Examples Illustrating Regular and Singular Perturbation Concepts

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


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


Singular Perturbation Formal Power Series Power Series Expansion Shock Layer Outer Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Robert E. O’MalleyJr.
    • 1
  1. 1.Department of Applied MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations