20}{c}} {{{k}_{1}}} \\ \to \\ \leftarrow \\ {{{k}_{{ - 1}}}} \\ \end{array} SE\xrightarrow{{{{k}_{2}}}}P + E. $$
Robert E. O’Malley

Chapter 3. Singularly Perturbed Boundary Value Problems

Consider the two-point problem εy+a(x)y+b(x)y=f(x) on 0≤x≤1 where a(x)>0 and with the boundary values y(0) and y(1) prescribed. We shall suppose that a, b, and f are arbitrarily smooth, and we shall prove that the asymptotic solution will exist, be unique, and have the form
$$y\left( {x,\varepsilon } \right) = Y\left( {x,\varepsilon } \right) + \xi \left( {x/\varepsilon,\varepsilon } \right) $$
where the outer expansion Y(x, c) has a power series expansion
$$Y\left( {x,\varepsilon } \right) \sim \sum\limits_{j = 0}^\infty {{Y_j}\left( x \right){\varepsilon ^j}} $$
and the initial layer correction \(\xi \left( {\tau,\varepsilon } \right) \) has an expansion
$$\xi \left( {\tau , \in } \right)\sim \sum\limits_{{j = 0}}^{\infty } {{{\xi }_{j}}\left( \tau \right){{ \in }^{j}}} $$
such that each ξj(and its derivatives) will tend to zero as the stretched variable \(\tau = x/\varepsilon \) tends to infinity. We shall base our proof on the existence of both a smooth asymptotic solution of the differential equation and of a (linearly independent) asymptotic solution which features an initial layer of nonuniform convergence near x = O. Recognizing that the asymptotic solution is an additive composite function of the slow “time” x and the fast time x/E generalizes to multitime expansions in many asymptotic contexts [cf. Nayfeh (1973) and, especially, Kevorkian and Cole (1981).
Robert E. O’Malley


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