Asymptotic Analysis and Singular Perturbation Theory

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It follows that dψ dX = −U0 sec ψ. Using this result, the definition of ψ, and the equation dU0 dψ dU0 dX = dX dψ , we find that U0, X satisfy the following ODEs: The boundary conditions on X(ψ) are The solution for U0 is dU0 sin ψ = , dψ U0 dX ψ = −cos . dψ U0 X(θ0) = 0, X(ψ) → ∞ as ψ → 0 + . U0(ψ) = 2(k − cos ψ), where k is a constant of integration. The solution for X is X(ψ) = 1 √ 2 θ0 ψ cos t √ k − cos t dt, where we have imposed the boundary condition X(θ0) = 0. The boundary condition that X → ∞ as ψ → 0 implies that k = 1, and then U0(ψ) = 2 sin ψ 1 , X(ψ) = 2 2 Evaluating the integral for X, we get X(ψ) = 1 θ0 cosec 2 ψ t = t − sin 2 2 log tan t θ0 t + 2 cos 4 2 = log tan θ0 4 + 2 cos θ0 2 ψ θ0 ψ dt cos t sin t dt. 2 ψ ψ − log tan − 2 cos 4 2 . The asymptotic behaviors of U0 and X as ψ → 0 + are given by U0(ψ) = ψ + o(1), X(ψ) = log tan θ0 4 + 2 cos θ0 2 70 ψ − log − 2 + o(1). 4
It follows that the outer expansion of the leading-order boundary layer solution is U0(X) ∼ 4 tan θ0 4 e −4 sin2 (θ0/4) e −X as X → ∞. Rewriting this expansion in terms of the original variables, u ∼ εU0, r = 1 − εX, we get θ0 u(r, ε) ∼ 4ε tan e 4 −4 sin2 (θ0/4) −1/ε r/ε e e . The inner expansion as r → 1− of the leading order intermediate solution in (4.11) is ε u(r, ε) ∼ λ 2π er/ε . These solutions match if Thus, we conclude that λ = 4 tan u(0, ε) ∼ 4 tan θ0 e 4 −4 sin2 (θ0/4) √ 2πεe −1/ε . θ0 e 4 −4 sin2 (θ0/4) √ 2πεe −1/ε 71 as ε → 0. (4.13)
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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
Page 23 and 24: Chapter 2 Asymptotic Expansions In
Page 25 and 26: If, as is usually the case, the gau
Page 27 and 28: where denotes the C n -norm. We def
Page 29 and 30: It follows that where Since we have
Page 31 and 32: 2.2.4 Nonuniform asymptotic expansi
Page 33 and 34: Chapter 3 Asymptotic Expansion of I
Page 35 and 36: we find that the optimal truncation
Page 37 and 38: and making the change of variable (
Page 39 and 40: 3.3 The method of stationary phase
Page 41 and 42: Applying this argument n times, we
Page 43 and 44: The standard normalization for the
Page 45 and 46: We assume for simplicity that the i
Page 47 and 48: where w = − t2/3v . 31/3 It follo
Page 49 and 50: 3.5.1 Multiple integrals Propositio
Page 51: When λ > 0, we can deform the inte
Page 54 and 55: plex C, and the complex breaks down
Page 56 and 57: with the substrate is balanced by t
Page 58 and 59: For more information about enzyme k
Page 60 and 61: Thus, the solution involves two ter
Page 62 and 63: where c is an arbitrary constant of
Page 64 and 65: (a) if a(x) > 0, then the boundary
Page 66 and 67: The inner solution near x = −1 is
Page 68 and 69: If the fluid interface is a graph z
Page 70 and 71: where the prime denotes a derivativ
Page 72 and 73: We seek an asymptotic expansion of
Page 77 and 78: Chapter 5 Method of Multiple Scales
Page 79 and 80: Using this expansion in the equatio
Page 81 and 82: where the overline denotes the aver
Page 83 and 84: 5.3 The method of averaging Conside
Page 85 and 86: where f : R n ×T → R n is a Lips
Page 87 and 88: 5.5 The WKB method for ODEs Suppose
Page 89: Thus, the amplitude of the oscillat
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