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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Series expansion of the determinant for a matrix near the identity
The problem is to find the second order term in the series expansion of the expression $\mathrm{det}( I + \epsilon A)$ as a power series in $\epsilon$ for a diagonalizable matrix $A$. Formally, we ...
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Possible analytical solution to $g''(x)=\alpha\left[g(x)^3-g(x)\right]+\beta g(x) e^{-\kappa x}$
kind of related to a previous question of mine. I am describing a physical phenomena related to charged molecules and am interested in the following quantity:
$$\xi=\int_0^{\infty}\left[1-g(x)^2\right]...
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Matrix function converges, how about the eigenvalues?
Suppose I have a matrix function $A(t)$ with $$\lVert A(t) - B\rVert \le ct^\alpha$$ in some matrix norm (this will work for any norm, I guess). So, in a sense $A(t)\rightarrow B$ for $t\rightarrow 0$ ...
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Reference: Continuity of Eigenvectors
I am looking for an appropriate reference for the following fact.
For each $X \in \mathbb{R}^{n \times n}_{\text{sym}}$ (symmetric matrix),
there exist $\varepsilon, L > 0$, such that
for ...
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Linear Stability Analysis of ODEs/PDEs
I'm looking for a systematic understanding/approach to linear stability analysis of differential equations. I'm interested in an arbitrary (non-linear) PDE, $\mathcal L u=0$ (or system of PDEs ...
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Perturbative solution to $x^3+x-1=0$
I would like to calculate the real solution of
$$
x^3+x-1=0
$$
by resumming a perturbation series. To this end, I considered
$$
x^3+\epsilon x-1=0,
$$
$\epsilon$ being a perturbation parameter.
The ...
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Can we approximate a.e. invertible matrices with everywhere invertible matrices in $L^2$ sense?
Let $\mathbb{D}^n=\{ x \in \mathbb{R}^n \, | \, |x| \le 1\}$ be the closed unit ball, and let $A:\mathbb{D}^n \to \mathbb{R}^{n^2}$ be real-analytic on the interior $(\mathbb{D}^n)^o$ and smooth on ...
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Method of dominant balance and perturbation
Approximate the solutions of $$\epsilon x^4 + (x-1)^3=0$$
I can't perform a singular perturbation because if I let $\epsilon=0$ then I lose a root. My professor suggests The Method of Dominant ...
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Does the modulus of a linear operator change continuously with the operator?
Let $X,Y$ be real Banach spaces, and let $B(X,Y)$ be the space of bounded linear operators.
Given $T \in B(X,Y)$ the modulus of $T$ is defined to be
$$
\gamma(T):=\inf \{ \,\|Tx\| \, \, | \, \, d(x,\...
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Can epsilon be a matrix?
Question
In the following expression can $\epsilon$ be a matrix?
$$ (H + \epsilon H_1) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) = (E |m\rangle + \epsilon E|m_1\rangle + \...
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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 friends/...
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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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Behavior of the spectral radius of $A_1^k\ldots A_j^k$ when $k\to \infty$
First formulation of my problem :
Let $A_1,\cdots,A_j$ be hermitian definite positive matrices of dimension $n$ with all their eigenvalues in $(0;1]$. We also add the condition that $\|A_1\cdots A_j\|...
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Region of attraction of simple ODE with perturbation
There are a few nice discussions about ROA covering a few subtopics:
Region of attraction of : $x'=-y-x^3,y'=x-y^3$ via Lyapunov Function
Region of attraction and stability via liapunov&#...
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How to identify secular terms in multiscale expansion?
How can one identify secular terms while doing multiscale expansion?
For e.g. in an initial value problem, can any term where t appears can be counted as secular term?
What about; $t e^{-t}$, is it ...
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