# 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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### Question related to isolated eigenvalue of a Hermitian operators

This picture is from the paper of F. J. Narcowich "Narcowich, F.J., 1980. Analytic properties of the boundary of the numerical range. Indiana University Mathematics Journal, 29(1), pp.67-77."...
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### Invertibility of the product of matrices when the norm is less than 1

I am reading through a book about numerical linear algebra. It is challenging but I understand the concepts after some research. However, there is one thing that I can't figure out and it's about the ...
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### Expansion of $y=\sqrt{1+x+\frac{\varepsilon}{\varepsilon+x}}$

I'm reading "Introduction to Perturbation Methods", Second Edition, by Mark H. Holmes. In ch 2.2.5 "Matching Revisit" it explains the approach to match the outer and boundary layer ...
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### Leading order matching of $\epsilon x^py'' + y' + y = 0$

Question: The function $y(x)$ satisfies $$\epsilon x^py'' + y' + y = 0,$$ in $x\in [0,1]$, where $p<1$, subject to the boundary conditions $y(0) = 0$ and $y(1)=1$. Find the rescaling for the ...
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### Nondimensionalizing Fourth Order Differential Equation for an Elastic Beam Under Tension

I am going through the textbook A First Look At Perturbation Theory 2nd ed. by James G. Simmonds and James E. Mann Jr. Exercise 1.14 states: "An elastic beam of section modulus $EI$, resting on ...
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### Derivative of Spectral Radius of Matrix $\exp(A(t))$

I am faced with the practical problem of solving a system $$\rho(\exp(A(t))) = 1$$ numerically, where $\rho$ signifies the spectral radius of the matrix $A(t) = B+ \frac{C}{t},$ $t \in (0, \infty)$. ...
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### Can we say anything about how $\delta x$ and $\delta y$ are related to each other?

I have an equation of the form $$\frac{x^2}{y} = F(r),$$ where F is a function of $r$. This equation has a solution $r(x,y)$. Suppose we perturb this solution to $r(x + \delta x, y + \delta y)$ for ...
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### Develop perturbation solutions of a cubic polynomial

Question: Develop perturbation solutions to $$x^3 + (3+4\epsilon + \epsilon^2)x^2 + (3 + 9\epsilon + 7\epsilon^2 + 2\epsilon^3)x + 1 + 5\epsilon + 8\epsilon^2 + 5\epsilon^3 + \epsilon^4 = 0$$ finding ...
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### Perturbation with positive diagonal

Suppose I have some matrix $A$ and I perturb it by some small amount: $A \to A + \delta A$. I am interested in determining the sign of the quantity $a^\dagger \delta A a$. Here is what I know: $a$ is ...
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### Theorem (Rellich) - Perturbation Theory

I got stuck with part of a proof of: The steps are all clear to me, until it is said: "Therefore $f_w$ is the eigenvector of $A+wB$ associated with the eigenvalue lying within for all small ...
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### Poisson Equation for a perturbed sphere - both exterior and interior solutions

I am trying to solve a Heat transfer problem in a slightly ellipsoidal geometry using the Poission Equation, and Cauchy matching conditions on the boundary between the interior (which is a finite ...
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### Asymptotics of a nonlinear PDE

Consider the partial differential equation with boundary conditions \frac{\partial u}{\partial t} = \varepsilon \bigg( (u+1)\frac{\partial u}{\partial x}\bigg), \qquad \frac{\partial ...
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### Where does this factor of $\pi$ come from in the period of small oscillations about equilibrium points?

I am working through some exercises in Arnold's Mathematical Methods of Classical Mechanics book, specifically the second problem on page 20. For context, $T(E)$ is the period of motion along a closed ...
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### Perturbations of an integrable system with no resonant tori

I think the following is probably a trivial application of KAM theory, which I know little about, so I am hoping someone can give me an answer pointing me in the right direction. Suppose I have a ...
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### Number of distinct discrete eigenvalues of a self adjoint operator after compact perturbation increases.

Let $T$ be a self adjoint operator on a complex separable Hilbert space $H$. Let $K$ be self adjoint compact operator. So, $T+K$ is also self adjoint operator. I want to know for what $K$ the number ...
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Consider $xe^{x-1}+x-2-\epsilon=0$ Assume the solution can be expanded in terms of $\epsilon$, i.e. $x=a_{0}+a_{1}\epsilon+a_2\epsilon^2+... (1)$ Substitute (1) into the equation, we have $(a_{0}+a_{1}... 0 votes 0 answers 33 views ### Relation between elements of essential spectrum and the eigenvalue of a self adjoint operator Relation between elements of essential spectrum and the eigenvalue of a self adjoint operator. In particular, is there any way we can say that the element of essential spectrum is an eigenvalue of ... • 78 6 votes 0 answers 82 views ### Method of Dominant Balance with high order system This question comes from Bender and Orszag's Asymptotic Methods and Perturbation Theory. I'm practicing applying the method of dominant balance to study behavior as$x\to \infty$for systems which ... • 1,106 0 votes 0 answers 22 views ### Transversality of stable and unstable manifolds in a travelling wave equation I am trying to understand the persistence of a heteroclinic orbit$(u^*,v^*,c^*)$in the FitzHugh-Nagumo equation. I use geometric singular perturbation theory as described by Jones in "Geometric ... 0 votes 1 answer 75 views ### Perturbation of a maximal dissipative operator by a non-negative self-adjoint operator. Let$A$be a maximal dissipative operator in a Hilbert space$\mathcal{H}$, and consider$B$a self-adjoint operator such that $$\langle B\xi,\xi \rangle \geq 0\ , \quad \xi\in \mathcal{H}\ .$$ Does$...
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Let $A_0:\mathcal{H} \supset \mathcal{D}(A_0) \to \mathcal{H}$ be a densely defined (by $\mathcal{D}(A_0)$ I denote a domain of an operator $A_0$), closed and self-adjoint operator acting on a ...