# Examples Illustrating Regular and Singular Perturbation Concepts

• Robert E. O’MalleyJr.
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 89)

## Abstract

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
$$\begin{array}{*{20}c} {y(x, \in ) = y(0)\cos \left( {\sqrt \in x} \right) + \frac{1} {{\sqrt \in }}y'(0)\sin (\sqrt \in x)} \\ { - \frac{1} {{\sqrt \in }}\cos (\sqrt { \in x} )\int_0^x {\sin } (\sqrt \in t)f(t)dt} \\ { + \frac{1} {{\sqrt \in }}\sin (\sqrt \in x)\int_0^x {\cos (\sqrt \in t)f(t)dt.} } \\ \end{array}$$

## Keywords

Singular Perturbation Formal Power Series Power Series Expansion Shock Layer Outer Solution
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