Perturbation theory describes a range of tools and techniques to find approximate solutions to problems containing small parameters.

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Find the inner solution

I need to show that the leading order inner solution is given by the below. Thus far, I have rescaled and showed the boundary layer is of order $\epsilon^{\frac{3}{4}}$. Hence at leading order I ...
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19 views

How to show a function is $<<$ to an integral [on hold]

For those integrals (integrating with respect to $s$), say $A(x)$, with $x\rightarrow 0$ that you have to perform IBP but the number of iterations will never end. After a number of iterations, we have ...
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17 views

Justify Why Boundary Layer Exists at x=0

With part a of this question (I asked about latter parts before), does it suffice to find the outer solution and show that since both boundary conditions cannot be met, then there exists a boundary ...
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1answer
89 views

Show $R(x)=o(x^3)$

I got $$R(x)=4! \, x^4 \int _0^{\infty} \frac{1}{(1+xt)^{5}}e^{-t} \, \, dt$$ is this correct? I have no idea what to do for the last part of ii
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16 views

Method of stationary phase when the stationary point is neither minimum nor maximum.

I am trying to evaluate the leading order behaviour of $I(x) = \int_{0}^{1} e^{ix(t-sin(t))} dt$, using the method of stationary phase. The way we have been taught to solve these types of integrals is ...
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1answer
22 views

How to know which boundary condition to use

With asymptotic methods for ODEs where you have like an inner, outer region and you are given two boundary condition, how do you know which condition to use when constructing the inner/outer solution? ...
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34 views

Dominant Balance with epsilon small

Consider the boundary value problem $$ε \frac{d^2y}{ dx^2} + (1 + x) \frac{dy }{dx} + y = 0$$ subject to $y(0) = 0$, $y(1) = 1$, for $0 \le x \le 1$, $ε ≪ 1$. By considering the rescaling $x = x_0 + ...
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1answer
15 views

How to identify secular terms in multiscale expansion?

How can one identify secular terms while doing multiscale expansion? For e.g. in an initial value problem, can any term where t appears can be counted as secular term? What about; $t e^{-t}$, is it ...
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47 views

Does one need a differential equation to do boundary layer theory?

I'm trying to understand permutation theory and in particular the $$z=\frac{x}{\epsilon}$$ substitution to get an inner solution. Here is my toy example: $$f(x)=\sqrt{x} - x^{1/\epsilon}$$ and I'm ...
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37 views

Leading Order $\epsilon \frac{\mathrm{d}^2y }{\mathrm{d} x^2} + 12x^{\frac{1}{3} }\frac{\mathrm{d} y}{\mathrm{d} x}+y= 0 $

I am required to find the leading order outer and inner solutions and then the constants by asymptotic matching. I have shown there exists a boundary layer at x=0 and hence have use the condition$ ...
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30 views

What is the fundamental difference between matched asymptotic expansion and multiple scale analysis?

I was wondering about the fundamental difference between the matched asymptotic expansion and the method of multiple scales. They both work extremely well for singularly perturbed problems. Do they ...
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23 views

solution of small perturbation in fluid dynamics

Suppose we have a 2D flow in a narrow gap as shown in the picture The flow is governed by Navier-Stokes equation $\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} = - ...
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1answer
37 views

When is a balance assumption consistent?

From Asymptotic analysis and perturbation theory by Paulsen: Find the behavior of the function defined implicitly by $$x^2+xy-y^3=0$$ as $x\to\infty$. [...] The final case to try is to ...
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1answer
82 views

Why is $\varepsilon x^5 \sim -x$?

I'm trying to understand what's going on in this lecture on perturbation (the link brings you to 1h 08m 12s). The original problem is to find the real root of $$x^5+x=1.$$ We have inserted ...
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2answers
39 views

Algebraic approximation to differential equation

Suppose I have a first order differential equation of the form: $\frac{dx}{dt} = \frac{1}{\tau}f(x,t)$ where $f(x,t)$ is a nonlinear function. In the limit where the time constant $\tau$ is small, ...
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18 views

Writing down the equations for the first-order perturbation for the wave height.

Plane waves satisfy Laplace's equation $\nabla^2\phi = 0$ for the velocity potential $\phi$. On the surface of the wave $y=h(x,t)$ the boundary conditions are given as $\frac{\partial \phi}{\partial ...
2
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1answer
43 views

Pendulum loss/gain of time per day given : $\ddot{\phi}+\frac{g}{l}\sin{\phi}=0$ and max displacement $5^{\angle}$

Here is what i am given: The oscillations of a pendulum are described by the equation: $$\ddot{\phi}+\frac{g}{l}\sin{\phi}=0$$ where $\phi$ is the angle between the pendulum and the vertical axis, ...
2
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0answers
42 views

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
39 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 ...
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1answer
18 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
17 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
25 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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13 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
51 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 $0 < \epsilon \ll 1$. ...
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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
38 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 ...
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1answer
53 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
27 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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26 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 $\epsilon=0$, ...
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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
25 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
59 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 ...
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2answers
62 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
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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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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
65 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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38 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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52 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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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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20 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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35 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
51 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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254 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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34 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 ...