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Erschienen in:
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1991 | OriginalPaper | Buchkapitel

Examples Illustrating Regular and Singular Perturbation Concepts

verfasst von : Robert E. O’Malley Jr.

Erschienen in: Singular Perturbation Methods for Ordinary Differential Equations

Verlag: Springer New York

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

Metadaten
Titel
Examples Illustrating Regular and Singular Perturbation Concepts
verfasst von
Robert E. O’Malley Jr.
Copyright-Jahr
1991
Verlag
Springer New York
DOI
https://doi.org/10.1007/978-1-4612-0977-5_1