All Questions
Tagged with perturbation-theory approximation
25
questions
0
votes
1
answer
49
views
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
1
vote
0
answers
55
views
Applying multi-scale analysis directly on the exact solution
Usually, multiple scales analysis (e.g. Poincaré-Lindstedt method or other multi-scale expansions) is applied on an ODE. Suppose we start from the exact solution of the ODE, how do we obtain the ...
- 121
19
votes
3
answers
800
views
Possible analytical solution to $g''(x)=\alpha\left[g(x)^3-g(x)\right]+\beta g(x) e^{-\kappa x}$
kind of related to a previous question of mine. I am describing a physical phenomena related to charged molecules and am interested in the following quantity:
$$\xi=\int_0^{\infty}\left[1-g(x)^2\right]...
- 309
1
vote
0
answers
21
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Large $t$ solution to $\partial_t S(x,v,t) = v \partial_x S + (a+v) \partial_v S + \partial_v^2 S$
I have the partial differential equation
$$\partial_t S(x,v,t) = v \partial_x S - (v + a ) \partial_v S + \partial_v^2 S$$
subject to the following boundary conditions:
$$S(x,v,0)=1,$$
$$S(0+,|v|,t)=...
- 1,454
1
vote
1
answer
152
views
Bound eigenvalues $A$ and $B$ with 2-norm $\|A-B\|$
Suppose $\| A-B\|_2 \leq\epsilon$, can we bound the difference in eigenvalues of $A$ and $B$ as well? For instance, would it hold that $|\lambda(A)-\lambda(B)|\leq\epsilon$ as well? Or alternatively, ...
- 113
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 $$n=...
- 105
0
votes
0
answers
47
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Perturbative solution of $\frac{d^2\varphi}{d\xi^2}=\sin \varphi$
I'm trying to solve
$\frac{d^2\varphi}{ds^2}=\lambda^{2}\sin \varphi$ using perturbation methods which in non dimensional form can be written as $\frac{d^2\varphi}{d\xi^2}=\sin \varphi$.
With ...
0
votes
0
answers
56
views
Approximating a function which has small derivatives
Suppose I have some function $f(t) = f(a(t),b(t),c(t))$ which I want to evaluate at all times $t$.
Now, I know that in general function $c(t)$ varies very quickly ($\dot{c}(t)$ is large) whilst the ...
- 508
1
vote
1
answer
47
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What is the asymptotic solution to the roots of $x^n+a_1 \epsilon x^{n-1} +\cdots+ a_{n-1}\epsilon^{n-1}x+a_n \epsilon^n$?
The polynomial I'm working with is:
$$
\lambda^n
+\frac{\epsilon}{p \ 1!}\lambda^{n-1}
+\frac{\epsilon^2}{p^2 2!}\lambda^{n-1}
+\dots
+\frac{\epsilon^{n-1}}{p^{n-1} (n-1)!}\lambda
+\frac{\epsilon^{n}}...
1
vote
1
answer
52
views
Approximating $z^2(1-(\varepsilon z) + (\varepsilon z)^2) +1 = 0 $ solutions when $\varepsilon \rightarrow 0$
Given the perturbation algebraic problem $z^2(1-(\varepsilon z) + (\varepsilon z)^2) +1 = 0 $ solutions when $\varepsilon \rightarrow 0$ , I wish to evaluate the first order terms of the 4 roots.
...
- 1,390
1
vote
2
answers
176
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Finding leading order approximation to $\frac{d^2y}{dx^2}-\epsilon y=x$
The conditions provided were $y(0)=1$ and $y'(0)=1$.
Since this equation is regular, I can neglect the term involving $\epsilon$, giving $\frac{d^2y}{dx^2}=x$.
I tried to solve this in order to ...
- 171
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
0
votes
1
answer
937
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Perturbation theory and approximation - help! [duplicate]
I am having a bit of difficulties with expanding perturbation series. I need to find a first and second order perturbative approximation to the root of $1+(x^2+\epsilon)^{1/2}=e^x$.
I first tried the ...
- 1
2
votes
0
answers
247
views
Perturbation theory for almost degenerate matrices
I have a nearly degenerate matrix $A$ and a perturbation $\epsilon B$, where $\epsilon$ is a small positive number. I know the eigenvalues and eigenvectors of $A$, but as I said, some of the ...
- 6,771
7
votes
1
answer
884
views
Perturbing a polynomial with repeated real roots to get distinct real roots
Consider a real polynomial $f$ of degree $d$ which has $d$ real roots not necessarily distinct. In general, can we accomplish the following?
For every $\epsilon>0$, can we perturb each ...
- 215