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

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what happens when expansion parameter is of the order of dynamical variable itself?

Lets consider following differential equation, $\epsilon \frac{dy}{dt} = ....$ In principle one can use Method of matched asymptotic expansion or Method of multiple scales to solve such singular ...
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22 views

Multiscale analysis with non-integer exponents

I am dealing with the following non-linear differential equation: $$\frac{d^2 x}{d t^2}=2\varepsilon\frac{d x}{d t}-\left(\frac{d x}{d t}\right)^3-x$$ I found that $x=0$ is the only one fixed point ...
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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 ...
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22 views

asymptotic matched expansion with transiently blowing inner solution

I have been trying to solve following set of equations with method of matched asymptotic expansion, $\frac{dy(t)}{dt}=k z(t) - 3 \alpha y(t) - y(t)^2 + \mu (M-z(t))^2$ $\epsilon ...
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30 views

Perturbed density of eigen-states of a 3 diagonal matrix

How does the density of eigen-states ($D(\lambda)$ is defined as $D(\lambda) d\lambda$ = Number of states in the range $\lambda ... \lambda + d\lambda$) of the following tridiagonal matrix ($A$) ...
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64 views

For what values of $\varepsilon$ has $x^2-1 = \varepsilon e^x$ how many solutions?

Consider the equation $x^2-1 = \varepsilon e^x$. How many solutions does it have dependent of the value of $\varepsilon$? By plotting i guess the number is either 0,1,2 or 3. How does one rigorously ...
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14 views

Subspace Perturbation

For two positive semidefinite matrices $A,B\in\mathbb{R}^{n\times n}$, with dominant $r$ dimensional subspaces $U,V\in\mathbb{R}^{n\times r}$ and eigenvalues $\Sigma_A, \Sigma_B$, what can we say ...
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22 views

Perturbation theory and variable exchange of poisson-boltmann equation in spherical coordinates

I'm trying to understand this article. I think he has missing terms in his equations, and I can't understand how he derived equations 8-10. The math should be straight forward, and this make ...
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31 views

two variable perturbation analysis of nonlinear set of differential equations.

I have following set of equations, $\frac{dy(t)}{dt}=k z(t) - 3 k y(t) - y(t)^2 + \epsilon_1 (M-z(t))^2$ $\epsilon_2 \frac{dz(t)}{dt}=Mz(t) - z(t) y(t) - 2 \epsilon_2 y(t) + 2 \epsilon_1 \epsilon_2 ...
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15 views

Understanding Van Dyke's matching rule example

In Hinch's Perturbation Methods book, in the first example of matched asymtoptics example he introduces Van Dyke's matching rule. In the example, he has the equation: $$\epsilon f_{xx}(x) + f_{x}(x) ...
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96 views

Non linear second order ODE

I really need help solving this : $$y_{xx}-\left(y^{3}-y\right)-\varepsilon\frac{1}{2}\left(1-y^{2}\right)=0 $$ With boundary conditions : $$ y(\pm \infty )=-1 $$ I need to find a solution that ...
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78 views

evaluating $ \int\limits _{0}^{1}\frac{1}{\sqrt{x+\varepsilon}}dx $

I came across this : I'm trying to evaluate it up to $ o(\epsilon) $ $$ F\left(\varepsilon\right)=\int\limits _{0}^{1}\frac{1}{\sqrt{x+\varepsilon}} \, \mathrm{d}x $$ I've trying considering to look ...
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31 views

An advection problem with weak diffusion in asymptotic analysis.

Consider the following advection problem with weak diffusion: $$ \varepsilon\partial_{x}^2 u=\partial_{t}u+\partial_{x}u, $$ for $−\infty < x < \infty$, and $t > 0$ where $u(x, 0) = ...
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26 views

Second order perturbed equation

I've been studying asymptotic behavior on Ordinary Differential Equations. While doing some excercises I found out one excercise which has had me thinking for a while, so I am asking humbly for your ...
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38 views

Using Multipule Scale Analysis to solve a non-linear differential equation

I would like to know if there are other methods to solve equations such as this one below. I don't really understand the theory behind the multiple scale analysis and why it works I understand some of ...
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35 views

random matrix perturbation in linear system

Let $\Phi$ be a $m\times n~ (m<n)$ matrix whose entries are i.i.d. normal Gaussian variables, i.e., $\Phi_{i,j}\sim \mathcal{N}(0, 1)$. Project a vector $\hat{x}$ to $\Phi$ we have $y=\Phi ...
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27 views

Positive matrix perturbed by one with negative eigenvalues

This is firstly a reference request in order to understand and study the following problem. Consider the sum of matrices $$ \sum_{k=1}^mc_kA_k,\hspace{3cm} (1) $$ where each $A_k$ (they are $k$ ...
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8 views

Perturbed quadratic form definiteness

I have a rectangular matrix $[\mathbf \Pi]\in \mathbb R^{m\times n}$, where $m>n$, and $\mathrm {rank}([\mathbf \Pi])=k<n$. Its left null space basis are the columns of $[\mathbf Z]$. It's ...
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88 views

What more can be said about $\max_{v^\mathsf{T} v=1} \frac{v^\mathsf{T} B v}{v^\mathsf{T} A v}$?

Assume we have a positive semidefinite matrix $A$. Another matrix $B$ is equal to $A$ except it's $i$th row and$i$th column is zeros and element $B_{ii}=(n-1)A_{ii}$. i.e. \begin{align} B&=A-e_i ...
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1answer
33 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 ...
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44 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 ...
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22 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 ...
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47 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 ...
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44 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 ...
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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 ...
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2answers
40 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 ...
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51 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 ...
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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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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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56 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
55 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
23 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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94 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
82 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
38 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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31 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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15 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
42 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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60 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
27 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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1answer
46 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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125 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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24 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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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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41 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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51 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
81 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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140 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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148 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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13 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 ...