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Domain perturbation problem on a ring?

Find the electric potential $\phi$, satisfying $\nabla^2 \phi=0$ between the two cylinders $r=a$, on which $\phi=0$, and $r=b>a$, on which $\phi=V$. Suppose that the inner cylinder is perturbed ...
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33 views

Question related to the ballistic motion

A point mass will move in the gravitational field of the Earth according to the equation $$\ddot R =-\frac{GM_eR}{|R|^3},$$ where $R$ is the position vector of the point mass measured from the ...
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0answers
29 views

Asymptotic expansion of root of $\epsilon x \tan(x)=1$

Indicate a range of roots of $\epsilon x \tan(x)=1$ for which it is impossible to get an approximation using expansions. Since $\epsilon$ is small, I think for the equation to hold, we need ...
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2answers
37 views

A further question on asymptotic expansions of all real roots of xtan(x)=ϵ

I have asked a related question here How to find asymptotic expansions of all real roots of $x \tan(x)=\epsilon?$, however, when I discussed with my adviser today, he argued the solution is flawed. ...
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1answer
17 views

Difference between a convergent series and an asymptotic series?

Can someone let me know the difference between a convergent series and an asymptotic series with an example? Can both the series be the same at some situations? In what situations an asymptotic series ...
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2answers
63 views

How to find asymptotic expansions of all real roots of $x \tan(x)/\epsilon=1?$

Find expansions of all the real roots of $$x\tan(x)=\epsilon?$$ (You have to consider the first root separately) It is really bothering me. If I assume $x=x_0+x_1\epsilon +x_2\epsilon^2$ and do ...
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2answers
41 views

Asymptotic expansion of exp of exp

I am having difficulties trying to find the asymptotic expansion of $I(\lambda)=\int^{\infty}_{1}\frac{1}{x^{2}}\exp(-\lambda\exp(-x))\mathrm{d}x$ as $\lambda\rightarrow\infty$ up to terms of order ...
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1answer
31 views

Converting a cubic to a perturbation problem

I'm trying to learn about Perturbation, but feel like I'm confused before I've even started. Right now I'm focused on using them to find solutions to polynomial equations. The initial example I've ...
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28 views

Growth factor problem

I am trying without success to understand how two formulae in appendix B of this paper are derived. Equation B1 is an equation for perturbations, obtained from regular perturbation theory: ...
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0answers
7 views

Equivalent definition of the Kreiss constant

Let $A$ be an $n\times n$ matrix. The $\epsilon$-pseudospectrum of $A$ is defined to be $\{z\in\mathbb{C}:\|(zI-A)^{-1}\|\geq\frac{1}{\epsilon}\}$, where the norm considered is the operator norm ...
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1answer
31 views

Locating boundary layers for pertubation problem

Consider the BVP: $\epsilon \dfrac{d^2y}{dx^2}-(x^2-2)y=-1 \\ \text{where} -1<x<1 \;\text{and} \; y(-1)=y(1)=0, \; 0<\epsilon<<1$ I am trying to show the existence of a boundary ...
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2answers
54 views

How to bound error when approximating ODE

I have a question regarding how to bound the error, if one changes the "right hand side" of an ODE. For example, the equation of a simple pendulum in polar coordinates is something like ...
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1answer
26 views

How to prove Kahan's example on componentwise pertubation theory?

In Matrix Computations (4th edition) by Gene H. Golub and Charles F. Van Loan, Problem 3.5.3 asks the following problem (and citing Kahan, William. "Numerical linear algebra." Canadian Math. Bulletin ...
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0answers
24 views

Bloch front solutions for Complex Ginzburg Landau equation

I am trying to figure out how to get an approximation of the form $$B(z) = B_0 (z) + \epsilon B_1 (z), \;\; \text{where } z = x - ct $$ for the temporally forced Complex Ginzburg Landau equation ...
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1answer
43 views

why does this linear differential equation does not gives correct equilibrium in one limit

I have a linear set of equation, $\frac{dx(t)}{dt}= 4 \frac{1-a^{-1}}{a} y(t) - 8 (1-a^{-1}) x(t)$ and $\frac{dy(t)}{dt} = b x(t) - \frac{b}{a} y(t)$ with initial conditions $x(0)=x_0$ and ...
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1answer
81 views

Reference: Continuity of Eigenvectors

I am looking for an appropriate reference for the following fact. For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric matrix), there exist $\varepsilon, L > 0$, such that for ...
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0answers
14 views

Non-linear perturbation definition

What exactly is the definition of a nonlinear perturbation when applied to a background spacetime metric? I have seen so called "linear perturbations" which look like $$ds^2 = -(1+2\Phi)dt^2 ...
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0answers
108 views

Classical perturbation theory + KAM theory

In classical canonical perturbation theory of many degrees of freedom we encounter the problem of small divisors when attempting to find a solution for the generating function of the canonical ...
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0answers
15 views

Does the magnus convergence test not hold for the factorization of second order differential operators?

Given the operator \begin{align} H = V(x)-\partial_x^2 \end{align} and given an eigenfunction $\phi_0(x)$ such that $H\phi_0=0$ with a zero eigenvalue, I can factor $H$ into \begin{align} H = h_+h_- ...
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1answer
33 views

How to use the Magnus Series Convergence Test for complex matrix?

I have a two by two functional complex matrix $A$ belonging to the Magnus differential equation \begin{align} Y'(x)=A(x)Y(x) \end{align} I read from wiki that a convergence test for real $A$ can be ...
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0answers
40 views

Singular value perturbation vs. eigenvalue perturbation

Suppose $C = B + \mathrm{i} c AA^\dagger$, where $^\dagger$ denotes the conjugate-transpose of a matrix. Here $c > 0$ is a free parameter, $B$ is hermitian and $AA^\dagger$ is obviously positive ...
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1answer
62 views

Perturbation, straightforward expansion

Consider the equation: $$\ddot{u} + \frac{\omega_0^2u}{1+u^2} = 0$$I want to determine the straightforward expansion for small but finite $u$. what form should the expansion take? Normally the ...
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1answer
119 views

Eigenvalues of symmetric matrix with skew-symmetric matrix perturbation

If $A$ is diagonalizable, using the Bauer-Fike theorem, for any eigenvalue $λ$ of $A$, there exists an eigenvalue $μ$ of $A+E$ such that $|\lambda-\mu|\leq\|E\|_2$ (the vector induced norm). Here I ...
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1answer
112 views

Perturbation theory of the eigenvalues about the symmetric matirx

From Weyl's theorem, i.e.: Let $A$ and $E$ be $n\times n$ real symmetric matrices. Let $\alpha_1\geq\ldots\geq\alpha_n$ be the eigenvalues of $A$ and $\hat{\alpha}_1\geq\ldots\geq\hat{\alpha}_n$ be ...
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11 views

Model of a Swinging SkyScraper: Dimensions, Characteristic Scales, and Non-dimensionalization

$d^2y$$/$$dt^2$ $+$ $ay$ $-$ $by^3$ $=0$, $y(0)$ = $y_0$, $dy$$/$$dt$$(0)$ $=$ $0$ where $y$ is the horizontal displacement of the top of the building, and $a$ and $b$ are positive constants. The ...
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0answers
22 views

Using the WKB approximation to find the values of different positive Eigenvalues $E_n$

Consider $$y''(x)+EQ(x)y=0, Q(x)>0 \mbox{ subject to } y(0)=y(\pi)=0$$ The WKB approximation is (which i've proved) is: $$y(x) = CQ^{-0.25}(x)\sin{(\sqrt{E}\int_0^x\sqrt{Q(t)}dt)}$$ Then the ...
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1answer
35 views

Perturbative Solution to Boundary-layers Problem

Could anyone help me with the part marked in red? Why do we have a series expansion in there? And why does the limit epsilon -> 0 gives eq_in = Y''(w)+Y'(w)=0
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0answers
23 views

Perturbation Theory - WKB Approximation

I'm not familiar with the big-O notation, so I have trouble understanding the part marked in red in the pictures below. Could anyone help? Thanks.
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1answer
57 views

Rayleigh-Bénard convection

I have a nondimensionalized linear perturbation system relevant to the appropriate pure conduction solution for Rayleigh-Bénard convection in upper planetary atmospheres under the compressible gas ...
1
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1answer
23 views

estimating the roots of $ \epsilon z^n + p(z)$

I have a polynomial $p(z)$ of degree $n-1$ with known roots $z_1, \dots, z_{n-1}$. How I add the monomial term $a z^n$. What are the roots of $$ p_1(z) = p(z) + \epsilon z^n $$ In terms of the ...
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0answers
123 views

analytical solution to nonlinear ODE

I have following set of equations, and wants to do a perturbation theory to find approximate solution of my dynamical equations. $ \frac{dx}{dt} = -\beta x y + 2 \alpha y - 2 \gamma x^2 \\ ...
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1answer
19 views

Perturbative solution to an initial-value problem

Could anyone help me with the part marked in red? I have trouble understanding the reasoning behind the technique used, and also the steps to go from (7.1.8) and (7.1.9) to (7.1.10). Thank you!
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3answers
43 views

Why do the coefficients of a equation, expressed in terms of a small parameter epsilon, have to be 0?

I have trouble understanding this very elementary example of perturbation theory, especially the part marked in red ("It is because epsilon is variable that we can conclude that the coefficient of ...
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1answer
55 views

Trying to show $\int_0^1 e^{-xt}sin(t) dt \sim \frac{1}{x^2}$

I am using Laplace's Method and I am trying to show $$I =\int_0^1 e^{-xt}sin(t) dt \sim \frac{1}{x^2}$$ $h(t) = -t$ has a maximum at $0$ and as it is a simple function there is no need to expand it. ...
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0answers
20 views

Does it make sense to solve polynimals using pertubation theory?

I was curisious to see if pertubation theory could be used to solve polynimals. For example \begin{align} x^5+b(\epsilon x)+1=0 \end{align} I started with the following expansion \begin{align} x = ...
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0answers
14 views

Applying perturbed matrix to unperturbed eigenvector

Suppose we've got a matrix $P$ and a perturbed version $\hat{P}=P+E.$ Given that $v$ is an eigenvector of $P$ with $Pv=0,$ I'd like to get as sharp a bound as possible on $\hat{P}v$ (in terms of ...
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0answers
29 views

A basic analysis/O.D.E/perturbation theory question

Consider a system of equations $$x'=f(x,y,\epsilon)$$ $$y'=\epsilon g(x,y,\epsilon)$$ I have seen in the book to claim the following: As $\epsilon -> 0$ the limit is $$x'=f(x,y,0)$$ $$y'=0$$ I ...
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1answer
98 views

Find the leading order uniform approximation when the conditions are not $0<x<1$

$$\epsilon y''+y'\sin x+y\sin 2x = 0$$ with boundary conditions $y(0)=\pi$ and $y(\pi)=0$ as $\epsilon \rightarrow 0$. I don't know how to find out where the boundary layer is? I thought initially it ...
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1answer
50 views

Obtain the leading order uniform approximation of the solution to: $\epsilon y'' +(1+x)^2y'+y=0$?

Obtain the leading order uniform approximation of the solution to: $\epsilon y'' +(1+x)^2y'+y=0, y(0)=0 y(1)=1$ as $\epsilon \rightarrow 0$. I am completely lost. Am i right in doing this? Since ...
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0answers
26 views

Boundary Layer Theory

We consider the differential equation $$ (x-\epsilon y) y' + xy = e^{-x},~y(1) = 1/e $$ this is from an example in a book, and I am trying to make sense of the explanation. As the convention is for ...
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1answer
88 views

How does perturbation method guarantee its solution for the perturbed pde $\Delta u + \epsilon u^2 =0$

My question is quite simple: Suppose we are given a PDE of with a boundary condition $$ \Delta u + u^2 =0 $$ where $u=u(r,\theta), 0<r<1$ and $u(1,\theta) = \cos\theta$ with $0 \leq \theta \leq ...
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32 views

perturbation of trace norm

The definition of trace norm is the summation of singularities of that matrix. I need to calculate the trace norm of matrix with the form $$A = I + r$$ where the ...
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0answers
20 views

Where to learn perturbation theory for pde (in introductory level)? [Reference Request]

Recently I've been reading the text by Falow 'PDEs for Scientist and Enginieers'. In the latter sections is contained 'Perturbation method'. This one gives only kind of computational techniques; no ...
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0answers
9 views

Perturbation theory for eigenvectors?

It is known from the Hoffman and Wielandt Theorem that if $A,B$ are normal matricies and $\lambda_1,\dots,\lambda_n$ are the eigenvalues of $A$ and $\lambda_1',\dots,\lambda_n'$ are the eigenvalues of ...
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3answers
33 views

For small $z, (1 + z)^{−2} \sim 1 − 2z$…

I came across the following statement while reading Holmes book on Perturbation Methods - To reduce the differential equation, recall that, for small $z, (1 + z)^{−2} \sim 1 − 2z$ I don't know ...
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0answers
35 views

Bound on Signal Amplitude for subspace methods (MUSIC, ESPRIT)

MUSIC and ESPRIT are methods that use subspace decomposition to identify signal Parameters. Subspace decomposition is achieved either by SVD or Eigen Value Decomposition. Subspace decomposition ...
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36 views

Asymptotic parameter for a transcendental equation

I need to find the roots of the following equation $(x^2-a^2)(x^2+a^2)\sin(b^2-x^2)-b^2 \cos(\sqrt{b^2-x^2})=0$. Say $\mathcal{A}=(x^2-a^2)(x^2+a^2)$. I assume that as long as $x$ is away from its ...
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0answers
40 views

What is the difference between perturbation theory and numerical analysis?

What is the difference between perturbation theory and numerical analysis? Both subjects are trying to obtain the approximate answer. What are they study specifically?
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1answer
23 views

Series expansions and perturbation

My professor said that $ f \left( y_1(x)+ \epsilon y_2(x)+... \right)= f(y_1(x)) +f'(y_1(x))\> (\epsilon y_2(x)+...) + ...$ but I have no idea how the series continues. Has anyone seen this ...
3
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2answers
237 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 ...