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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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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Non linear second order ODE up to $O(\epsilon) $ for $v_{xx}-\left(v^{3}-v\right)-\varepsilon\frac{1}{2}\left(1-v^{2}\right)=0$

I really need help solving this : $$v_{xx}-\left(v^{3}-v\right)-\varepsilon\frac{1}{2}\left(1-v^{2}\right)=0 $$ With boundary conditions : $$ v(\pm \infty )=-1+\frac{1}{4}\epsilon $$ I need to ...
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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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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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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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Impact of perturbation on the eigen-values of 3 diagonal matrix [closed]

Lets consider a 3-diagonal matrix as following: $$ A(i,i) = 2 $$ $$ A(i,i+1) = -1 $$ $$ A(i,i-1) = -1 $$ The eigen-values of this system is known easily. How eigen-values would change if we add ...
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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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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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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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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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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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32 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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42 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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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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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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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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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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34 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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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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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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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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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
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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77 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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132 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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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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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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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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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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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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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 ...
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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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126 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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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!