Perturbation theory is a tool to find approximate solutions to equations that contain small parameters.

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Uniform approximation. Two boundary layers?

Find uniform approximation up to order $O(\epsilon)$: $$ \begin{cases} \epsilon y''+\epsilon y' - y^2=-1-x^2 \\ y(0)=2 \\ y(1)=2 \end{cases} $$ At $\epsilon=0$ solutions $\pm \sqrt{1+x^2}$ don't ...
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1answer
32 views

Singular Perturbation Theory problem for slightly unstable pde

I'm learning about singular perturbation theory for solving an approximate solution to a pde and I'm a little confused on how to apply it. The problem I'm trying to work on is $\frac{\partial ...
0
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1answer
16 views

Determing stretching variable in inner expansion of boundary layer problem

I am studying perturbation theory, and I have a problem when reading the book "Introduction to Perturbation Methods" by M.H. Holmes. This is about boundary layer. We know when seeking inner expansion, ...
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1answer
16 views

If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional?

If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional, for any $\lambda$ in A's spectrum? P.S: What I know now is that the spectrum of A is discrete.
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2answers
24 views

Roots of a perturbed equation

I'm looking to show that the equation $$\displaystyle \psi(\delta) := e^{\alpha g(\delta)} - \delta$$ has a real root for $\alpha$ sufficiently small that converges to $\delta = 1$ as $\alpha ...
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12 views

Renormalization Group

I am studying singular perturbation technique right now. Can anyone suggest introductory books on singular perturbation using renormalization group method? I have several books on perturbation theory ...
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42 views

Using the perturbation method to solve an initial value problem.

I am in a differential equation class and I am doing a project involving the perturbation method and this certain question is puzzling me. The question states.... Calculate the first order ...
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1answer
46 views

Singular perturbation problem (ODE)

I have found the following singular perturbation problem, $\epsilon u_{xx} + |u_x|u_x + u = 0$, $x>0$; with initial conditions, $u(0) = \epsilon^2$, $u_x(0) = 0$, where $0 < \epsilon \ll 1$. ...
3
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1answer
39 views

Which singular perturbation method should be used for this system?

Consider the system $$ \varepsilon \dfrac{dx}{dt} = -(x^3 - ax + b)$$ $$ \dfrac{db}{dt} = x - x_a$$ where $\varepsilon \ll 1$. Applying regular perturbation methods isn't suitable because when ...
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1answer
36 views

Find the first two terms in the perturbation expansion of the solution

I want to find the $\mathcal{O}(1)$ and $\mathcal{O}(\epsilon)$ terms in the pedestrian expansion $y = y_0 + \epsilon y_1 + \epsilon ^2 y_2 + \dots$, where $y$ satisfies the following second order ...
2
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1answer
52 views

Inner solution of singular perturbation problem

Consider singular perturbation problem $$\epsilon \left[\frac{d}{dx}\left(h^3p\frac{dp}{dx}\right)\right]=\frac{d}{dx}(hp)$$ $$p(0)=p(1)=1$$ where $h(x)$ is a positive smooth function with $h(0)\ne ...
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1answer
26 views

Lagerstrom-Cole equation

Consider this boundary value problem $$\epsilon u''+uu'-u=0,\quad u(0)=A\in\mathbb{R},\quad u(1)=3.$$ This differential equation is known as Lagerstrom-Cole equation. I trying to construct asymptotic ...
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25 views

convergence of nonlinear PDE with parameter

We have a nonlinear PDE: $L(u^{\epsilon}, \epsilon)=f$. Here $u^\epsilon$ is the unknown function, $\epsilon$ is a parameter. When $\epsilon>0$, this is a hyperbolic PDE, and when $\epsilon=0$, ...
3
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0answers
18 views

WKB problem with 4 turning points?

I was recently given a problem that asked to find the solvability conditions for $$\epsilon^2y''=(W(x)-E)y;\quad y\rightarrow0\text{ as }|x|\rightarrow0$$ where $W$ was some piecewise linear, ...
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1answer
24 views

Asymptotic Inner and Outer Expansion for a Function

In the question above, I understand that to compute the outer layer you take x = O(1). Thus this means in the asymptotic expansion the first term disappears since it is so small. However, there is ...
4
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1answer
57 views

Boundary layers: approximately satisfying BC

I am working on a boundary layer problem for a second order linear ODE. A simpler problem which I think still illustrates the issue I am having is $$\varepsilon y''-y'+y=0,y(0)=0,y(1)=1$$ where ...
3
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2answers
61 views

Why this equation does not develop a boundary layer?

The equation I am talking about is $$ \epsilon y''(x)+y(x)+1=0,y(0)=0,y(1)=1 $$ The $+1$ is not essential as $y(x)$ can be decomposed into $1 + y_1$, but is kept here for a more direct comparison ...
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1answer
30 views

Problem with constants in Solving first order differential equation with perturbation

$$y'+\lambda \ y^4 +y=0$$ Where $\lambda$ is very small and $y(1)=-0.5$. I've tried to solve it by Substituting: $y=y_0+\lambda y_1 +\lambda^2 y_2+\cdots$ but I had problems with the ...
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1answer
36 views

Find leading order matched asymptotic expansion for $\epsilon y'' + y y' - y = 0; y(0) = -1, y(1) = 0$

My attempt: The outer problem for leading order term: $$y_0 y_0' - y_0 = 0$$ This has solution: $y_0(x) = 0$ or $y_0(x) = x + c$. I notice that $y(x) < 0$ on this interval, so I assume the ...
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2answers
60 views

Solving a non linear differential equation using Perturbation

$$yy'+x+\lambda \frac{x}{y}=0$$ Where $\lambda$ is very small and $y(0)=R$. How can we solve it with Substituting: $y=y_0+\lambda y_1 +\lambda^2 y_2+\cdots$ Also, can we solve it without this ...
2
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34 views

Perturbation method that could capture the global behavior?

Today I encountered this regular perturbation problem (from a habit of doing random math problems on boring classes): $$y''(x)=-\epsilon y(x)^2-\sin (x),y(0)=0,y'(0)=1$$ A regular perturbation ...
2
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1answer
73 views

Eigenvalue-related statements

How can I prove that the following statements are equivalent? $\lambda$ is an eigenvalue of $A+\delta A$, where $\|\delta A\|_{2}\leq \epsilon$ $\exists u\in \mathbb{C}^{m}$ such that $\|(A-\lambda ...
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1answer
64 views

Series Expansion within a fraction

I'm currently reading "The cumulant lattice Boltzmann equation in three dimensions: Theory and validation" from Geier et. al. and have some trouble in a proof. We have given multivariat cumulants ...
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33 views

Is there a connection between the Davis-Kahan sin theta Theorem and the smallest nonzero singular value of a symmetric Matrix

I'm trying to understand a step in a proof in the paper "Consistency of Spectral Clustering in stochastic Block Models" from J.Lei and A.Rinaldo. In Lemma 5.1 they use the Davis Kahan-sin $\Theta$ ...
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51 views

perturbation theory on multiple variables

I'm facing what I expect will turn out to be quite a simple question in first order perturbation theory but so far I am very confused. I have a system of equations which relates functions g(x) and ...
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1answer
53 views

Perturbation of the Upper Incomplete Gamma Function

The Upper Incomplete Gamma function, for $t \in \mathbb{R}$, is defined as: \begin{equation} \Gamma(α,β)=\int_{β}^{\infty}t^{α-1}e^{-t}dt \end{equation} For the problem which I am studying it takes ...
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37 views

Book Recommendations on Perturbation Theory

I am interested in studying Quantum Electrodynamics and figure I should begin by learning Perturbation theory and Asymptotic expansions. If anyone could recommend some books, that would be very ...
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19 views

Perturbation theory for a symetric rank-one update

I know perturbation theory of the eigenspectrum/singular value decompostion of a symetric matrix $A$ under a symetric perturbation $E$, that besides being symetric has no other structure. Is there ...
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11 views

Bounding the perturbation between eigenvectors

Can somebody explain this part of the proof of a deduction from the Davis-Kahan $\sin \theta$ theorem? I understand how to get from: $||P_{u_1} - P_{v_1}|| \le \epsilon$ to $||P_{u_1}v_1 - v_1|| \le ...
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33 views

Spectral norm of lower triangular perturbation

Suppose $A\in R^{n×n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. \begin{equation} A=I+L \end{equation} All diagonal entries of $L$ are equal to $0$, so that, ...
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1answer
49 views

Bounding lower triangular perturbation

Suppose $A\in R^{n\times n}$ is a matrix equal to sum of the Identity matrix and a lower triangular matrix $L$. Diagonal entries of $L$ are $0$. \begin{equation} A=I+L \end{equation} Define spectral ...
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248 views

Can epsilon be a matrix?

Question In the following expression can $\epsilon$ be a matrix? $$ (H + \epsilon H_1) ( |m\rangle +\epsilon|m_1\rangle + \epsilon^2 |m_2\rangle + \dots) = (E |m\rangle + \epsilon E|m_1\rangle + ...
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32 views

Spectral radius of perturbed bipartite graphs

I am looking into how perturbation(s) on a bipartite graph affect its spectrum (specifically its spectral radius or largest eigenvalue). Actually I'm not exactly looking into bipartite but the ...
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33 views

Determine perturbation around saddles for 2D system

Consider the system $$ \dot{x} = \mu + x^2 - xy\\ \dot{y} = y^2 - x^2 -1 $$ with $\mu \neq 0$ and small. I need to determine the Taylor/perturbation expansion of the two saddles $a^+$ and $a^-$ up to ...
3
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1answer
63 views

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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1answer
46 views

The differential of a symmetric matrix in terms of its eigen-decomposition

Given a (square) symmetric matrix $A$, I would like to write its first order perturbation in terms of its eigenvalue decomposition $$A=Q\Lambda Q^T$$ I'm thinking about this problem in terms of ...
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32 views

averaging of differential equations with periodic coefficients

Consider the scalar ODE $$\dot x = -a(t)x,$$ where $a(t) = a(t+T)$, and the corresponding averaged ODE $$\dot {\bar x} = -\bar a \bar x,$$ where $\bar a = 1/T \int_0^{T}a(t)dt$. The question is how to ...
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48 views

Eigenvalue perturbation of a block companion matrix

Consider the following matrix $$F(\epsilon )= \begin{pmatrix} -H_1 & I & O \\ -H_2-\epsilon^2 K & -\epsilon D & I \\ -H_3 & O & O \end{pmatrix}$$ Where, $I$ and $O$ are ...
2
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0answers
52 views

Solving for a $v$ in $\sum a_i e^{b_i (z^2+d_i) + c_i v}$

I have an equation in complex domain, $$P(e^u,e^v)=\sum_{i=1}^{N} a_i e^{b_i u + c_i v}=0 \;\;\;\text{(A)}$$ and by redefining, at the roots (I'm only showing work for one root), the first ...
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0answers
28 views

Perturbation of a linear homogeneous equation system

Let $A$ be a $n\times(n+1)$ matrix, full row rank. Let $\tilde A=A+\Delta A$ be a perturbation of $A$, again with full row rank. I am interested what is known about bounds on the angle between the ...
0
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1answer
56 views

Solve this differential equation using perturbation method [duplicate]

Consider the following problem: $$\frac{\text{d}^2y}{\text{d}t^2} + y + \epsilon\ y^3 = 0\ \ \ \ \text{for}\ \ t \geq 0, y(0) = A, y'(0)=0$$ Compute an approximate solution by substitution method: ...
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28 views

How does one in general analyze the convergence of the following series?

The following question is inspired in the following videos: https://www.youtube.com/playlist?list=PL43B1963F261E6E47 Say one has a general second order linear differential equation $y''+Qy=0$ for ...
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1answer
33 views

Taylor expanding $f(y+\epsilon U1 + {\epsilon}^{2} U2,t,\epsilon)$ in $\epsilon$

How would one Taylor expand $\epsilon f(y+\epsilon U1 + {\epsilon}^{2} U2,t,\epsilon)$ in $\epsilon$? Somehow the professor obtained the first few terms to be: $\epsilon f(y+\epsilon U1 + ...
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1answer
41 views

Inner solution and outer solution question [closed]

Find the 2 term outer solution and one term inner solution for (using matched expansions) $$ (1+\epsilon)x^2y'=\epsilon((1-\epsilon)xy^2-(1+\epsilon)x+y^3+2\epsilon y^2), \space \space \space \space ...
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1answer
40 views

Single Perturbation inner approximation characteristic equation.

I'm given $\epsilon y'=y=e^{-t}$. I went through and found the outer approximation and then proceeded to find the inner approx. I rescaled and balanced the equation to get: $Y'+Y=e^{- \tau\epsilon}$ ...
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1answer
41 views

Finding a composite solution to an ODE (boundary layer problem)

Given $\epsilon \frac{d^2u}{dt^2}-a(t)\frac{du}{dt}+b(t)u=0$, where $a(t)>0$, $u(0)=1$, $u(1)=1$, and assuming that the boundary layer is at $t=1$, and the boundary layer variable is ...
2
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1answer
84 views

Singular Perturbation Approx. for $\epsilon y'' + \frac{2 \epsilon}{t} y'-y=0$

Use singular perturbation techniques to find the leading order uniform approximation to the solution to the boundary value problem $$\epsilon y'' + \frac{2 \epsilon}{t} y'-y=0$$ $0<t<1$ and ...
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14 views

campbell-baker-hausdorff with one small matrix

Let $A$ and $B$ be non-commuting matrices. (Probably for the purposes of this question it is fine to assume that they are Hermitian.) I am interested in computing $\log(e^{A} e^{t B})$ in a formal ...
2
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2answers
58 views

Exact solution of Second order ODE

We have the second order differential equation $\epsilon \dfrac{d^{2}y}{dx^{2}} + \dfrac{dy}{dx} +y = 0$ with boundary values $y(0)=0,\, \, \, y(1)=1$. I would like to get the exact solution in ...
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59 views

Poincaré perturbation method on Mathieu's equation

I need to solve the IVP $\frac{d^2u}{dt^2}+[\omega^2+2\epsilon \cos(2t)]u=0$; $u(0)=1$, $\frac{du}{dt}(0)=0$ using the Poincaré method of perturbation. However, I have no idea how to start. We weren't ...