All Questions
Tagged with perturbation-theory asymptotics
198
questions
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69
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Finding eigenvalues of a turning point ODE using WKB method
Question: Using the WKB method, provide an approximation for the eigenvalue, $\lambda$, of the problem:
$$y'' + \pi^2\lambda y(1+2\cos \pi x)\sin^2(\pi x/2) = 0,~0\le x\le 1,~y(0)=y(1)=0.$$
Compute ...
- 3,965
3
votes
1
answer
100
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Asymptotic expansion of $\int_{\xi}^{\infty}\frac{e^{-\alpha t}}{t}dt$
Question: Provided $\xi \ll 1$, find the first three terms of the asymptotic expansion of the integral $$I(\xi) = \int_{\xi}^{\infty}\frac{e^{-\alpha t}}{t}dt.$$
My approach: Assuming $\alpha>0$ ...
- 3,965
2
votes
1
answer
70
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Leading-order approximation of $\int_0^{\infty} e^{t-z(t^4-2t^2)}\sin^2(2\pi\nu t)~dt$
Question: For $z \gg 1$, find the leading-order approximation to the integral,
$$\int_0^{\infty} e^{t-z(t^4-2t^2)}\sin^2(2\pi\nu t)~dt,$$
allowing for any value of the parameter $\nu > 0$.
My ...
- 3,965
1
vote
1
answer
70
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Determine the two-term expansion for large roots of the transcendental equation $\tan(x) =\frac 1x$
For this problem, I was given a hint that when $x$ is large, $\frac 1x$ is nearly zero, and $x \sim n\pi$ where $n$ is a large integer.
Initially tried the taylor expansion, but that didn't work out.
...
- 11
3
votes
0
answers
45
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behavior of SDE as parameter goes to infinity (Ornstein-Uhlembeck?)
In a physics paper, I saw the following (weird) heuristic argument:
Let $\theta,v>0$ be constants. Starting from the SDE
\begin{equation}
dX_t=D(X_t)(U'(X_t) -\theta(X_t-vt))dt +\sqrt{2D(X_t)}dW_t
...
- 305
1
vote
1
answer
96
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Two Timing (Multiple Time Scales) with Coupled IVPs
Question: Find the leading-order approximation for times of order $\epsilon^{-1}$ to $$\ddot{x} + x = y,~~\dot{y} = \epsilon(xy - 2y^2),~~x(0) = 1,~~\dot{x}(0) = 0,~~y(0)=1.$$
My approach: Let the ...
- 3,965
1
vote
1
answer
56
views
Leading order matching of $\epsilon x^py'' + y' + y = 0$
Question: The function $y(x)$ satisfies $$\epsilon x^py'' + y' + y = 0,$$ in $x\in [0,1]$, where $p<1$, subject to the boundary conditions $y(0) = 0$ and $y(1)=1$. Find the rescaling for the ...
- 3,965
1
vote
2
answers
97
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Asymptotic expansion of $I(\alpha;\epsilon) = \int_0^{\infty}\frac{dx}{(\epsilon^2+x^2)^{\alpha/2}(1+x)}$
Question: Evaluate the first two terms of as $\epsilon \to 0$ of $$I(\alpha;\epsilon) = \int_0^{\infty}\frac{dx}{(\epsilon^2+x^2)^{\alpha/2}(1+x)},$$ for $\alpha = \frac{1}{2}, 1, 2$, if $$C(\alpha) = ...
- 3,965
1
vote
1
answer
61
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Asymptotic expansion of $\int_0^{\pi/2}\frac{\sin^2\theta}{(1-m^2\sin^2\theta)^{1/2}}d\theta$
Question: Evaluate the first two terms of
$$I(m) = \int_0^{\pi/2}\frac{\sin^2\theta}{(1-m^2\sin^2\theta)^{1/2}}d\theta$$ as $m\to 1^-$.
My approach: Setting $m = 1-\epsilon$ and $k = \sin\theta$, we ...
- 3,965
1
vote
1
answer
115
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Asymptotic approximation of an integral using splitting range
Question: Evaluate the first two terms as $\epsilon\to 0$ of $$\mathcal{N}(z)=\int_z^1\frac{dx}{\sqrt{x^3+\epsilon}}$$ where $0\le z <1$.
My approach: First, let us calculate $\mathcal{N}(z=0)$. It ...
- 3,965
0
votes
1
answer
62
views
Perturbed real roots of an exponential-polynomial equation
Question: Develop three terms of the perturbation solutions to the real roots of
$$(x^3 + 2x^2 + x)e^{-x} = \epsilon,$$
identifying the scalings in the expansion sequence $\delta_0(\epsilon)x_0 + \...
- 3,965
0
votes
0
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48
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Finding regular and singular roots of a cubic perturbed polynomial using rescaling
Question: Find the rescalings for the roots of
$$\epsilon^5 x^3 - (3 - 2\epsilon^2 + 10\epsilon^5 - \epsilon^6)x^2 + (30 - 3\epsilon -20 \epsilon^2 + 2\epsilon^3 + 24\epsilon^5 - 2\epsilon^6 - 2\...
- 3,965
0
votes
1
answer
91
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Asymptotics of a nonlinear PDE
Consider the partial differential equation with boundary conditions
\begin{equation}
\frac{\partial u}{\partial t} = \varepsilon \bigg( (u+1)\frac{\partial u}{\partial x}\bigg), \qquad \frac{\partial ...
- 635
6
votes
0
answers
88
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Method of Dominant Balance with high order system
This question comes from Bender and Orszag's Asymptotic Methods and Perturbation Theory.
I'm practicing applying the method of dominant balance to study behavior as $x\to \infty$ for systems which ...
- 1,106
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How to do this perturbation expansion?
I got the following expansion in the context of studying $\phi^4 $ quantum field theory. This is the solution for exact 2-point propagator in the ladder-rainbow approximation. The expansion is -
$\mu^{...
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