All Questions
Tagged with perturbation-theory boundary-layer
18
questions
1
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1
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56
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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
0
votes
0
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131
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Small parameter expansion of solution to ODE
I'm working with an ODE of the form
\begin{equation}
\begin{split}
C'(z) &= (z^2 + \epsilon^2)C'' \\
C(a) &= b \\
C(1) &= 1 \\
\end{split}
\end{equation}
where $0 < a,b < 1$, and $\...
- 351
1
vote
0
answers
53
views
Spotting distinguished limits from Robin boundary conditions
I am currently working on a problem involving solving PDEs and using boundary layers. In this problem, $f(x,y)$ is the main dependent variable, $f(x,y)=f_R (X_R ,y)$ and $f(x,y)=f_B(x,Y_B)$ where $X_R$...
- 173
2
votes
2
answers
335
views
WKB for a boundary value problem with two layers
The equation $\epsilon y''-x^4y'-y=0,\ y(0)=y(1)=1$, can be solved by boundary layer analysis and turns out it has two layers of size $O(\epsilon) $ and $O(\sqrt{\epsilon})$ at $1,0$ accordingly. Is ...
- 291
1
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1
answer
280
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WKB for non-homogeneous ODE
Consider the ODE $$\epsilon^2 y'' + \epsilon x y' - y = -1, \; y(0) = 0, \; y(1) = 3$$
I've seen the WKB method applied to homogeneous (linear) ODEs, but here we have the $-1$ term. I could perhaps do ...
- 301
1
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1
answer
303
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Inner and outer expansions
I'd really appreciate some help with the following question.
Find inner and outer expansions, correct up to and including terms of O(ε), for the function
$$ f(x;ε) = \frac{e^{-\frac{x}{ε}}}{x} + \...
- 101
3
votes
2
answers
293
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ODE with nested boundary layers
Problem:
Consider the equation
$$\varepsilon^3 \frac{d^2y}{dx^2} + 2x^3 \frac{dy}{dx} - 4\varepsilon y = 2x^3 \qquad \qquad y(0) = a \;, \; y(1)=b$$
in the limit as $\varepsilon \rightarrow 0^+$, ...
- 4,945
0
votes
1
answer
204
views
Matched Asymptotic Expansion- Boundary Layer Problem
$\epsilon y''=e^{\epsilon y'}+y$ for $0<x<1$
where $y(0)=1$ and $y(1)=-1$.
Compute the first term matched asymptotic expansion for the equation.
The outer expansion expansion gives me $y(x)=-...
- 361
0
votes
1
answer
469
views
How to find inner solution in the method of matched asymptotic expansions
Consider the equation of the form $\epsilon y^{\prime \prime}+a y^{\prime}=0$ on $x\in[0,1]$ with $a\in\mathbb{R}$, $0<\epsilon\ll1$,$y(0) =\alpha$, and $y(1)=\beta$. Show that if $a >0$ then ...
- 393
1
vote
1
answer
296
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Matched Asymptotic Expansion- Boundary Layer
In this problem
$\epsilon y''+(x+\frac{1}{2})y'+y=0$ for $0 < x <1 $
$y(0)=2, y(1)=3$
In the outer expansion,
$y\approx y_0(x)+\epsilon y_1(x)+\cdots$
I found the $O(1)$ problem to be: $(...
- 361
0
votes
1
answer
88
views
$y_t+-\epsilon .y_{xx}+ M.y_x=0\, ;(x,t) \in (0,1)\times(0,T)$ Boundary layers
I was reading an article about pertubation in advection-transport equations, nad so they have defined the following equation with the perturbation ($\epsilon). $
$$y_t+-\epsilon .y_{xx}+ M.y_x=0\, ;(...
- 1,415
2
votes
4
answers
235
views
Boundary layer in time
Consider the initial value problem $\varepsilon x'' + x' + tx = 0$ where $x(0) = 0$ and $x'(0) = 1$. I'm solving this problem using a matched asymptotic expansion. First, I let $$x(t, \varepsilon) = \...
1
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1
answer
945
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Matching expansion of an ODE: $\epsilon y'' + xy' + y = 0$
I am trying to solve a boundary layer problem using matched expansion
$$\epsilon y'' + xy' + y = 0$$
where the boundary condition is
$$y(0) = 1, y(1) = 1$$
and $x\in (0,1)$.
So far, I have the outer ...
- 53
1
vote
1
answer
870
views
Obtain the leading order uniform approximation of the solution
Obtain the leading order uniform approximation of the solution to $ \epsilon y′′-x^2y′-y=0$.
The boundary conditions are $y(0)=y(1)=1$.
Since $a(x)<0$ the boundary layer is at $x=1$.
The outer ...
- 15
1
vote
1
answer
125
views
Boundary layer type with initial value problem
Consider the initial value problem
$\sqrt{\epsilon} \, u'' + u' - u = e^{2t}$ , with $u(0)=1$, and $u'(0)=1/\sqrt{\epsilon}$.
I am trying to use a matched asymptotic expansion to find the leading ...
- 343