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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Why is $y = \sin \left(\frac{y^s}{t} - k \right)$ outstandingly approximated by $y = - \sin k$, even for large $k$?
Equations of the form $$y = \sin \left(\frac{y^s}{t} - k \right)$$ are surprisingly well approximated as $$y = - \sin k$$ for a large range of $s$ $t$ and $k$ values:
Why? I understand why this is so ...
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How to perform the singular perturbation analysis for this reaction-diffusion PDE?
Suppose I want to determine the transient solution of the following IBVP involving a nonlinear reaction-diffusion PDE:
$\partial u(x,t)/\partial t = D \partial ^2 u(x,t)/\partial x^2 - kR(u(x,t)) $
$u(...
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Interpretation of an equation arising in matrix perturbation on the inner product of eigenvectors, weighted by eigengaps
I have a question about an equation that is so simple that I feel like it should have a name and be analyzed, but I can't find a reference for it, so I am hoping someone here has seen this before. I ...
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Limit case of Bernstein's inequalities for Markov chain with spectral gap
Context:
Let $\pi$ be a (potentially continuous) probability distribution.
Let $\mathcal{L}^2(\pi)$ be the set of square-integrable function (real-valued) with respect to $\pi$, equipped with the ...
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analyzing the sensitivity of two matrix expressions
I'm working on analyzing the sensitivity of two matrix expressions. I'd like to formally show that one is more sensitive to perturbations in a covariance matrix C than the other.
We are given:
$$\...
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Dominant balance in finding a uniform approximation to a BVP
My question comes from Example 3.11 (p.184) of Applied Mathematics 4th ed by J. David Logan. The example is on finding a uniform approximation to a boundary value problem. Here's the relevant snippet ...
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matched asymptotic expansion for the ODE $u_t = 1+\varepsilon/\log(u)$
We consider the ODE with $t>0$
$$ u_t = 1+\frac{\varepsilon}{\log(u)}, \qquad u(0)=k, $$
where $0<u<1,$ which we analyse for $\varepsilon\ll1$. A figure of the numerical solution is attached ...
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2nd order expansion alternative $f(x)|_{x=a}\approx f(a)+f'(a)(x-a)+f''(a)(1-\cos(x-a))$: Would it be better?
2nd order expansion alternative $f(x)|_{x=a}\approx f(a)+f'(a)(x-a)+f''(a)(1-\cos(x-a))$: Would it be better?
Working in the following question (5th added later) I got this idea it show to work better ...
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Perturbation to Singular Values & Vectors
Consider the following singular value decomposition of a matrix $Y$:
$$
Y = U \Sigma V^\dagger
$$
And now consider perturbing $Y \to Y + \delta Y$. As a result, the singular value decomposition also ...
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Is Tensor Rank Decomposition Continuous?
It is known that for matrices, under the correct conditions, the eigenvalues of the matrix are continuous when considered as functions of some perturbation. See for example, here . Similarly, under ...
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Comparison of exact and perturbed limits of solving a PDE
I have the following 'nonlinear heat equation' where I am trying to compare the exact solution with the answer which one would get using lowest-order perturbation theory:
$$ {\frac {\partial }{\...
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Is this a concidence, or a general result in perturbation theory with degeneracy?
Today, a smart girl want to solve
$$x^2-4.01x+4=0$$
perturbatively
We know $x=2$ is the degenerate solution of
$$x^2-4x+4=(x-2)^2=0$$
If we assume $x=2+\epsilon$ for small $\epsilon$ is our solution, ...
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Analytic perturbation of eigenvalues
Let $(-\delta,\delta)\ni t\mapsto A(t)$ be analytic, where $A(t)$ is a self-adjoint operator on a finite-dimensional Hilbert space $\mathcal{H}$. According to Rellich's theorem, there exist analytic ...
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Lebesgue measure of the set of invariant tori in KAM theorem
Fasano & Marmi Analytical Mechanics, states the K.A.M. theorem in section 12.6, page 528 as follows:
Theorem 12.12 (KAM) Consider a quasi-integrable Hamiltonian system
$$H(J, χ, ε) = H_0(J) + εF(...
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Higher-Order Approximations for the Long-Term Dynamics of a Coupled Oscillator with Slowly Varying Parameters
I am studying the multiscale dynamics of a coupled two-degree-of-freedom oscillator with a slowly varying parameter. This system arises in my research on adiabatic invariants and is described by the ...
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