Asymptotic Analysis and Singular Perturbation Theory

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and the constant α, with dimensions of 1/length, is given by α 2 √ mk = . The energy levels E ε n of a slightly anharmonic oscillator with potential V ε (x) = 1 2 kx2 + ε k α 2 W (αx) + O(ε2 ) as ε → 0 + where ε > 0 have the asymptotic behavior E ε n = ω n + 1 2 + ε∆n + O(ε 2 ) where ∆n = 2 W (ξ)Hn(ξ)e−ξ2 dξ H2 n(ξ)e−ξ2 . dξ as ε → 0 + , For an extensive and rigorous discussion of spectral perturbation theory for linear operators, see [11]. 1.4 Nondimensionalization The numerical value of any quantity in a mathematical model is measured with respect to a system of units (for example, meters in a mechanical model, or dollars in a financial model). The units used to measure a quantity are arbitrary, and a change in the system of units (for example, to feet or yen, at a fixed exchange rate) cannot change the predictions of the model. A change in units leads to a rescaling of the quantities. Thus, the independence of the model from the system of units corresponds to a scaling invariance of the model. In cases when the zero point of a unit is arbitrary, we also obtain a translational invariance, but we will not consider translational invariances here. Suppose that a model involves quantities (a1, a2, . . . , an), which may include dependent and independent variables as well as parameters. We denote the dimension of a quantity a by [a]. A fundamental system of units is a minimal set of independent units, which we denote symbolically by (d1, d2, . . . , dr). Different fundamental system of units can be used, but given a fundamental system of units any other derived unit may be constructed uniquely as a product of powers of the fundamental units, so that for suitable exponents (α1, α2, . . . , αr). [a] = d α1 1 dα2 2 . . . dαr r (1.13) Example 1.8 In mechanical problems, a fundamental set of units is d1 = mass, d2 = length, d3 = time, or d1 = M, d2 = L, d3 = T , for short. Then velocity 12
V = L/T and momentum P = ML/T are derived units. We could use instead momentum P , velocity V , and time T as a fundamental system of units, when mass M = P/V and length L = V T are derived units. In problems involving heat flow, we may introduce temperature (measured, for example, in degrees Kelvin) as another fundamental unit, and in problems involving electromagnetism, we may introduce current (measured, for example, in Ampères) as another fundamental unit. The invariance of a model under the change in units dj ↦→ λjdj implies that it is invariant under the scaling transformation for any λ1, . . . λr > 0, where Thus, if ai → λ α1,i 1 λα2,i 2 . . . λ αr,i r ai i = 1, . . . , n [ai] = d α1,i 1 d α2,i 2 . . . d αr,i r . (1.14) a = f (a1, . . . , an) is any relation between quantities in the model with the dimensions in (1.13) and (1.14), then f has the scaling property that λ α1 1 λα2 2 . . . λαr r f (a1, . . . , an) = f λ α1,1 1 λα2,1 2 . . . λ αr,1 r a1, . . . , λ α1,n 1 λ α2,n 2 . . . λ αr,n r an . A particular consequence of the invariance of a model under a change of units is that any two quantities which are equal must have the same dimensions. This fact is often useful in finding the dimension of some quantity. Example 1.9 According to Newton’s second law, force = rate of change of momentum with respect to time. Thus, if F denotes the dimension of force and P the dimension of momentum, then F = P/T . Since P = MV = ML/T , we conclude that F = ML/T 2 (or mass × acceleration). Example 1.10 In fluid mechanics, the shear viscosity µ of a Newtonian fluid is the constant of proportionality that relates the viscous stress tensor T to the velocity gradient ∇u: Stress has dimensions of force/area, so T = 1 2 µ ∇u + ∇u T . [T ] = ML T 2 1 M = . L2 LT 2 13
Page 1: Asymptotic Analysis and Singular Pe
Page 4 and 5: 4.1.1 Outer solution . . . . . . .
Page 6 and 7: Once we have constructed such an as
Page 8 and 9: For example, setting ε = 0 in (1.2
Page 10 and 11: Solving the first equation, we get
Page 12 and 13: and A ε : R n → R n is a linear
Page 14 and 15: Energy eigenstates are wavefunction
Page 18 and 19: The velocity gradient ∇u has dime
Page 20 and 21: 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
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The inner solution near x = −1 is
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If the fluid interface is a graph z
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where the prime denotes a derivativ
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We seek an asymptotic expansion of
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It follows that dψ dX = −U0 sec
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Chapter 5 Method of Multiple Scales
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Using this expansion in the equatio
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where the overline denotes the aver
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5.3 The method of averaging Conside
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where f : R n ×T → R n is a Lips
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5.5 The WKB method for ODEs Suppose
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Thus, the amplitude of the oscillat
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Asymptotic Analysis and Singular Perturbation Theory

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