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The Method of Envelopes

W. L. Miranker and M. van Veldhuizen
Mathematics of Computation
Vol. 32, No. 142 (Apr., 1978), pp. 453-496
DOI: 10.2307/2006158
Stable URL: http://www.jstor.org/stable/2006158
Page Count: 44
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The Method of Envelopes
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Abstract

The differential equation $$\frac{dx}{dt} = \frac{A}{\epsilon}x + g(t, x)$$ where $A = \begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix} and $\epsilon > 0$ is a small parameter is a model for the stiff highly oscillatory problem. In this paper we discuss a new method for obtaining numerical approximations to the solution of the initial value problem for this differential equation. As $\epsilon \rightarrow 0$, the asymptotic theory for this initial value problem yields an approximation to the solution which develops on two time scales, a fast time $t$ and a slow time $\tau = t/\epsilon$. We redevelop this asymptotic theory in such a form that the approximation consists of a series of simple functions of $\tau$, called carriers. (This series may be thought of as a Fourier series.) The coefficients of the terms of this series are functions of $t$. They are called envelopes and they modulate the carriers. Our computational method consists of determining numerical approximations to a finite collection of these envelopes. One of the principal merits of our method is its accuracy for the nonlinear problem.

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