Perturbation theory is a tool for finding an approximate solution to a problem but starting from the exact solution of a related problem.

learn more… | top users | synonyms

2
votes
1answer
31 views

Building matrices from eigenvalues

I saw a question some time ago, asking about the eigenvalues of the matrix $$A=\begin{pmatrix}5&-3&0\\-3&5&0\\0&0&2\end{pmatrix}$$ which were then shown to be ...
0
votes
0answers
38 views

Derivation of perturbation series

I'm a little bit confused about the derivation of the perturbation series. I know from my quantum mechanics course that for a perturbed operator, eigenvalue is a series that is depend on the ...
0
votes
0answers
16 views

Multi time scales analysis on nonlinear system of ODEs

So I have this coupled set of nonlinear ODEs that I want to do a multi time scales perturbation analysis on. $ u'(t)+\frac{C \epsilon u(t)^2}{Cl}-\frac{2 \epsilon p(t)}{Cl}-\frac{2 q_1'(t)}{Cl}=0 ...
0
votes
0answers
29 views

Boundary layer problem

This question is taken from Bender & Orszag "perturbation methods" $y' = (1 + X^{-2}/100)y^2 - 2y + 1$ ,$y(1)=1$ first we can see that if we set $\epsilon=100x^{2}$ we can translate the above to ...
0
votes
0answers
41 views

Approximation of integral as integral range tends to 0

I would like to approximate $\int_{0}^{x}t^{-2}e^{t}\mathrm{d}t$ (maybe find the first two terms) as $x\rightarrow0$. I can't seem to do it by "divide and conquer" or any method. Any suggestion would ...
0
votes
2answers
27 views

Finding the roots and the rescaling of an equation

This question is taken from Hinch's book on perturbation. I need to find the rescalings $x=\delta X$ and the roots of the equation $\epsilon^2x^3+x^2+2x+\epsilon=0$ I have found to possible ...
0
votes
2answers
29 views

Solving Bernoulli equation transformation

I'm trying to solve the Bernoulli's equation via perturbation method but I need some help understanding how its done: We start off with $y'=-y+\epsilon y^2$ with $y(0)=1$. Then how is it possible ...
0
votes
0answers
47 views

Solve $y'' + \epsilon y' + 1 = 0$ with initial conditions $y(0) = 0$ and $y'(0) = 1$

Let $\epsilon << 1$. I guess I'm trying to use perturbation method but I've been getting really weird numbers when I'm determining the initial conditions. Can someone perhaps help me with ...
1
vote
0answers
21 views

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 ...
2
votes
0answers
44 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 ...
2
votes
0answers
47 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 ...
1
vote
2answers
51 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. ...
0
votes
1answer
18 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 ...
2
votes
2answers
84 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 ...
1
vote
2answers
48 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 ...
2
votes
1answer
36 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 ...
0
votes
0answers
29 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: ...
0
votes
0answers
8 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 ...
0
votes
1answer
37 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 ...
1
vote
2answers
56 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 ...
2
votes
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 ...
0
votes
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 ...
5
votes
1answer
100 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 ...
0
votes
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 ...
2
votes
0answers
119 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 ...
0
votes
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_- ...
1
vote
1answer
36 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 ...
1
vote
0answers
44 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 ...
1
vote
1answer
70 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 ...
6
votes
1answer
125 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 ...
4
votes
1answer
118 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 ...
0
votes
0answers
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 ...
0
votes
0answers
24 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 ...
0
votes
1answer
36 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
0
votes
0answers
31 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.
2
votes
1answer
62 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
vote
1answer
26 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 ...
0
votes
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 \\ ...
0
votes
1answer
22 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!
3
votes
3answers
46 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 ...
0
votes
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. ...
0
votes
0answers
21 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 = ...
0
votes
0answers
18 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 ...
0
votes
1answer
101 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 ...
1
vote
1answer
56 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 ...
0
votes
0answers
30 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 ...
4
votes
1answer
97 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 ...
1
vote
0answers
34 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 ...
1
vote
0answers
22 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 ...
0
votes
0answers
13 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 ...