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Questions tagged [perturbation-theory]

Perturbation theory describes a range of tools and techniques to find approximate solutions to problems containing small parameters.

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Can we choose smoothly the singular vectors of a matrix?

Let $X$ be the space of all real $n \times n$ matrices, with strictly negative determinant, and pairwise distinct singular values. $X$ is an open subset of the space of all real square matrices. Is ...
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Find a two term asymptotic expansion of the following problem

I want to find a two term asymptotic expansion, for small $\epsilon$, of the solution of the following problem: $$ y'' - \epsilon y' - y = 1\tag{1}, \, y(0) = 0, \,y(1) = 1 $$ My approach: I assume ...
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Matching expansion of an ODE: $\epsilon y'' + xy' + y = 0$

I am trying to solve a boundary layer problem using matched expansion $$\epsilon y'' + xy' + y = 0$$ where the boundary condition is $$y(0) = 1, y(1) = 1$$ and $x\in (0,1)$. So far, I have the outer ...
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33 views

Perturbation Methods ODE Example

I am currently doing a Masters-level class in Perturbation Methods and I am stuck on a question. I have done similar questions but the middle term is confusing me. It could be that I'm rescaling wrong ...
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Expected value of eigenvalues of perturbed matrix

Let $A$ be an arbitrary $n \times n$ real matrix and $P$ be a random perturbation matrix with zero-mean, i.i.d. entries. Can we say anything about the eigenvalues of $A+P$? In particular, are there ...
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Definition of perturbation using norms

Can someone give me the definition of a perturbation vector? I know that it is a vector composed of $x_i$ that have a small value and are independent, but I know I can define it by using norms because ...
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34 views

Solution Poisson Equation by Homotopy Perturbation Method

I need to solve the following Boundary Value problem $$\frac{\partial^2w }{\partial x^2}+\frac{\partial^2w }{\partial y^2}=c$$ Boundary conditions are $$w(x,h)=0$$ $$w\left(\frac{\pm h}{\sqrt3},...
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Stationary Phase with controlled phase $|\int_0^Te^{\frac{i}{\epsilon}\phi_j(t)}dt| = O(\epsilon)$

My question concerns the stationary phase theorem when we only know control information on the phase. Suppose that $\phi_a(t)$ and $\phi_b(t)$ are phases such that $$\phi_a(t) = a_1 + a_2t$$ $$\...
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Can we perturb a low rank map to a full rank map in a smooth way?

Let $f:\mathbb{R}^n \to \mathbb{R}^n$ be smooth. Can we find, for every $\epsilon>0$, a $C^1$ map $\tilde f:\mathbb{R}^n \to \mathbb{R}^n$ of full rank such that $\|df-d\tilde f\|_{C^0}<\...
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Showing that $\lambda - (A + B)$ has dense range

Let $A$ be the generator of a $C_0$-semigroup $(T(t))_{t \geq 0}$ of contractions on a Banach space $X$ and $B \in \mathcal L(X)$ a bounded operator. To apply some approximation formula I want to show ...
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Perturbation expansion within trig function

I'm trying to find an approximate solution to a nonlinear differential equation. It involves something to the effect of $\frac{d\Psi}{ds} = \sin{\Psi} + \dots$ , where $\Psi$ is a small variable. If ...
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Perturbation solution Dirac equation

I'd like to know how to solve the Dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\...
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What should I do with: $\frac{d}{dx} \frac{d}{d \xi}$ in multiple scale perturbation for second-order eqs?

What should I do with: $\frac{d}{dx} \frac{d}{d \xi}$ in multiple scale perturbation for second-order eqs? I'm doing my first ever exercise in perturbation theory for "multiple scales". The eq. is: ...
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Perturbation of Diracs equation (first order)

I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\...
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73 views

Mandelbrot set perturbation theory: When do I use it?

I have read the post on Perturbation of Mandelbrot set fractal. I will also be referring to the PDF by K.I. Martin on this topic. My question is to do with the precision laid out in the Martin paper. ...
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How to find the critical point for this coulomb field

Two equal positive charges are at distance $d$, $-d$ from the origin on the $y$ axis. What is the distance on the $x$ axis beyond which a small perturbation in $y$ will move a particle away from the $...
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Effect of perturbation on eigenvalues

Given a diagonal matrix $A$ with diagonal elements $a_1,\dotsc,a_n$ what can we say about the eigenvalues of the perturbation $A+E(t)$ satisfying $E(t) = O(t^k)$ so that there exists a constant $K>...
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In perturbation theory, does it make sense to consider added error to “shift” the function and alter no. of roots?

In perturbation theory, does it make sense to consider added error to "shift" the function and alter no. of roots? That is consider e.g. $f(x)=x^2+\epsilon$. For $\epsilon=0$ there's a single root $x=...
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Inverting a power series matrix

Let's assume I want to invert a matrix function $M=M(x)$, which is expressed as a power series of the small parameter $\epsilon$ $$M = M_0(x) + M_1(x) \epsilon + M_2(x) \epsilon^2 + \mathcal{O}(\...
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Perturbing a matrix with asymptotic constraints

I want to expand the following function in series of $\epsilon = a/l$: $$f=\frac{h \cos\alpha + l\sin\alpha}{l \cos \alpha + c \sin \alpha}$$ I know from the physics of the problem that also $$\frac{...
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A perturbed function has a stationary point, while the unperturbed has not

I am looking for extrema of the function $$ B(x,y) = \frac{1}{x} \Big[ F(y) - \epsilon [\log(x-y) +1] \Big]$$ limited to the the domain $\Omega = \{y \ge 0, x \geq y\} $ $F$ is a twice ...
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Perturbation theory of the eigenvalues about a symmetric matrix: Reality of Eigenvalues

Let $A$ be an $n\times n$ real symmetric matrix, and let $E$ be a real matrix. Is it true that if the perturbation matrix $E$ is small in some norm, then the eigenvalues of $\hat A := A+E$ are all ...
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System of equations and perturbation methods

I would like to characterise how the solution of a nonlinear system of equations change if a perturbation term is added. Namely, I have the system \begin{array}{lcl} 2y [F(x) - \varepsilon \frac{1}{y(...
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Is the set of matrices with distinct singular values connected and dense?

$\newcommand{\GLp}{\operatorname{GL}_n^+}$ $\newcommand{\SO}{\operatorname{SO}_n}$ Consider the set $$ X=\{ A \in \GLp \,\,|\,\, \text{ all the singular values of } A \text{ are distinct } \},$$ where ...
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Consider the singularly perturbed cubic equation $\epsilon x^3+x^2-1=0$

Question: "Consider the singularly perturbed cubic equation $\epsilon x^3+x^2-1=0$ where $\epsilon >0$ is a small parameter. Obtain three-term astymptotic expansions for the two regularly ...
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first order Taylor expansion term of a function multiplied by a dot product of gradients.

I need to do the following Taylor expansion. Let's suppose we have a function $f(n,|\nabla n|^2)(\nabla (|\nabla n|^2),\nabla n)$, where by $(\nabla (|\nabla n|^2),\nabla n)$ I mean the dot product ...
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Regular Perturbation to obtain solution to: $y''+y= \epsilon y^2$

The following perturbation problem: $y''+ y = \epsilon y^2, y(0,\epsilon)=1, y'(0,\epsilon)=0$ So far I have deduced that: $y_0''+y_0=0, y_0(0)=1, y_0'(0)=0$, and so $y_0(x)= cos(x)$ For the second ...
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Using Regular Perturbation, obtain approximate solution:

Consider the following ODE: $y'+ y = \epsilon y^2, y(0,\epsilon)=1$ Assuming the perturbation amsats gives: $y'_0 + y_0 =0, y_0(0)=1,$ $y'_1 + y_1 = y^2_0$, $y_1(0)=0$ How is: $y_0(x)=e^{(-x)}$ ...
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Solve by means of regular perturbation to obtain an approximate solution up to and including $\mathcal{O}(\epsilon^2)$

I have to solve the following ODEs: $y''+ y = e^{\epsilon \cos x}$, $y(0,\epsilon)=y(1,\epsilon)=0$; $y''+ y = \epsilon y^2$, $y(0,\epsilon)=1$, $y'(0,\epsilon)=0$. I am having trouble ...
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Lower bound for the speed convergence of the spectrum

Let $A$ be a (complex unital) Banach algebra and $a \in A$ with spectrum $\sigma(a) = \{ 0 \}$ ($a$ is quasinilpotent). For $b \in A$ and $\varepsilon > 0$ consider the linear perturbation $a + \...
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Obtain first two terms in the asymptotic approximation $\epsilon x^3+x^2+2x-3=0$

Consider the singularly perturbed cubic equation $$\epsilon x^3+x^2+2x-3=0$$ where $\epsilon >0$ is a small parameter. This equation has two regularly perturbed roots and one singularly perturbed ...
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58 views

WKB Approximation to Schrödinger Equation

Consider the Schrödinger equation: $$y''(x)+EQ(x)y(x)=0,\tag{1}$$ where $E>0, Q(x)>0, y(0)=y(\pi)=0. $ Use WKB approximation to obtain $$y(x) \sim CQ^\frac{-1}{4}(x)\sin\Big(\sqrt(E) \int_{0}^{...
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Conditions for Boundedness of Spectral Measures of Perturbations of Self-Adjoint Operators?

Suppose $A$ is an unbounded self-adjoint operator in a Hilbert space $H$ with discrete spectrum $$\lambda_0 < \lambda_1 < \cdots$$ bounded below with lowest eigenvalue $\lambda_0$, lowest ...
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Change in eigenvalues due to perturbation to a correlation matrix

Let $A$ be a $m \times n$ matrix defined as $ A = \Big[\frac{a_1}{\|a_1\|} \cdots \frac{a_n}{\|a_n\|}\Big]$ and $a_k \in \mathbb{R}^{m\times 1}$ where $k \in [1,\dots,n]$. Now, we define a ...
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Solve for Differential Equations by using regular perturbation method

carry out regular perturbation calculation for $\epsilon$ satisfying $$x''(t)+x(t)=\epsilon x^2(t)$$ correct to the second order in the small parameter $\epsilon$. then use the result to perform a ...
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74 views

Obtain the leading order uniform approximation of the solution

Obtain the leading order uniform approximation of the solution to $ \epsilon y′′-x^2y′-y=0$. The boundary conditions are $y(0)=y(1)=1$. Since $a(x)<0$ the boundary layer is at $x=1$. The outer ...
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Eigen-vector and eigen-function perturbation

In the following question we assume the eigenvalues are sorted in descending orders and the eigen-vectors and eigen-functions are sorted accordingly. Given a symmetric $n\times n$ matrix $P$ and a ...
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1answer
85 views

Perturbation of complex square root function

Let $z$ and $w$ be complex numbers. Than it is claimed that $$ |\sqrt{z+w} - \sqrt{z}| \leq \frac{|w|}{\sqrt{|z| + |w|}} $$ or $$ |\sqrt{z+w} + \sqrt{z}| \leq \frac{|w|}{\sqrt{|z| + |w|}} $$ I am ...
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26 views

Boundary layer type with initial value problem

Consider the initial value problem $\sqrt{\epsilon} \, u'' + u' - u = e^{2t}$ , with $u(0)=1$, and $u'(0)=1/\sqrt{\epsilon}$. I am trying to use a matched asymptotic expansion to find the leading ...
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148 views

Perturbation theory to speed up Julia fractal drawing

I have a really underpowered platform here and want to draw a Julia (and possibly Mandelbrot and Burning ship) fractals using a 8.24 fixed point class. I use iteration count for coloring and need to ...
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29 views

Newton Polygon Method (Newton-Kruskal)

$ϵ^{1-α}z^3-z^2+ϵ^{α}z-ϵ^{2α+\frac12}$. The solution to this in my notes now says: "The corners of the curve $\min\{1-α,0,α,2α+\frac12\}$ are at $α=-\frac12, 0, 1$." How does one arrive at this ...
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Leading Order Behaviour using Watson and Laplace

Find the leading order behaviour of $$I(x)=\int_0^1 \sin te^{-x\sinh^4 t}dt, \enspace x\rightarrow\infty$$ Thus evaluate $I(10)$ and compare it with its numerical value $I(10)\approx0.13564$ So far, ...
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Perturbation to the coefficients of a polynomial

I am reading 'Trefethen and Bau: Numerical Linear Algebra' book and I came across the following problem- Assume that the polynomial $p(x) = \sum_{k=0}^n a_k x^k$ with real coefficients has $n$ ...
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Is it valid to partially neglect some terms in a higher order contribution?

Suppose I have a function $f(x,\alpha)$ which has a convergent expansion in the small parameter $\alpha$: $ f(x,\alpha)=\sum_{n=1\dots \infty,i=1\dots k} g(x,n,i)\alpha^n $ This expansion has $k$ ...
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Does a norm ball around a real matrix contain all possible pairs of complex spectrum?

We consider the normed vector space $(M_n(\mathbb R), \|\cdot\|_F)$, i.e., real matrices with Frobenius norm. Let $A \in M_n(\mathbb R)$ be a diagonalizable matrix and have all eigenvalues to be real. ...
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Picking the correct Ansatz for valid solutions in Asymptotic Methods

I am trying to find the solution to the following equation, $\epsilon x^3 -x^2 +x-\epsilon^{\frac{1}{2}}=0$, for the first two non-zero solutions as $\epsilon \to 0^+$. I have used the principal of ...
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34 views

Conditions necessary for a boundary layer to exist

Determine values of $a$ for which the problem: $\epsilon y^{''} + y^{'}+ae^y=0,$ $ y(0)=y(1)=0$ has a solution with a boundary layer structure. I am familiar with the procedure for tackling this ...
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Analytic perturbation theory and its applications - Avrachenkov

I am reading Avrachenkov's book - 'Analytic perturbation theory and its applications'. One of the problems contained in books is as follow: If we substitute the series expansions $$A(z)=A_0+zA_1+z^...
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How can apply a perturbation method in a system of equations?

If a have a system of equation $$ \frac{d u}{dt} = 1- u e^{\epsilon(q-1)}$$ $$ \frac{d q}{dt} = u e^{\epsilon(q-1)}-q$$ $u(0)=q(0)= 0$ How Can I apply the perturbation methot $\epsilon$ small ...
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128 views

Differentiability of largest eigenvalue for a $C^1$ function

I encountered following interesting statement: If $f: [a, b] \to M_n(\mathbb C)$ is a $C^1$ ( $C^1$ in the interior and left/right differentiable over the end points) function over an interval $[a, b]...