Asymptotic Analysis and Singular Perturbation Theory

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with the solution y0(t) = cos t. The next-order perturbation equations are with the solution y1(t) = 1 32 y ′′ 1 + y1 + y 3 0 = 0, y1(0) = 0, y ′ 1(0) = 0, 3 [cos 3t − cos t] − t sin t. 8 This solution contains a secular term that grows linearly in t. As a result, the expansion is not uniformly valid in t, and breaks down when t = O(ε) and εy1 is no longer a small correction to y0. The solution is, in fact, a periodic function of t. The straightforward expansion breaks down because it does not account for the dependence of the period of the solution on ε. The following example illustrates the difficulty. Example 5.1 We have the following Taylor expansion as ε → 0: cos [(1 + ε)t] = cos t − εt sin t + O(ε 2 ). This asymptotic expansion is valid only when t ≪ 1/ε. To construct a uniformly valid solution, we introduced a stretched time variable τ = ω(ε)t, and write y = y(τ, ε). We require that y is a 2π-periodic function of τ. The choice of 2π here is for convenience; any other constant period — for example 1 — would lead to the same asymptotic solution. The crucial point is that the period of y in τ is independent of ε (unlike the period of y in t). Since d/dt = ωd/dτ, the function y(τ, ε) satisfies ω 2 y ′′ + y + εy 3 = 0, y(0, ε) = 1, y ′ (0, ε) = 0, y(τ + 2π, ε) = y(τ, ε), where the prime denotes a derivative with respect to τ. We look for an asymptotic expansion of the form y(τ, ε) = y0(τ) + εy1(τ) + O(ε 2 ), ω(ε) = ω0 + εω1 + O(ε 2 ). 74
Using this expansion in the equation and equating coefficients of ε 0 , we find that The solution is ω 2 0y ′′ 0 + y0 = 0, y0(0) = 1, y ′ 0(0) = 0, y0(τ + 2π) = y0(τ). y0(τ) = cos τ, ω0 = 1. After setting ω0 = 1, we find that the next order perturbation equations are y ′′ 1 + y1 + 2ω1y ′′ 0 + y 3 0 = 0, y1(0) = 0, y ′ 1(0) = 0, y1(τ + 2π) = y1(τ). Using the solution for y0 in the ODE for y1, we get y ′′ 1 + y1 = 2ω1 cos τ − cos 3 τ = We only have a periodic solution if and then It follows that 2ω1 − 3 cos τ − 4 1 cos 3τ. 4 ω1 = 3 8 , y1(t) = 1 [cos 3τ − cos τ] . 32 y = cos ωt + 1 32 ε [cos 3ωt − cos ωt] + O(ε2 ), ω = 1 + 3 8 ε + O(ε2 ). This expansion can be continued to arbitrary orders in ε. The appearance of secular terms in the expansion is a consequence of the nonsolvability of the perturbation equations for periodic solutions. Proposition 5.2 Suppose that f : T → R is a smooth 2π-periodic function, where T is the circle of length 2π. The ODE y ′′ + y = f 75
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Asymptotic Analysis and Singular Pe
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4.1.1 Outer solution . . . . . . .
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Once we have constructed such an as
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For example, setting ε = 0 in (1.2
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Solving the first equation, we get
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and A ε : R n → R n is a linear
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Energy eigenstates are wavefunction
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and the constant α, with dimension
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The velocity gradient ∇u has dime
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Imposing the requirement that R d
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Chapter 2 Asymptotic Expansions In
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If, as is usually the case, the gau
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Page 33 and 34: Chapter 3 Asymptotic Expansion of I
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Page 39 and 40: 3.3 The method of stationary phase
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Page 47 and 48: where w = − t2/3v . 31/3 It follo
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Page 77: Chapter 5 Method of Multiple Scales
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Page 83 and 84: 5.3 The method of averaging Conside
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Page 87 and 88: 5.5 The WKB method for ODEs Suppose
Page 89: Thus, the amplitude of the oscillat
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Asymptotic Analysis and Singular Perturbation Theory

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