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
Enthalten in: Professional Book Archive
Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
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 $$ \begin{array}{*{20}c} {y(x, \in ) = y(0)\cos \left( {\sqrt \in x} \right) + \frac{1} {{\sqrt \in }}y'(0)\sin (\sqrt \in x)} \\ { - \frac{1} {{\sqrt \in }}\cos (\sqrt { \in x} )\int_0^x {\sin } (\sqrt \in t)f(t)dt} \\ { + \frac{1} {{\sqrt \in }}\sin (\sqrt \in x)\int_0^x {\cos (\sqrt \in t)f(t)dt.} } \\ \end{array} $$