All Questions
Tagged with perturbation-theory numerical-methods
30
questions
0
votes
2
answers
45
views
How to approach this singular perturbation problem?
I have set myself the following singular perturbation problem:
For small values of $\varepsilon > 0$ find the two roots closest to $x=0$ for the equation.
$${x^4} - \,\,{x^2} + \,\,\varepsilon (x +...
- 147
1
vote
2
answers
110
views
Method to solve this ODE $x^{(6)}+2Ax^{(4)}+A^2x^{(2)}+B^2x = 0$
I have to solve this ODE: $$x^{(6)}+2Ax^{(4)}+A^2x^{(2)}+B^2x = 0$$
where the upper index in brackets () indicates the order of the time derivative, $A = 4(m^2-2eHs_z)$ and $B= 4meH$, both are ...
- 95
1
vote
0
answers
22
views
How to Nicely (Ideally Uniformly) Approximate Inhomogeneous System of Two ODEs
I have a nonlinear system of ODEs of the form:
$$
\dot{k} = k(t)^α-gk(t)-c(t)\\
\dot{c} = \frac{\alpha k(t)^{\alpha-1}-g-\rho + (1-\gamma)\frac{g}{A_0+gt}}{\gamma}c(t)
$$
with associated initial ...
- 11
6
votes
2
answers
318
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Relative Condition Number of Atan(x) vs Atan2(y,x)
I'm trying to think through the sensitivity of atan2 to errors in its inputs and I've run into a disconnect that I don't quite understand.
I know that you can compute the relative condition number of ...
- 583
2
votes
1
answer
38
views
Bound $\frac{\lVert x-y \rVert}{\lVert x \rVert}$ with perturbation
Firstly, I want to say I could not find a proper title sorry. I have a question.
Let $Ax =b$ and $(A+\Delta A)y = b + \Delta b $.
Suppose $\lVert \Delta A \rVert \leq \epsilon \lVert E \rVert, \lVert ...
- 121
0
votes
1
answer
430
views
A couple of questions about implementing perturbation for the Mandelbrot set
I followed this thread on perturbation of the Mandelbrot set iterations:
Perturbation of Mandelbrot set fractal
I was wondering what accuracy these different variables need to be calculated to (high ...
- 103
0
votes
1
answer
89
views
What is the solution of $\cosh(x)=(1-\epsilon) x \quad (\epsilon >0)$
In a program of mine, I am solving numerically in the real domain (with limited precision) the equation
$$\cosh(x)=a\,x$$ which does not present any difficulty. The problem is that $a$ is a function ...
- 268k
0
votes
1
answer
101
views
Solving IVP exactly with an epsilon variable
I am unsure of how to interpret the question:
Given $y''+(1+\epsilon)y=0, y(0)=0,y'(0)=1$. Solve exactly.
The context of the problem is that we are practising solving IVP with regular perturbation ...
- 597
1
vote
0
answers
34
views
How to evaluate the accuray of quadratic eigenvalue problem (QEP)?
When solving the QEP, we transform it into a GEP and then use qz algorithm to handle it.
But there are several formulations of GEP, how to evaluate the accuracy and stability of the solution?
I ...
- 79
0
votes
0
answers
73
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Definition of perturbation using norms
Can someone give me the definition of a perturbation vector? I know that it is a vector composed of $x_i$ that have a small value and are independent, but I know I can define it by using norms because ...
0
votes
2
answers
63
views
Using Regular Perturbation, obtain approximate solution:
Consider the following ODE:
$y'+ y = \epsilon y^2, y(0,\epsilon)=1$
Assuming the perturbation amsats gives:
$y'_0 + y_0 =0, y_0(0)=1,$
$y'_1 + y_1 = y^2_0$, $y_1(0)=0$
How is: $y_0(x)=e^{(-x)}$ ...
- 41
1
vote
1
answer
48
views
Solve by means of regular perturbation to obtain an approximate solution up to and including $\mathcal{O}(\epsilon^2)$
I have to solve the following ODEs:
$y''+ y = e^{\epsilon \cos x}$, $y(0,\epsilon)=y(1,\epsilon)=0$;
$y''+ y = \epsilon y^2$, $y(0,\epsilon)=1$, $y'(0,\epsilon)=0$.
I am having trouble ...
- 41
1
vote
1
answer
248
views
WKB Approximation to Schrödinger Equation
Consider the Schrödinger equation: $$y''(x)+EQ(x)y(x)=0,\tag{1}$$
where $E>0, Q(x)>0, y(0)=y(\pi)=0. $
Use WKB approximation to obtain $$y(x) \sim CQ^\frac{-1}{4}(x)\sin\Big(\sqrt(E) \int_{0}^{...
- 137
3
votes
2
answers
399
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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
0
votes
0
answers
57
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How does the following expression scale with respect to $\epsilon$?
Let $\mathbf{V}$ be a $2N \times 2N$ positive definite matrix.
Let $\mathbf{V}^\star=\mathbf{V}+E$ where $||E||_\infty\leq\epsilon$ and $|\epsilon|<<1$. How does the following expression scale ...
- 1