# Questions tagged [perturbation-theory]

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

500 questions
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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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### 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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### 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: $y_0(x)=e^{(-x)}$ ...
### 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 ...