All Questions
Tagged with perturbation-theory roots
20
questions
1
vote
1
answer
41
views
Roots of an equation (perturbation theory)
Consider $xe^{x-1}+x-2-\epsilon=0$
Assume the solution can be expanded in terms of $\epsilon$, i.e. $x=a_{0}+a_{1}\epsilon+a_2\epsilon^2+... (1)$
Substitute (1) into the equation, we have
$(a_{0}+a_{1}...
-1
votes
1
answer
77
views
Singular perturbation theory: largest root of $f(y)=\epsilon y (y^2-1)^2-(2y^2-1)$ as $\epsilon\to0$
Recently, I have received questions regarding the small $\epsilon$ expansion for the largest real root of
$$\tag{1}
f(y)=\epsilon y (y^2-1)^2-(2y^2-1)
$$
I am posting this self-answered question as a ...
- 4,817
1
vote
0
answers
88
views
Asymptotic behavior of the zeros of a polynomials for large values of a parameter
Consider a polynomial in $r$ of the form
$$
r^4+p_3(\lambda)r^3+p_2(\lambda)r^2+p_1(\lambda)r+p_0(\lambda),
$$
where the $p_i$ are polynomials in the parameter $\lambda$. I use degree four to simplify ...
- 1,139
2
votes
3
answers
296
views
Asymptotic expansion, solving roots to an equation with dominant balance, what went wrong in my approach?
So I wanted to compute the asymptotic expansion of the roots to, as $\epsilon \to 0$,
$$\epsilon x^3-x^2+2x-1=0$$
Now when I tried to find $x\sim x_0+\epsilon x_1+\epsilon^2x_2+...$ I ran into trouble ...
- 3,886
1
vote
1
answer
99
views
zero's of the " T-zeta function " ? ( $ \zeta_t(s) = \prod_{p = 2}^{\infty} \frac{p^s}{p^s - t}$)
In an old diary I read this :
Consider the following functions :
Let $s,t,v$ be complex.
V-zeta function :
$$ \zeta_v(s) = \sum_{n = 1}^{\infty} \frac{1}{n^s + v} $$
T-zeta function :
$$ \zeta_t(s) = \...
- 16.4k
2
votes
1
answer
982
views
Perturbation to the coefficients of a polynomial
I am reading 'Trefethen and Bau: Numerical Linear
Algebra' book and I came across the following problem- Assume that the polynomial $p(x) = \sum_{k=0}^n a_k x^k$ with real coefficients has $n$ ...
- 81
1
vote
2
answers
288
views
How to find roots by perturbation methods for this problem?
I have trouble to find roots for the following problem:
$$\epsilon^{-1}x^3=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$
I think this is a singular perturbation problem for $\epsilon$ showing up in the highest ...
- 683
1
vote
1
answer
70
views
Verifying a computation in a basic asymptotic expansion
Just want to check that I'm not going crazy because I don't have someone to work with on this. For the perturbed quadratic
$$ x^2 - \epsilon x - 1 = 0 \hspace{5em} (*) $$
I want to insert the ansatz
$$...
- 35.1k
5
votes
2
answers
267
views
Perturbation series for $x^5+\varepsilon x-1=0$
I want to find a closed form for the perturbation coefficients $a_n$ defined by the perturbative solution
$$
x(\varepsilon)=1+\sum_{n=1}^\infty a_n \varepsilon^n
$$
to the quintic equation
$$
x^5+\...
- 6,753
11
votes
2
answers
527
views
Perturbative solution to $x^3+x-1=0$
I would like to calculate the real solution of
$$
x^3+x-1=0
$$
by resumming a perturbation series. To this end, I considered
$$
x^3+\epsilon x-1=0,
$$
$\epsilon$ being a perturbation parameter.
The ...
- 6,753
1
vote
0
answers
342
views
Two-term asymptotic approximation for roots of an equation
Find a two-term asymptotic expansion, for $\varepsilon \to 0$, of each solution $x$ of
\begin{equation}
x^{2+\varepsilon} = \frac{1}{x+2\varepsilon},\qquad (x > 0)
\end{equation}
My approach:
First ...
- 31
2
votes
1
answer
110
views
Why is $\varepsilon x^5 \sim -x$?
I'm trying to understand what's going on in this lecture on perturbation (the link brings you to 1h 08m 12s).
The original problem is to find the real root of $$x^5+x=1.$$ We have inserted $\...
- 7,720
1
vote
2
answers
217
views
Roots of a perturbed equation
I'm looking to show that the equation $$\displaystyle \psi(\delta) := e^{\alpha g(\delta)} - \delta$$ has a real root for $\alpha$ sufficiently small that converges to $\delta = 1$ as $\alpha \...
- 13
1
vote
0
answers
60
views
Please check this perturbation solution of polynomial root and truncation order.
I have a quintic polynomial where the coefficients depends on a parameter $c$, i.e.
$$
a_0(c)+a_1(c)x+a_2(c)x^2+a_3(c)x^3+a_4(c)x^4+x^5
$$
I know that the roots of the polynomial are real and non-...
- 402
9
votes
2
answers
6k
views
Method of dominant balance and perturbation
Approximate the solutions of $$\epsilon x^4 + (x-1)^3=0$$
I can't perform a singular perturbation because if I let $\epsilon=0$ then I lose a root. My professor suggests The Method of Dominant ...
- 3,234