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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find an approximate solution, up to the order of epsilon
The question is to find an approximate solution, up to the order of epsilon of following problem.
$$y'' + y+\epsilon y^3 = 0$$
$$y(0) = a$$
$$y'(0) = 0$$
I tried to solve the given problem using ...
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Find location and width of boundary layer
Consider the boundary value problem $$\varepsilon (2y+y'')+2xy'-4x^2=0$$ subject to $y(-1)=2$ and $y(2)=7$, for $-1 \leq x \leq 2$, $\varepsilon \ll 1$.
Find the location and width of the boundary ...
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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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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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Books on Perturbation Methods
I am having problems finding descent books on perturbation methods. I am looking for a book which covers; asymptotic expansions, matched Asymptotic expansions, Laplace's Method, Method of steepest ...
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How to solve an ODE with $y^{-1}$ term
My major is not Mathematics, but I came across the following ODE for $y(x)$:
$$\left(y^3y^{\prime\prime\prime}\right)^\prime+\frac{5}{8}xy^\prime-\frac{1}{2}y+\frac{a}{y}=0,$$
where the prime denote ...
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Boundary layer problem
This question is taken from Bender & Orszag "perturbation methods"
$y' = (1 + X^{-2}/100)y^2 - 2y + 1$ ,$y(1)=1$
first we can see that if we set $\epsilon=100x^{2}$ we can translate the above to ...
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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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Can we choose smoothly the singular vectors of a matrix?
Let $X$ be the space of all real $n \times n$ matrices, with strictly negative determinant, and pairwise distinct singular values. $X$ is an open subset of the space of all real square matrices. Is ...
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Asymptotic expansions for the roots of $\epsilon^2x^4-\epsilon x^3-2x^2+2=0$
I'm trying to compute the asymptotic expansion for each of the four roots to the following equation, as $\epsilon \rightarrow 0$:
$\epsilon^2x^4-\epsilon x^3-2x^2+2=0$
I'd like my expansions to go ...
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Approximating definite integral over infinitesimal interval (reformulated)
Pursuant to helpful comments by user254433, I have decided to take another swing at this problem while reformulating it with a simplified example.
(Reformulated) General Problem: Generally speaking, ...
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Matrix inverse of $A + \epsilon I$, where $A$ is invertible
Let $A$ be a square invertible matrix, and $\epsilon$ a small positive quantity. To first-order in $\epsilon$, what is the inverse of $A + \epsilon I$, where $I$ is the identity matrix?
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Perturbation Methods-Multiscale expansions
I was looking at some of the problems from the chapter Multiple-scale expansion in Introduction to Perturbation Methods by Mark H. Holmes. I came across this question to find the first term expansion:
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Burgers' equation / Perturbation theory
Let the Burgers' equation:
$$u_t+u u_x=\epsilon u_{xx} ,\qquad x\in\mathbb{R}, \; t>0.$$
With the initial condition: $u(x,0)=1_{\{x<0\}}(x).$
Studying it in the theory of perturbation:
I did ...
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Asymptotic expansion of exp of exp
I am having difficulties trying to find the asymptotic expansion of $I(\lambda)=\int^{\infty}_{1}\frac{1}{x^{2}}\exp(-\lambda\exp(-x))\mathrm{d}x$ as $\lambda\rightarrow\infty$ up to terms of order $O(...
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