All Questions
Tagged with perturbation-theory boundary-value-problem
17
questions
2
votes
0
answers
101
views
Poisson Equation for a perturbed sphere - both exterior and interior solutions
I am trying to solve a Heat transfer problem in a slightly ellipsoidal geometry using the Poission Equation, and Cauchy matching conditions on the boundary between the interior (which is a finite ...
- 31
1
vote
0
answers
58
views
Singular perturbation theory involving exponentials
Suppose we're given the following second order ODE
\begin{align}
-\epsilon x''(t)=e^{x(t)}-1
\end{align}
with boundary conditions
\begin{align}
x(0)=0,\quad x(1)=1.
\end{align}
Suppose $\epsilon>0$ ...
- 1,055
0
votes
0
answers
131
views
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
112
views
Non linear spring mass IVP
I’ve read a solution to a question in the Applied Mathematics book , but it seems incorrect to me . So here is the question :
Non-dimensionalize and find the leading order solution to the IVP:
$ m y’’ ...
- 141
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
0
votes
1
answer
202
views
Solution Poisson Equation by Homotopy Perturbation Method
I need to solve the following Boundary Value problem
$$\frac{\partial^2w }{\partial x^2}+\frac{\partial^2w }{\partial y^2}=c$$
Boundary conditions are
$$w(x,h)=0$$
$$w\left(\frac{\pm h}{\sqrt3},...
2
votes
0
answers
137
views
eigenfunctions of Dirichlet laplacian in perturbed domain of the disk
Let $\varepsilon$ a small positive real and $\Omega_\varepsilon$ an open set of $\mathbb{R}^2$ such that $$\Omega_\varepsilon=\{(x,y) \mid x = R(\theta,\varepsilon) \cos(\theta)\quad y=R(\theta,\...
- 704
0
votes
1
answer
108
views
Composite approximation of BVP
$$\epsilon \ddot y + (1+x)\dot y = 0 \;\;\;\;\;\; y(0) = a \;\;\; y(1) = b$$
How do I find the "leading order composite approximation" of this BVP? I'm not totally sure I'm approaching this right, ...
- 353
1
vote
1
answer
274
views
Boundary Values and Initial conditions for Linear Stability analysis (Fluid Dynamics)
Lets say we have a system of partial differential equation $\Delta(x,y,t,u^{(n)})=0$ (Navier-Stokes Equations) with a given stationary solution $u_s(x,y)$ for a inviscid flow. Note that $u$ is a 2D-...
- 15.9k
2
votes
1
answer
177
views
How to know which boundary condition to use
With asymptotic methods for ODEs where you have like an inner, outer region and you are given two boundary condition, how do you know which condition to use when constructing the inner/outer solution?
...
- 3,751
0
votes
1
answer
485
views
Dominant Balance with epsilon small
Consider the boundary value problem $$ε
\frac{d^2y}{ dx^2}
+ (1 + x)
\frac{dy }{dx}
+ y = 0$$ subject to $y(0) = 0$, $y(1) = 1$, for $0 \le x \le 1$, $ε ≪ 1$.
By considering the rescaling $x = x_0 + ...
- 3,751
1
vote
1
answer
1k
views
Find leading order matched asymptotic expansion for $\epsilon y'' + y y' - y = 0; y(0) = -1, y(1) = 0$
My attempt:
The outer problem for leading order term: $$y_0 y_0' - y_0 = 0$$
This has solution: $y_0(x) = 0$ or $y_0(x) = x + c$.
I notice that $y(x) < 0$ on this interval, so I assume the ...
- 1,819
0
votes
1
answer
512
views
Finding a composite solution to an ODE (boundary layer problem)
Given $\epsilon \frac{d^2u}{dt^2}-a(t)\frac{du}{dt}+b(t)u=0$, where $a(t)>0$, $u(0)=1$, $u(1)=1$, and assuming that the boundary layer is at $t=1$, and the boundary layer variable is $T=\frac{1-t}{...
- 9,708
3
votes
1
answer
825
views
Singular Perturbation Approx. for $\epsilon y'' + \frac{2 \epsilon}{t} y'-y=0$
Use singular perturbation techniques to find the leading order uniform approximation to the solution to the boundary value problem
$$\epsilon y'' + \frac{2 \epsilon}{t} y'-y=0$$
$0<t<1$ and $y(...
- 259
0
votes
1
answer
240
views
Locating boundary layers for pertubation problem
Consider the BVP:
$\epsilon \dfrac{d^2y}{dx^2}-(x^2-2)y=-1
\\ \text{where} -1<x<1 \;\text{and} \; y(-1)=y(1)=0, \; 0<\epsilon<<1$
I am trying to show the existence of a boundary ...
- 293