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0answers
6 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
7 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
31 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
11 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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0answers
25 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
27 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?
1
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1answer
21 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 ...
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2answers
107 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 ...
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0answers
46 views

Boundary Layer, leading order, Pertubation Theory, Differential Equations

I have got the following problem, taken from Multiple Scale and singular perturbation methods, Kevorkian & Cole book, page 94, exercise 1.b.: Find the leading order of the problem: $\varepsilon ...
1
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1answer
73 views

Continuity of the spectral radius

Let $M \in \mathbb{R}^{n\times n}$ be a nonnegative irreducible matrix with strictly positive entries on its main diagonal. Then $M$ is primitive and by the Perron-Frobenius Theorem we know that the ...
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0answers
30 views

How to use weakly nonlinear analysis?

I'm doing a PhD and my work so far has involved linear stability analysis. I believe I have a grasp on that. Now, however, my supervisor wants me to work with weakly nonlinear analysis, that is, ...
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0answers
27 views

How to show an aymptotic expansion is uniformly valid?

I have an equation $$ nt = u - \epsilon \sin(u) $$ which asks for the first four terms in the asymptotic solution. Hence if the solution is $u_0 + \epsilon u_1 + \cdots.$, expand $\sin(u)$ around ...
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1answer
39 views

Recommended graduate book for perturbation theory

I'm mostly interested in perturbation theory for linear algebra problems (such as finding eigenvalues) but any book on the subject can help. Thanks.
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0answers
14 views

Perturbation Theory for Interacting Quantum Mechanical System

Hello all! I am rather stuck at the start of this question; once I can get going, I should be ok. The issue that I'm having is that I don't know (/ can't work out) what Hamiltonian I am supposed to ...
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0answers
29 views

Uniform perturbative solutions to the Mathieu equation

The Mathieu equation is a second-order linear differential equation given by $$y''(t) + [a - 2q\cos(2t)]y(t) = 0$$ There are two special functions defined as linearly independent solutions to ...
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0answers
18 views

eigenvalues perturbed stochastic matrix

Let us given a stochastic matrix $P$ and a perturbed version $\tilde{P}=P+E$. Are the (right/left) eigenvectors and eigenvalues of these matrices close if $P$ and $\tilde{P}$ are close in the ...
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3answers
71 views

Solving a Perturbed Cubic Equation

Consider a cubic equation $(1 + \epsilon)x^3 - 2ax^2 + (a - 3\epsilon)x + 2\epsilon = 0$ where $\epsilon > 0$ and $a \gg 1$. In the limit of $\epsilon \rightarrow 0$, $x(x^2 - 2ax + a) = 0$ so ...
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1answer
90 views

Asymptotic expansion on 3 nonlinear ordinary differential equations

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation} \begin{equation} \frac{ds}{dt}= ...
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0answers
26 views

Derivative of terminal state w.r.t. the inital conditions.

Let $x\in R^n$ and consider the system $$ \dot{x}=f(t,x) \;\;\mbox{with}\;\; x(0)=x_0 $$ and suppose that we know it's exact or very accurate solution $x(t)$ for the time interval $[0,T]$. I'm ...
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1answer
90 views

How did Le Verrier calculate Neptune's position?

In the Wikipdia article on Neptune the discovery is described as a mathematical achievement: Subsequent observations revealed substantial deviations from the tables, leading Bouvard to ...
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0answers
42 views

Analyze rotation of satellite orbit due to transverse acceleration.

Consider a small satellite which moves in a 2D elliptical orbit around a much larger body (e.g. the Sun) under the influence of Newtonian gravitational acceleration $$Ar=G.M/d^2$$ QUESTION:- Is ...
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0answers
25 views

To prove a pertubation series converge

This is probably easy, however I can't figure out... This is an introductory problem for C Bender's book. The equation is $x^3 - (4 + \epsilon)x + 2 \epsilon = 0$. By assuming the solution is in ...
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0answers
32 views

Quantum Mechanics: time-dependent perturbations

I must solve this problem, but I'm not good at non constant differential equation system. Can somebody help me?
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0answers
56 views

perturbation theory solution of forced Duffing's equation

Question: Find the leading order of the asymptotic expansion for large t: $\frac{d^2x}{dt}+\varepsilon\beta\frac{dx}{dt}+x+\varepsilon x^3=Fcos(\frac{1}{3}\big(1+\varepsilon\omega)t\big)$ I have ...
1
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1answer
30 views

Perturbation of complex polynomials

Let $f(z)=\sum\limits_{k=0}^N a_kz^k$ be a (monic) complex polynomial and $\{\xi_{k}\}_{k=1}^{N}$ be the roots of $f$ (with multiplicities). Let $\{\tilde{\xi_{k}}\}_{k=1}^{N}$ be the perturbed ...
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0answers
31 views

Finding a leading order approximation for a system of ODE (multiple scales)

I need to find the leading order approximation which is valid for times $t=ord(\frac{1}{\epsilon} ) $ as $\epsilon \to 0$ to the solution $x(t,\epsilon)$ and $y(t,\epsilon)$ satisfying: ...
7
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1answer
217 views

First-term approximation for singular perturbation of ODE (with two turning points)

I'm reading "Introduction to Perturbation Methods" by Mark Holmes, and I came across an exercise that I don't know how to approach. As I decided to independently read this book, I have no ...
2
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1answer
56 views

Leading order approximation to differential equation

Find a leading order approximation to the solution of $\epsilon y'' + 2 y' + e^y = 0$, $y(0)=y(1)=0$ as $\epsilon \to 0$. I know there is a boundary layer near $x=0$ and not at $x=1$ so I can ...
2
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1answer
50 views

Extending solutions of an ODE past a singular point

In the course of my studies, I'm looking at at the ODE: \begin{equation} (f^3(x))'''=\frac{1}{6}xf(x),\quad f(0)=1,\,\,f'(0)=0 \end{equation} Where $f''(0)$ is a parameter left undetermined. In ...
3
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3answers
116 views

Asymptotic expansions for the roots of $\epsilon^2x^4-\epsilon x^3-2x^2+2=0$

I'm trying to compute the asymptotic expansion for each of the four roots to the following equation, as $\epsilon \rightarrow 0$: $\epsilon^2x^4-\epsilon x^3-2x^2+2=0$ I'd like my expansions to go ...
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0answers
32 views

Matched asymptotic expansion with interior layer (corner layer?)

I am given with the following: $$\epsilon y'' + e^x (xy' -y ) = x^2,\quad -1<x<1,\quad 0< \epsilon \ll 1,\text{ and }y(-1)=1,\ y(1)=-1 $$ We learned that in such a case we can expect an ...
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2answers
76 views

First-order perturbation of a spring with some stiffness

I am learning about perturbation theory in ODE. I have the ODE $$\ddot{y}(t) + \alpha \epsilon \dot{y}(t) + y(t) = 0,\\y(0) = y_0\neq 0\\ \dot{y}(0) = 0,$$ where $\epsilon>0$ is small. (This ...
1
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1answer
52 views

BVP and Perturbation Methods

I asked this question but no one answered this... This is a question on one of my ODE past papers: You are given the non linear boundary value problem $$ y^{\prime\prime}+y^2=0,\ \text{subject ...
2
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1answer
21 views

Critical eigenvalues for smooth family of 2x2 matrices?

Consider the following simple setup: we have a smooth family of symmetric $2\times 2$ matrices $A(t)$, with normalized eigenpairs $(\lambda_1(t),v_1(t))$ and $(\lambda_2(t),v_2(t))$. Suppose there ...
0
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1answer
21 views

stability of essential spectra

Let $X$ be a Banach space. $A$ and $B$ are linear closed and densely defined operators and $\lambda\in\rho(A)\cap\rho(B)$ such that $(\lambda - A)^{-1}-(\lambda - B)^{-1}$ is a Frehholm perturbation ...
2
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1answer
42 views

Solving $\operatorname{ctg} x=x/b$

I have no problems finding first solution (both: $b \to 0$ and $b \to \infty$). My solutions on photos. I got stuck trying to find solution when $x \to \infty$. As I think, solution for $x$ will have ...
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0answers
30 views

How to pick the leaving variable for Perturbation method? (Linear programming)

I am studying Optimization, a math course. We are going over simplex method and its variances. One of which is called the perturbation method. From this example, O is the objective function and ...
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0answers
29 views

perturbation solution of two singular ODEs

I need some help with solving the following system of ODEs: $$\epsilon \frac{dx}{dt}=Ay +ABx(1-y)$$ $$\epsilon\frac{dy}{dt}=Bx(1-y)-y-\epsilon y$$ I'm confused by the fact that both equation are ...
1
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1answer
108 views

Stuck on perturbation theory for finding a root of polynomial, with rescaling

I have been given the polynomial $$\epsilon x^3+x-2=0,$$ where epsilon is very small and I need to find the roots using perturbation theory. So far I have found the first root, 2, using the direct ...
5
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1answer
119 views

Perturbation theory PDEs

I have the solution of a PDE of the form: $$ \Delta \Psi(r,\theta, \phi) = k \Psi(r,\theta,\phi)$$ on a set $\mathbb{R}^3 \backslash B(0,R)$. Hence, the actual solution is known there! Regarding ...
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0answers
136 views

Method of dominant balance and perturbation theory

We know perturbation theory express the desired solution of differential equations in terms of a formal power series in some "small" perturbation parameters: $y=y_0+\epsilon ^1 y_1+\epsilon ^2 ...
1
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1answer
60 views

non-homogeneous constant co-efficient 2nd order linear differential equation

I am doing a perturbation theory question and am having trouble with the (seemingly simple!) differential equation method of undetermined coefficients... I have reduced my given system so that now I ...
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2answers
110 views

Introductory text on perturbation theory for dynamical systems

I am working on my thesis which is about oscillations and as far as I realise I need to know about perturbation theory and methods in solving differential equations, specifically dynamical systems. A ...
2
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0answers
49 views

Strong versus weak coupling expansion to solve hard problems

For the quintic equation $$ x^5 + x = 1 $$ it can be seen that when taking the strong coupling limit to solve $$ x^5 + \epsilon x = 1 $$ perturbatively, summing the terms of all orders in ...
2
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0answers
88 views

Perturbation of eigenvalues

I am looking at a certain operator, that is a Hilbert-Schmidt integral operator from $L^2(X,d\mu)$ to $L^2(X,d\mu)$.I want to see how its eigenvalues or singular values change as its kernel is ...
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1answer
65 views

curves bounding discs

I'm interested in the following question. Please forgive me if my question is lacking in precision. I'm not a mathematician, and need some help getting started: If I have a smooth, simple curve ...
1
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1answer
548 views

Asymptotic Matching for boundary layer problem

The question asks to find a global approximation to the leading order of $\epsilon$. $\epsilon y'' + xy' + \epsilon y =0$, with boundary conditions $y(0)=1,y(1)=-1$. I assumed it's a boundary layer ...
1
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2answers
65 views

How can I do a perturbation on this ODE?

I have an ode of the form $$\frac{dy}{dx}=y(x)+\sqrt{y(x)+\epsilon\cdot f(x)}$$ and I would like to do a perturbation up to first order $\epsilon$. My advisor gave me an example for a simpler ODE, ...
2
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1answer
65 views

Derive a perturbation of period $2\pi$, to order $\epsilon$

I have the following problem: In the equation $\ddot{x}+\Omega^2x+\epsilon f(x) = \Gamma \cos t$, $\Omega$ is not close to an odd integer, and $f(x)$ is an odd function of x, with expansion, $$f(a\cos ...
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0answers
18 views

Subordinate operators

Let $A$ be a linear densely defined operator on a Hilbert space $H$ and $L$ is a selfadjoint operator with discrete spectrum such that $\mathcal{D}(L) = \mathcal{D}(L)$ and $$\|Tf\| \leq M ...