All Questions
Tagged with perturbation-theory ordinary-differential-equations
246
questions
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Degenerate perturbation theory to nonlinear equation
I want to use perturbation theory to find the solution to the following nonlinear equation:
$$x_i\left(\sum_{j=1}^Nx_j^2\right)-a x_i + \epsilon \sum_{j\neq i}^N J_{ij}x_j=0,$$where $i=1\cdots N$ and $...
- 101
2
votes
1
answer
61
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Approximating solution to vector recurrence relation with element-wise exponential
$$
\mbox{Let}\ A \in \mathbb{R}^{n \times n}\ \mbox{and}\ \gamma_{1}, \ldots, \gamma_{n} \in \mathbb{R}\ \mbox{with}\ \gamma_{i} > 0,\ \forall\ i.
$$
$$
\mbox{Define}\ \operatorname{f}: \mathbb{R}^{...
- 33
4
votes
2
answers
155
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Periodic perturbation of ODE
Recently, I have read Khasminskii's book (Stochastic Stability of Differential Equations). In Section 3.5, the author mentioned that the following is well-known in the theory of ODEs.
If $x_0$ is an ...
- 351
2
votes
0
answers
52
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Leading order perturbation to the solution of a dynamical system
I was reading the paper 'A Proposal on Machine Learning via Dynamical Systems', where I came across the following steps:
Consider a system-
$$\frac{dz}{dt} = f(A(t),z),$$ with $z(0) = x.$
So, the ...
- 21
0
votes
1
answer
49
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Approximate solution to ODE potentially using perturbation theory
On the wikipedia article for stokes drift (https://en.wikipedia.org/wiki/Stokes_drift) they show that for:
$\dot{\zeta} = u\sin(k\zeta - wt)$ has approximate solution (by perturbation theory):
$\zeta(\...
- 1,923
0
votes
0
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69
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Finding eigenvalues of a turning point ODE using WKB method
Question: Using the WKB method, provide an approximation for the eigenvalue, $\lambda$, of the problem:
$$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.$$
Compute ...
- 3,965
1
vote
1
answer
96
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Two Timing (Multiple Time Scales) with Coupled IVPs
Question: Find the leading-order approximation for times of order $\epsilon^{-1}$ to $$\ddot{x} + x = y,~~\dot{y} = \epsilon(xy - 2y^2),~~x(0) = 1,~~\dot{x}(0) = 0,~~y(0)=1.$$
My approach: Let the ...
- 3,965
5
votes
1
answer
120
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Why is this approximate solution correct?
Consider the following differential equation
$$ y''=-y + \alpha y |y|^2, $$
where $y=y(x)$ is complex in general and $\alpha$ is a real constant such that the second term is small compared to $y$ ($||^...
- 331
1
vote
1
answer
144
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Applying WKB Method for a Fourth Order Schrodinger Like Equation
I was trying to apply the WKB method to analyze the approximate eigenvalue condition for the Schrodinger like equation
\begin{equation}
\varepsilon^{4} y^{(4)} = (E-V(x))y, \quad y(\pm \infty) = 0, V(...
- 13
1
vote
0
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79
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Find the general solution of an ODE with a nonlinear perturbative term
Let's say I start with the linear differential equation
$$ y''=-y, $$
which has (for example) the two solutions $y=e^{\pm i x}$, therefore following the superposition principle the general solution is ...
- 331
6
votes
0
answers
88
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Method of Dominant Balance with high order system
This question comes from Bender and Orszag's Asymptotic Methods and Perturbation Theory.
I'm practicing applying the method of dominant balance to study behavior as $x\to \infty$ for systems which ...
- 1,106
1
vote
2
answers
110
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Method to solve this ODE $x^{(6)}+2Ax^{(4)}+A^2x^{(2)}+B^2x = 0$
I have to solve this ODE: $$x^{(6)}+2Ax^{(4)}+A^2x^{(2)}+B^2x = 0$$
where the upper index in brackets () indicates the order of the time derivative, $A = 4(m^2-2eHs_z)$ and $B= 4meH$, both are ...
- 95
0
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0
answers
55
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How to do this perturbation expansion?
I got the following expansion in the context of studying $\phi^4 $ quantum field theory. This is the solution for exact 2-point propagator in the ladder-rainbow approximation. The expansion is -
$\mu^{...
- 1
0
votes
1
answer
123
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Perturbation theory with coupled nonlinear differential equations
I’ve got a problem with a set of differential equations for which I’m trying to find fixed points (or rather restrictions for the parameters).
The equations have the form
(1) $\frac{d}{dt} R(t) = -B ...
- 1
5
votes
2
answers
244
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Solution to a nonlinear ODE
$$ a_1 y'''''+ a_2 y'''+\left(a_3 + y^2 \right) y' = 0 $$
where $a_1, a_2, a_3$ are constants with $a_1>0$ and $a_2,a_3 \in \mathbb{R}$.
Is there a general solution $y(x)$ to the above differential ...
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