All Questions
Tagged with perturbation-theory analysis
16
questions
1
vote
2
answers
110
views
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
votes
0
answers
88
views
Help to solve an ODE asymptotically
Consider the following ode for $x\rightarrow\infty$
$$\left(x f(x)^3 \left(\frac{(x f^\prime)^\prime}{x}\right)^\prime\right)^\prime=0$$
Assume the function $f(x)>0$ with $x>0$ and introduce an ...
- 73
2
votes
1
answer
509
views
Eigenvalue perturbation of hermitian matrices
For $\rho$ and $A$ hermitian matrices, $t$ a scalar parameter, $f_i(\rho)$ denoting the $i-$th eigenvalue of the matrix $\rho$ and $|x_i\rangle$ the normalized eigenvector of $\rho$ corresponding to $...
- 181
0
votes
1
answer
51
views
Averaging method (parameter variation) [closed]
Use the averaging method (parameter variation) to determine a uniform first-order approximation for:
$$ y''+y+ \epsilon y'^5=0$$
for $\epsilon\ll1$.
1
vote
1
answer
81
views
Asymptotic expansion in a polinomial
The problem is about Perturbation Theory
Find the first-order approximation for the roots of:
$$P^\epsilon(x)=\epsilon^2x^6-\epsilon x^4-x^3+8=0 $$ Here
you no longer know the exact solution by ...
2
votes
1
answer
59
views
Endpoint Perturbation Theory
So, suppose we want to evaluate the integral
$$\int_{a}^{b+\epsilon c}f(x)\, dx$$
where $f:\mathbb{R}\to\mathbb{R}$ is assumed to be smooth and regular in the integration region $[a,b+\epsilon c]\...
- 334
0
votes
0
answers
60
views
Order symbols: $f=O(g)$ then $f=o(g)$
I have a problem with an exercise from M. Holmes "Introduction to perturbation methods"
I'll write the definitions.
Def: Let $f=f(\epsilon),\quad g=g(\epsilon)$. We say $f$ is a "big Oh" of $g$ and ...
- 415
3
votes
2
answers
399
views
Picking the correct Ansatz for valid solutions in Asymptotic Methods
I am trying to find the solution to the following equation,
$\epsilon x^3 -x^2 +x-\epsilon^{\frac{1}{2}}=0$, for the first two non-zero solutions as $\epsilon \to 0^+$.
I have used the principal of ...
- 336
3
votes
0
answers
46
views
Perturbation evolution of a differential equation
Let $a$, $b$ be two real positive parameters with $a>b$, and consider the following nonlinear differential equation:
\begin{align}
\dot{x}_{\varepsilon}(t) = a - b\sin(x_{\varepsilon}(t))+\...
- 2,259
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
1
vote
0
answers
130
views
Perturbation theory - Differential equation
I'm having issues understanding/solving this question I got for an exam:
Consider this nonlinear differential equation (1): $x''(t) + \cos(x(t))\cdot x(t) = \sin(t)$
Suppose $x$ is small and make ...
- 434
0
votes
1
answer
183
views
approximate the solutions to a second order ordinary differential equation
The exercise is:
Approximate the solutions of $x'' + x + ε*x^3 = 0$, the initial value is $x(0) = 1$ and $x'(0) = 0$ up to order one. (Hint: It is not necessary to convert this second order equation ...
- 1
0
votes
1
answer
29
views
If $\lambda = e^{i\mu}$ and $\mu \simeq 0$, can we have $\lambda^{k} = 1$ for some $k <5$?
Suppose I have $\lambda = e^{i\mu}$, where $\mu = \mu(\epsilon)$ with $\epsilon$ a small parameter. Suppose $\mu$ can be expanded in series in $\epsilon$ as $\mu = \mu_{0} + \epsilon \mu_{1} + O(\...
- 1,560
0
votes
1
answer
47
views
Construct a single-valued function increasing arbitrarily quickly at a point $x=x_0$?
Title says it all.
(How) can we construct a function (a regular function, perhaps?) whose first derivative is arbitrarily large at a point $x = x_0 < \infty$?
- 562
2
votes
1
answer
228
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Extending solutions of an ODE past a singular point
In the course of my studies, I'm looking at at the ODE:
\begin{equation}
(f^3(x))'''=\frac{1}{6}xf(x),\quad f(0)=1,\,\,f'(0)=0
\end{equation}
Where $f''(0)$ is a parameter left undetermined. In ...
- 800