All Questions
Tagged with perturbation-theory calculus
33
questions
1
vote
0
answers
43
views
Degree with which a polynomial changes with some small change
Soft question: I was curious as to how one could measure the degree with which a polynomial is perturbed. More formally, let $P(x) \in \mathbb{C}$ be a polynomial and $\epsilon$ be a very small number,...
- 166
0
votes
1
answer
145
views
How to linearize a max function about a small change??
My question is the following:
Given a vector field $\mathbf{v}$, we have the following functional: $f\{\mathbf{v}\}=1/|\mathbf{v}|_{\text{max}}$, where $|\mathbf{v}|_\text{max}$ is the maximum of the ...
1
vote
1
answer
273
views
Perturbation solution in a Bessel equation
I have the following ODE for a function $y(x)$, over the real variable $x$, with $m>0$ an integer and $\epsilon$ a small parameter
$$ y'' + \frac{1}{x}y' + \frac{1}{x^{2}} y (\epsilon^{2} x^{2} -m^{...
- 1,309
1
vote
0
answers
38
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Fractional form that approximately defines exponential function
How can we prove that, in the limit of $|1/y| \leq |c|$, where $c,y$ are generally complex, the following approximation holds?
$$ \frac{1-yc}{1+yc} \rightarrow e^{-2/cy} $$
My attempt: if we call $...
- 1,309
0
votes
2
answers
68
views
Approximating the inverse of an exponent equation
Let $$m=a_{1}n^{\alpha}+a_{2}n^{\beta}$$ where $1>\alpha>\beta>0$ and $a_{1},a_{2}$ are positive constants, and we want to understand $n$ as a function of $m$ , the first order is clearly $$n=...
- 105
1
vote
1
answer
39
views
Placing perturbation series in a function problem
I have trouble under standing how to put perturbation series in a function.
In "eq-a" I would have assume $f(x_0) + \epsilon f'(x_0) + \epsilon^2 f''(x_0)$
In "eq-a"..how the term $ (\epsilon x_1 ...
- 129
3
votes
1
answer
285
views
Solution of non-homogeneous ODE is always bounded
I've been struggling with the following question:
$\ddot x + 2 \dot x +(1+e^{-t})x=t^{2}cos(t)$, show that all solution are bounded for $t>0$.
My problem: Im probably missing some theorem or ...
- 923
2
votes
1
answer
527
views
Meaning of power series expansion in perturbation theory
I encountered a power series expansion of $x$ in $\epsilon$ when solving for the general solution to Mathieu's equation in this paper.
$x(\xi, \eta) = x_0(\xi, \eta) + \epsilon x_1(\xi, \eta) + \...
- 469
2
votes
2
answers
2k
views
Find the two term asymptotic expansion of the solution
Find the two term asymptotic expansion of the solution of
$$
1 +\sqrt{x^2 + \epsilon} = e^x
$$
My approach: i tried solutions of the form $x = x_0+\epsilon ^ \alpha x_1$ and plug it in. Then using ...
- 79
2
votes
2
answers
69
views
Undetermined coefficients in a perturbative expansion
In order to familiarize myself with perturbation methods, I've been trying to derive the Lorentz transformations, given by
\begin{align*}
x \rightarrow \frac{x + vt}{\sqrt{1 - v^2}} & = (x + vt)(...
- 223
1
vote
1
answer
2k
views
Find a two term asymptotic expansion of the following problem
I want to find a two term asymptotic expansion, for small $\epsilon$, of the solution of the following problem:
$$
y'' - \epsilon y' - y = 1\tag{1}, \, y(0) = 0, \,y(1) = 1
$$
My approach: I assume ...
- 529
1
vote
0
answers
187
views
Regular Perturbation to obtain solution to: $y''+y= \epsilon y^2$
The following perturbation problem:
$y''+ y = \epsilon y^2, y(0,\epsilon)=1, y'(0,\epsilon)=0$
So far I have deduced that:
$y_0''+y_0=0, y_0(0)=1, y_0'(0)=0$, and so $y_0(x)= cos(x)$
For the second ...
- 41
1
vote
0
answers
57
views
Rules of linear perturbation theory
I am following a physics paper that uses linear perturbation theory. They take steps that imply things such as
$$\delta\left( \frac{1}{y} \right) = -\frac{\delta y}{y^2}$$
where $\delta$ is a small ...
- 1,552
0
votes
1
answer
553
views
Limit of a function ("order of magnitude")
Find $f(x)$ (in terms of exponentials) such that $$\lim_{x \to 0}\frac{e^{-\cosh\frac{1}{x}}}{f(x)}=A$$ where $A\in\mathbb{R}, A\neq0$
I have tried calculating Maclaurin series but I get zeroes ...
- 553
0
votes
1
answer
389
views
Perturbation solution of two coupled diff equations
I'm having a hard time trying to solve this coupled pair of differential equations by the perturbation method.
These are the equations:
Where Br you can treat as ε (base solution).
The solution ...
