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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perturbative expansion of vector's $L_2$ norm

Consider the vector $\vec{C} = \vec{A} + \epsilon \vec{B}$, in the limit $\epsilon \rightarrow 0$. I want to find a perturbative expansion for the quantity: $\frac{1}{\lVert C \rVert_2} = \frac{1}{\...
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Large $t$ solution to $\partial_t S(x,v,t) = v \partial_x S + (a+v) \partial_v S + \partial_v^2 S$

I have the partial differential equation $$\partial_t S(x,v,t) = v \partial_x S - (v + a ) \partial_v S + \partial_v^2 S$$ subject to the following boundary conditions: $$S(x,v,0)=1,$$ $$S(0+,|v|,t)=...
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How to perturb nonzero elements to get nonzero elements

Let $x \in \mathbb{R}^n$ be such that $x_i\neq 0$ for all $i\in \{1,\dots,n\}$. How can we perturb $x$ so that the perturbed vector $y$ has the same property, i.e., $y_i\neq 0$ for all $i\in \{1,\dots,...
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Converting an equation with "small displacements" retains a zeroth-order term -- can it become a PDE?

I have come across an equation that looks like $$ f\!\left(t + \tau, x\right) = f\!\left(t, x - \xi\right) - f\!\left(t, x + \xi\right) $$ where $\tau$ and $\xi$ are both small. Every bone in my body ...
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Decaying solution and periodic-$1$ solution to $y''(x)=y(x)-y(x)^2$

I came across a problem in the dynamic systems theory, which is very similar to the following simple example. Consider the simple model equation $$ y''(x)=y(x)-y(x)^2. \tag{*}$$ Let us denote by $y_\...
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Wilkinson's matrix eigengap

Task The $(2n+1) \times (2n+1)$ Wilkinson matrix is a symmetric tridiagonal matrix whose main diagonal is $$n, n-1, \ldots, 1, 0, 1, \ldots, n$$ and whose entries on the first diagonals above and ...
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Repeated Roots in Polynomial (Regular Perturbation)

The polynomial that I consider is as follows: $$ x^{4} + 12(2\epsilon - 1) x^{2} = 0 $$ I use regular perturbation which is as follows: $$ x = x_{0} + \epsilon x_{1} + \epsilon^{2} x_{2} $$ I ...
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Perturbation method to solve $\epsilon x^3=x+\epsilon$

I was given the following exercise: Find two-term approximation to the real roots of the equation $\epsilon x^3=x+\epsilon \tag*{}$ My attempt: Since this is a singular equation, we introduce a ...
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On the convergence of (eigen)values in perturbation theory ?!

Background: Let $H, H_0$ be real square matrices, we want to estimate the eigenvalues/vectors of $$H = H_0 + \epsilon R$$ where $0 \leq \epsilon \leq 1$, those of $H_0$ are assumed to be known. Let $\...
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Melnikov's method, homoclinic orbits, and bifurcation values

In nonlinear dynamics, Melnikov's approach provides an intriguing way to detect homoclinic bifurcations and bifurcation values, i.e., the values of the parameter at which a dynamical system exhibits ...
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Van Dyke's Rule for a Function

I have been studying perturbation theory and recently come across a question regarding van Dyke's rule which is of the form: Find and match the $(m,n) = (1,1), (1,2), (2,1)$ expansions for the ...
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Can I solve $\partial_t(\partial_t +a ) W(x,t) = (\partial_t + b)\hat{L} W(x,t)$ using the solution $\partial_t W_0 = \hat{L}W_0$?

I have a challenging partial differential equation which is second order in time: $$\partial_t(\partial_t +a ) W(x,t) = (\partial_t + b)\hat{L} W(x,t)$$ Here $L$ is some linear operator which involves ...
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unique equilibrium in a perturbed system

In today's lecture, I see that if $x_*$ is an asymptotic stable and hyperbolic equilibrium of the $\dot{x}=a(x), \, x\in\mathbb{R}^n$. But then prof said that "it's obvious" if we give a ...
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How does integration change in a multiple-scales approximation?

A multiple-scales approximation is usually performed on a differential equation in strong form. For example, if we assume that the function $u$ has two weakly-interacting time scales, $\tau=t$ and $T=...
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Singular value perturbation inequality

Need some help with this one. Given $A,E \in \mathbb{R}^{m\times n}$. Show that $$\sigma_{\max}(A+E) \leq \sigma_{\max}(A) + \|E\|_2 $$ The hint provided is: $$\sigma_{\min}(A)\|x\|_2 \leq \|Ax\|_2 \...
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Perturbation Theory Ambiguity?

I am trying to solve problem 2.1 in Schwartz, which is to derive the transformations $x \rightarrow \frac{x+vt}{\sqrt{1-v^2}}$ and $t \rightarrow \frac{t+vx}{\sqrt{1-v^2}}$ in perturbation theory. ...
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Regular perturbation on Fredholm equation

I want to find the three first terms in an expansion of $u(y)$ \begin{equation} u(y)=u_0(y) +\epsilon u_1(y)+\epsilon^2u_2(y)+\mathcal{O}(\epsilon^3) \space \space \space \space \space \space \space \...
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Number of negative eigenvalue of a perturbation by a bounded operator

Let $H=(H,(\cdot, \cdot))$ be a Hilbert space, $L:D(L) \subset H \rightarrow H$ be a self-adjoint operator (not necessarily bounded) and $A:H \rightarrow H$ be a bounded and symmetric operator. By ...
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Perturbation of the spectrum

In a lecture note, in the demonstration of a result it is said that "using analytic perturbation theory" it is possible to deduce certain things. The reference is Kato's classic book, "...
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Asymptotic behavior of the zeros of a polynomials for large values of a parameter

Consider a polynomial in $r$ of the form $$ r^4+p_3(\lambda)r^3+p_2(\lambda)r^2+p_1(\lambda)r+p_0(\lambda), $$ where the $p_i$ are polynomials in the parameter $\lambda$. I use degree four to simplify ...
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Eigenvalues of $\left[\begin{matrix}1&\epsilon&2\epsilon\\\epsilon&1&2\epsilon\\-2\epsilon&-2\epsilon&2\end{matrix} \right]$ as a series in $\epsilon$

Consider the matrix $$ M := \left[ \begin{matrix} 1 & \epsilon & 2\epsilon \\ \epsilon & 1 & 2\epsilon \\ -2\epsilon & - 2\epsilon & 2 \end{matrix} \right] $$ for some small ...
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Linear Algebra Question and Matrix Perturbation Problem Solution

I found the following formulation of a perturbation problem in my research (see attached image). In step 4c to 4d, the equation goes from: $$|I+A^{-1}e_k\Gamma^T| = 0$$ $$|1+\Gamma^TA^{-1}e_k| = 0$$ ...
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Obtaining an effective PDE whole solution is approximates to actual solution

Let's say I have a PDE. Let me introduce a parameter $\epsilon$ into it such that when $\epsilon = 0$, it is an easy PDE to solve. You probably already guessed where this is going ... perturbation ...
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Perturbed Gram Matrix

Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first cannonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix $$\sum_{t=1}^T(...
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Spotting distinguished limits from Robin boundary conditions

I am currently working on a problem involving solving PDEs and using boundary layers. In this problem, $f(x,y)$ is the main dependent variable, $f(x,y)=f_R (X_R ,y)$ and $f(x,y)=f_B(x,Y_B)$ where $X_R$...
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Preserving completeness and orthogonality relations while using perturbation theory for complex symmetric operators

I am working on the following problem right now: I want to built the resolvent for some finite-dimensional complex symmetric operator $M$ (you may call it just a complex symmetric matrix), so, I want ...
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Taylor expansion for a Bessel function with complex argument

If we have a Bessel function of the first kind and the $m$-th order as $J_{m}(x+i\epsilon x)$, where $m$ is integer, $x, \epsilon$ are real and $\epsilon$ is a small parameter ($0<\epsilon\ll 1$), ...
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Who has implemented the M2SPSA optimizer?

I ran across this paper, which is an update to the second-order simultaneous perturbation stochastic approximation (SPSA) black-box optimizer. Has anyone seen it implemented and/or is there a code ...
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Perturbation method for complex small variable in system of ODEs

Consider the following system of ODEs: $$ i\dot{C_1}(t)=\nu^*(\omega) e^{i(\omega-\omega_{10})t} C_2(t)\\ i\dot{C_2}(t)=\nu(\omega) e^{-i(\omega-\omega_{10})t} C_1(t) $$ with initial conditions $C_1(0)...
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Inverse of symmetric positive definite perturbation of symmetric positive definite matrix

Let $A$ be an $n\times n$ real invertible matrix, and $\delta A$ be a $n\times n$ matrix such that $A+\delta A$ is invertible. Then, it is known that $$ \frac{\left\|(A+\delta A)^{-1} - A^{-1}\right\|}...
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What is the smallest eigenvalue of this infinite, symmetric, tridiagonal matrix?

$A_{nm}$ ($n,m = 0, 1, 2, \ldots$) is a symmetric, tridiagonal matrix. The diagonal elements are $A_{nn} = a_n = n + 1$, and the off-diagonal elements are $A_{n,n+1} = A_{n+1,n} = b_n = \lambda \sqrt{\...
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Schrödinger operator whose potential analytically depends on a parameter – how does the spectrum change?

Let's say we have a self-adjoint operator $H_s$ on the Hilbert space $L^2(\Omega \subseteq \mathbb{R})$ defined by $$ H_s \, \psi(x) := -\psi''(x) + V_s(x) \, \psi(x) \: , $$ where $s \in \mathbb{R}$...
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Perturbation theory in math and quantum mechanics

This question is about my lets say lack of understanding. I can not make the connection between what i study and what i saw in lectures. For example in our QM lectures we saw perturbation theory like ...
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Asymp. Integrals

I am trying to get a two-terms asymptotic expansion {as $c\to 0 $ } for the following integral $$\int_{0}^{c} \sqrt{\frac{c^2-\zeta^2}{1-\zeta^2}} d\zeta$$ I have tried so far to substitute $\zeta=c\ ...
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Asymptotic integral

Let $$u= \frac{x}{\sqrt{4\pi\beta}}\int_{0}^t \frac{f(v)}{(t-v)^\frac{3}{2}}\exp(\frac{-x^2}{4\beta(t-v)}) dv $$ I want to show that $$\lim_{x\to 0}u=f(t)$$ Watson's lemma can be used as long as ${x\...
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Lorenz - why is the solution with noise in initial condition on the same curve?

I want to use Lorenz'63 for the purpose of testing some algorithms I'm currently researching. When I defined a system and solved it in Matlab, plot of x-coordinates for explicit and perturbed initial ...
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Perturbation Theory for the Definite Generalized Eigenvalue Problem

I have become confused on some basic questions concerning the perturbation theory of the generalized eigenvalue problem. I am likely missing something simple, but I have a couple of questions about ...
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regular or singular perturbation?

Can the problem of solving for $x$ the equation $\varepsilon x +t =0$, where $1<t<\infty$ and $\epsilon>0$ is a small parameter be termed a (very simple case of) singular perturbation problem ...
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Integral asymptotic approximation

I am trying to attain a two-term approximation for the following integral as $m$ goes to $1$ from below: $$I=\int_{0}^{\pi /2}\frac{\mathrm d\theta }{\sqrt{1-(m^2)\cdot\sin(\theta)^2 }}.$$ So far I am ...
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Reference for Perturbation theory

I need to study the eigenvalues and the eigenfunctions of a perturbated operator knowing the eigenvalues and the eigenfunctions of the initial operator, using the theory of perturbation and writing ...
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WKB form manipulation

If I have the equation $$y'' +\sin\left(\frac{1}{x}\right)y = 0$$ and I approximate $\sin(\frac1x)$ as $\frac1x$, how would I get epsilon squared in front of the first term (which is required for WKB ...
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WKB form substitution for V

If I have the equation $$y'' +\sin(1/x)y = 0$$ how would I replace $\sin(1/x)y$ in order to get epsilon squared in front of the first term? I have tried defining $X= \delta\sin(1/x)$, but that doesn't ...
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Prove perturbation theory breaks down

I want to find the $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon)$ terms in the pedestrian expansion $y = y_0 + \epsilon y_1 + \epsilon ^2 y_2 + \dots$, where $y$ satisfies the following second order ODE:...
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Expected Time of Absorption when Markov Chain is "Nearly" Irreducible

Suppose we have a Markov chain with states indexed $1$ through $n$ that is irreducible with a unique stationary distribution ${\bf \pi} \in \mathbb{R}^n$. The transistion probabilities of the Markov ...
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Finding mean of perturbed Gaussian using perturbation theory

I am trying to approximate the mean of the following distribution using perturbation theory: $$P(x) \propto \exp\left( \frac{1}{2\sigma^2}x^Tx + \alpha\frac{1}{2}(f(x)-y_0)^2 \right) $$ where $y_0$ ...
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Asymptotic expansion for $xe^{1/x}=e^{\lambda}$

I need to find a 3-term expansion as $\lambda\rightarrow\infty$ for each of the solutions to this transcendental equation. Plotting $xe^{1/x}$ I assume that there are 2 solutions, one around 0 and the ...
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Asymptotic expansion of $x\exp(1/x)=\exp(\lambda)$ as $\lambda\to\infty$

I am struggling to find a three-term expansion of the following equation $x\exp(1/x)=\exp(\lambda)$ as $\lambda\to\infty$ for each of the solutions for $x$. I graphed the functions $1/x$ and $\log(x)$ ...
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Deriving solution of a second order ode from a closely related one with a different scalar coefficient for the term in X(x)

I solved the wrong differential equation! I was looking at a parabolic pde in the style of a heat equations which upon separation of variables I wrote as \begin{equation} a x^2\left(\frac{d^2 X}{d x^...
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Is this a basic result of calculus of variation or perturbation theory?

If $x$ is a local minimum of the function $f$, then the minimum of $f+\lambda g$ at the first order in $\lambda$ is given by $x+\delta x$ with $\delta x=-\lambda g'(x)/f''(x)$. The proof is very ...
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Regular Perturbation theory for $L=L_0+\epsilon ix$, where $L_0$ is an unbounded self-adjoint operator, on $\mathbb R$

I am looking to understand the spectrum of the following operator: $L_{\epsilon}[u](x)=L_0[u](x)+i\epsilon x u(x)$ on $\mathbb R$. Here $L_0$ is a negative semi-definite self-adjoint operator that has ...

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