Questions tagged [perturbation-theory]
Perturbation theory describes a range of tools and techniques to find approximate solutions to problems containing small parameters.
856
questions
1
vote
0
answers
79
views
Find the general solution of an ODE with a nonlinear perturbative term
Let's say I start with the linear differential equation
$$ y''=-y, $$
which has (for example) the two solutions $y=e^{\pm i x}$, therefore following the superposition principle the general solution is ...
- 331
0
votes
0
answers
38
views
Bound for $\left\|\nabla^2 u\right\|_{L^2(\Omega)}$
Consider the elliptic equation
$$
\begin{aligned}
-\nabla \cdot A \nabla u=f, & \text { in } \Omega, \\
u=0, & \text { on } \partial \Omega.
\end{aligned}
$$
where $\Omega$ is a bounded domain....
- 445
0
votes
1
answer
62
views
Perturbed real roots of an exponential-polynomial equation
Question: Develop three terms of the perturbation solutions to the real roots of
$$(x^3 + 2x^2 + x)e^{-x} = \epsilon,$$
identifying the scalings in the expansion sequence $\delta_0(\epsilon)x_0 + \...
- 3,965
0
votes
0
answers
48
views
Finding regular and singular roots of a cubic perturbed polynomial using rescaling
Question: Find the rescalings for the roots of
$$\epsilon^5 x^3 - (3 - 2\epsilon^2 + 10\epsilon^5 - \epsilon^6)x^2 + (30 - 3\epsilon -20 \epsilon^2 + 2\epsilon^3 + 24\epsilon^5 - 2\epsilon^6 - 2\...
- 3,965
1
vote
0
answers
45
views
Develop perturbation solutions of a cubic polynomial
Question: Develop perturbation solutions to
$$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 ...
- 3,965
0
votes
0
answers
38
views
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 ...
- 101
1
vote
1
answer
105
views
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 ...
- 39
2
votes
0
answers
101
views
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 ...
- 31
0
votes
1
answer
91
views
Asymptotics of a nonlinear PDE
Consider the partial differential equation with boundary conditions
\begin{equation}
\frac{\partial u}{\partial t} = \varepsilon \bigg( (u+1)\frac{\partial u}{\partial x}\bigg), \qquad \frac{\partial ...
- 635
1
vote
0
answers
49
views
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 ...
- 53
1
vote
0
answers
24
views
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 ...
- 391
0
votes
0
answers
38
views
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 ...
- 78
1
vote
1
answer
41
views
Roots of an equation (perturbation theory)
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
42
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
88
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