Asymptotic Analysis and Singular Perturbation Theory

More documents

Recommendations

Info

Integrating the power series expansion of e−t2 term by term, we obtain the power series expansion of erf x, erf x = 2 √ x − π 1 3 x3 + . . . + (−1)n (2n + 1)n! x2n+1 + . . . , which is convergent for every x ∈ R. For large values of x, however, the convergence is very slow. the Taylor series of the error function at x = 0. Instead, we can use the following divergent asymptotic expansion, proved below, to obtain accurate approximations of erf x for large x: erf x ∼ 1 − e−x2 √ π ∞ n+1 (2n − 1)!! (−1) 2n 1 xn+1 as x → ∞, (2.4) n=0 where (2n − 1)!! = 1 · 3 · . . . · (2n − 1). For example, when x = 3, we need 31 terms in the Taylor series at x = 0 to approximate erf 3 to an accuracy of 10 −5 , whereas we only need 2 terms in the asymptotic expansion. Proposition 2.10 The expansion (2.4) is an asymptotic expansion of erf x. Proof. We write erf x = 1 − 2 √ π <strong>and</strong> make the change of variables s = t 2 , For n = 0, 1, 2, . . ., we define erf x = 1 − 1 √ π Fn(x) = ∞ x e −t2 dt, ∞ x2 s −1/2 e −s ds. ∞ x2 s −n−1/2 e −s ds. Then an integration by parts implies that Fn(x) = e−x2 − n + x2n+1 1 Fn+1(x). 2 By repeated use of this recursion relation, we find that erf x = 1 − 1 √ F0(x) π = 1 − 1 e √ π −x2 1 − x 2 F1(x) = 1 − 1 √ e π −x2 1 1 − x 2x3 + 1 · 3 F2(x) 22 24
It follows that where Since we have where = 1 − 1 √ e π −x2 1 x erf x = 1 − e−x2 √ π 1 1 · 3 · . . . · (2N − 1) − + . . . (−1)N 2x3 2Nx2N+1 N+1 1 · 3 · . . . · (2N + 1) + (−1) 2N+1 FN+1(x) . N n 1 · 3 · . . . · (2n − 1) (−1) + RN+1(x) n=0 N+1 1 RN+1(x) = (−1) √ π |Fn(x)| = ≤ 2 n x 2n+1 1 · 3 · . . . · (2N + 1) 2N+1 FN+1(x). ∞ x2 s −n−1/2 e −s ds ∞ 1 x 2n+1 ≤ e−x2 , x2n+1 x 2 e −s ds e |RN+1(x)| ≤ CN+1 −x2 , x2N+3 CN = 1 · 3 · . . . · (2N + 1) 2N+1√ . π This proves the result. 2.2.3 Generalized asymptotic expansions Sometimes it is useful to consider more general asymptotic expansions with respect to a sequence of gauge functions {ϕn} of the form where for each N = 0, 1, 2, . . . f(x) − f(x) ∼ ∞ fn(x), n=0 N fn(x) = o(ϕN+1). n=0 25
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: where denotes the C n -norm. We def
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 and 88:
5.5 The WKB method for ODEs Suppose
Page 89:
Thus, the amplitude of the oscillat
show all

Asymptotic Analysis and Singular Perturbation Theory

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

Delete template?

Save as template?