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

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Analytical Solution to Coupled Nonlinear ODEs

I am looking to solve several coupled nonlinear ODEs like this one: $\hspace{20mm} \frac{d x(t)}{dt} = C_1 \cdot x(t) + C_2 \cdot y(t) + C_3\cdot (x(t)^2 + y(t)^2) x(t),$ $\hspace{20mm} \frac{d y(t)...
3
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1answer
60 views

(Possible) Misunderstanding of perturbation method for finding solution of polynomial equation?

This is the strange moment that I get when I solve this equation: $$ \frac{w^4}{4} - \frac{w^3}{3} = \varepsilon, $$ where $\varepsilon$ is a small parameter. If I plot the graph $ w \mapsto \frac{w^...
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0answers
15 views

Cumulants of square of Poisson distribution

I'm writing up a derivation of an expression for mutual information between weakly interacting Poisson processes. I'm running into an expression that looks like this: $$\log\mathbb{E}\left[e^{\...
1
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1answer
17 views

Boundary Values and Initial conditions for Linear Stability analysis (Fluid Dynamics)

Lets say we have a system of partial differential equation $\Delta(x,y,t,u^{(n)})=0$ (Navier-Stokes Equations) with a given stationary solution $u_s(x,y)$ for a inviscid flow. Note that $u$ is a 2D-...
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Perturbation ODE with boundary layer problem

I encountered the following ODE and tried to solve using perturbation theory: $$y'=(1+\frac{1}{100x^2})y^2-2y+1$$ $$y(1)=1,\ x\in[0,1]$$ I am asked to find an approximation correct to $O(\epsilon)$. ...
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31 views

Sensitivity of Eigenvalues with invertibte matrix

Let $A$ a matrix having a set of eigenvectors $\{v_1,\ldots,v_n\}$ linearly independent with $\{\lambda_1,\ldots,\lambda_n\}$ eigenvalues associated. Let $\lambda$ eigenvalue of the perturbed matrix $...
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1answer
40 views

A first-term expansion of the solution of the ODE

Get a first-term expansion of the solution of $$εy''+y'+ y = 0, \text{where } y(0) = 0, y'(0) = 1$$ that is valid for large values of $t$. I tried the usual approach with two time scales $t$ and $s=ε^...
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20 views

Continuous Null Spaces

Consider a matrix $B(x) \in \mathbb{C}^{N \times M}$, $M<N$, which is a function of a single variable $x \in I$, where $I = [a,b]$ and $a < b$. It turns out that $B(x)$ is full rank for $x \in I$...
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1answer
103 views

Can I integrate an asymptotic expression?

Suppose that $y(x; \epsilon)$ is a real-valued function of $x \in [a,b] \subset\mathbb{R}$ depending on a real parameter $\epsilon$, and that \begin{align} \int_a^b dx \ y(x; \epsilon) =& 1 &&...
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25 views

Boundary perturbation (wave equation)

I have the following problem, \begin{equation} u_{tt} - \Delta u = 0, \, \, \text{with } x \in R \subset \mathbb{R}^N, \end{equation} \begin{equation*} u = 0 \, \, \text{at } \partial R. \end{equation*...
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61 views

Perturbation of the principal eigenvector of a PSD matrix

Setting: I have a $n \times n$ PSD matrix $A$ and $\tilde{A}=A+E$ be its symmetric perturbation such that $\|E\|_2=\epsilon.$ Let $(\lambda,u)$ be the principal eigenvalue, eigenvector pair of $A$ ...
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33 views

I'm comparing two different methods for solving the Navier Stokes equations. Why are my velocity results so different?

I want to use a code for modeling 2-D fluid flow out of a tank to understand a chemical process. The code has never been used for pressure boundary conditions, so I want to check that it works as it ...
0
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1answer
22 views

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 then ...
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0answers
20 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
97 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
24 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? ...
0
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1answer
35 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
22 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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38 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$ y(...
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31 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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30 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} = - \frac{\...
2
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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 ...
2
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1answer
85 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
41 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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19 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 t}...
2
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1answer
44 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, $l$...
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43 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
40 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 u}{\...
0
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1answer
20 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, ...
0
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1answer
18 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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45 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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2answers
65 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
41 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 $\...
2
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1answer
45 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 ODE:...
2
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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 h(...
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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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21 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, $``W"$-...
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1answer
29 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
63 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
63 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
32 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 integration ...
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1answer
40 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
62 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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0answers
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 ...