Questions tagged [perturbation-theory]
Perturbation theory describes a range of tools and techniques to find approximate solutions to problems containing small parameters.
856
questions
4
votes
1
answer
453
views
Perturbation Theory and Three-body Problems
I've recently been working on a perturbation theory course in my school, but to say the least, it has been entirely disappointing. I had originally taken it to try to learn more about the three-body ...
- 541
3
votes
1
answer
494
views
$\frac{dx}{dt}=-\lambda x +\epsilon x(t-a)$ series solution via Laplace method
Consider the following equation
$$\frac{dx}{dt}=-\lambda x +\epsilon x(t-a), \quad x(0)=1,\quad |\epsilon|\ll1$$
where $a$ and $\lambda$ are positive constants and $x(t-a)$ means the function $x(t)$ ...
- 4,583
3
votes
1
answer
531
views
Solving perturbed polynomial equations
Rather than asking the most general question possible, I will frame it in terms of what I believe is an illustrative example.
Let $\epsilon>0$ be a small parameter, let $a,b>0$ and $x\in [-\...
- 389
3
votes
0
answers
165
views
Stability of Orbits in Schwarzschild Spacetime
I'm looking at geodesics in the Schwarzschild geometry, and have come up against something I cannot prove. I've shown that for a particle moving on a geodesic with $r$ constant and $\theta=\pi/2$ we ...
- 5,235
1
vote
1
answer
1k
views
Perturbation Analysis for Linear Linearly-Perturbed ODEs
I have been struggling with an ostensibly simple problem, that is how to apply perturbation analysis principles on a system of linear differential equations with linear perturbation of the following ...
- 5,175
1
vote
1
answer
3k
views
Effect of normalization on eigenvectors of a matrix
I'm trying to understand the effect of certain operations on a matrix on eigenvectors and eigenvalues.
Let $X$ be a square matrix, I need to understand how eigenvectors change if;
Each column of $X$...
- 111
2
votes
1
answer
649
views
Integral expansion using the Watson's lemma
Given the integral:
$$I(s)=\int_{-a}^a \exp\{s \cos(t)\}dt$$
is it possible to find an expansion of $I(s)$ using the Watson's lemma?
Thanks in advance.
- 10.6k
1
vote
0
answers
326
views
Multiple perturbations to cubic equation
Suppose $\alpha\in(0,\frac12)$ and $\beta\in(0,\infty)$ are fixed. Initially I have $N\in\mathbb N\backslash\{0\}$ and $n\in\{0,\ldots,N\}$. I'd like to know, as a function $n$, the solution of the ...
- 221
2
votes
1
answer
275
views
Regular Perturbation Series Problem
I want to find the a 3 term perturbation solution of $$(1+x)^3 = \epsilon x\tag{i}$$
where $\epsilon \ll1$.
Direct substitution of the regular perturbation series $x = x_0 + \epsilon x_1 + \epsilon^...
- 1,494
2
votes
1
answer
2k
views
Method of matched asymptotic expansions
Consider the equation
$(x+1-\epsilon)\frac{dy}{dx}+(1-\frac{1}{4}\epsilon^2y)y=2(1-\epsilon x)$
with $y(1)=1$.
I am interested in finding an asymptotic expansion for the inner solution so I put $x=...
- 2,080
0
votes
1
answer
176
views
Perturbation problem equation
I have an equation:
$(1-\epsilon)x^2 -2x +1=0$ (regularly perturbed problem, we anticipate all roots to remain bounded when $\epsilon$ goes to $0$)
I substitute $x=\displaystyle \sum_{n\geq0}C_n \...
- 21
4
votes
0
answers
132
views
How to bound the change between optimal points when perturbing an objective function?
Let $A,B \in \mathbb{R}^{n \times n}$ be two positive semi-definite matrices and let $a > 0$ be a constant.
Consider the following maximization problem
$$
\max_{x \in \mathbb{R}^n, \gamma}\ x^T \...
- 897
2
votes
1
answer
102
views
How to relate the following problem to known results on the eigenvectors?
Let $A \in \mathbb{R}^{n \times n}$ be a positive definite matrix and let $a > 0$ be some constant.
I am interested in the following quantity
$$
\hat x = {\rm arg} \max_{x \in R^n}\ x' \left( \frac{1}...
- 897
2
votes
1
answer
572
views
differential equations
I am trying to work through an example from Jordan and Smiths book Nonlinear Ordinary Differential Equations. It's example $6.1$ on page $195$. The question reads:
Obtain an approximate solution ...
- 1,473
5
votes
2
answers
1k
views
The WKB method: motivation
Given a differential equation of the form
$\epsilon \frac{d^ny}{dx^n} + \sum_{k=0}^{n-1} a_k(x)\frac{d^ky}{dx^k}=0$
Then the WKB method says to choose the ansatz $y\sim exp({\frac{i\phi(x)}{\...
- 2,909