Asymptotic Analysis and Singular Perturbation Theory

More documents

Recommendations

Info

and By the chain rule, we have It follows that y satisfies the PDE Then We seek an expansion d dt = ϕτ ∂θ + ε∂τ , y ′′ = (ϕτ ) 2 yθθ + ε {2ϕτ yθτ + ϕττ yθ} + ε 2 yττ . (ϕτ ) 2 yθθ + ω 2 y + ε {2ϕτ yθτ + ϕττ yθ} + ε 2 yττ = 0. y(θ, τ, ε) = y0(θ, τ) + εy1(θ, τ) + O(ε 2 ). (ϕτ ) 2 y0θθ + ω 2 y0 = 0. Imposing the requirement that y0 is a 2π-periodic function of θ, we find that which is satisfied if The solution for y0 is then ϕ(τ) = (ϕτ ) 2 = ω 2 , τ 0 ω(σ) dσ. y0(θ, τ) = A(τ)e iθ + c.c., where it is convenient to use complex exponentials, A(τ) is an arbitrary complexvalued scalar, and c.c. denotes the complex conjugate of the preceeding term. At the next order, we find that ω 2 (y1θθ + y1) + 2ωy0θτ + ωτ y0θ = 0. Using the solution for y0 is this equation, we find that ω 2 (y1θθ + y1) + i (2ωAτ + ωτ A) e iθ + c.c. = 0. The solution for y1 is periodic in θ if and only if A satisfies It follows that so that 2ωAτ + ωτ A = 0. ω|A| 2 τ = 0, ω|A| 2 = constant. 84
Thus, the amplitude of the oscillator is proportional to ω −1/2 as its frequency changes. The energy E ofthe oscillator is given by E = 1 2 (y′ ) 2 + 1 ω = 1 2 ω2 |A| 2 . Thus, E/ω is constant. The quantity E/ω is called the action. It is an example of an adiabatic invariant. 85 2 y2
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 16 and 17:
and the constant α, with dimension
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
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 74 and 75: It follows that dψ dX = −U0 sec
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: 5.5 The WKB method for ODEs Suppose
show all

Asymptotic Analysis and Singular Perturbation Theory

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?