# Tagged Questions

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

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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: $$\log\mathbb{E}\left[e^{\... 1answer 17 views ### Boundary Values and Initial conditions for Linear Stability analysis (Fluid Dynamics) Lets say we have a system of partial differential equation \Delta(x,y,t,u^{(n)})=0 (Navier-Stokes Equations) with a given stationary solution u_s(x,y) for a inviscid flow. Note that u is a 2D-... 0answers 47 views ### Perturbation ODE with boundary layer problem I encountered the following ODE and tried to solve using perturbation theory:$$y'=(1+\frac{1}{100x^2})y^2-2y+1y(1)=1,\ x\in[0,1]$$I am asked to find an approximation correct to O(\epsilon). ... 0answers 31 views ### Sensitivity of Eigenvalues with invertibte matrix Let A a matrix having a set of eigenvectors \{v_1,\ldots,v_n\} linearly independent with \{\lambda_1,\ldots,\lambda_n\} eigenvalues associated. Let \lambda eigenvalue of the perturbed matrix ... 1answer 40 views ### A ﬁrst-term expansion of the solution of the ODE Get a ﬁrst-term expansion of the solution of$$εy''+y'+ y = 0, \text{where } y(0) = 0, y'(0) = 1that is valid for large values of t. I tried the usual approach with two time scales t and s=ε^... 0answers 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... 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 &&... 0answers 25 views ### Boundary perturbation (wave equation) I have the following problem, $$u_{tt} - \Delta u = 0, \, \, \text{with } x \in R \subset \mathbb{R}^N,$$ \begin{equation*} u = 0 \, \, \text{at } \partial R. \end{equation*... 0answers 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 ... 0answers 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 ... 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 ... 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 ... 1answer 97 views ### Show R(x)=o(x^3) I gotR(x)=4! \, x^4 \int _0^{\infty} \frac{1}{(1+xt)^{5}}e^{-t} \, \, dt$$is this correct? I have no idea what to do for the last part of ii 0answers 16 views ### Method of stationary phase when the stationary point is neither minimum nor maximum. I am trying to evaluate the leading order behaviour of I(x) = \int_{0}^{1} e^{ix(t-sin(t))} dt, using the method of stationary phase. The way we have been taught to solve these types of integrals is ... 1answer 24 views ### How to know which boundary condition to use With asymptotic methods for ODEs where you have like an inner, outer region and you are given two boundary condition, how do you know which condition to use when constructing the inner/outer solution? ... 1answer 35 views ### Dominant Balance with epsilon small Consider the boundary value problem$$ε \frac{d^2y}{ dx^2} + (1 + x) \frac{dy }{dx} + y = 0$$subject to y(0) = 0, y(1) = 1, for 0 \le x \le 1, ε ≪ 1. By considering the rescaling x = x_0 + ... 1answer 22 views ### How to identify secular terms in multiscale expansion? How can one identify secular terms while doing multiscale expansion? For e.g. in an initial value problem, can any term where t appears can be counted as secular term? What about; t e^{-t}, is it ... 0answers 47 views ### Does one need a differential equation to do boundary layer theory? I'm trying to understand permutation theory and in particular the$$z=\frac{x}{\epsilon}$$substitution to get an inner solution. Here is my toy example:$$f(x)=\sqrt{x} - x^{1/\epsilon}$$and I'm ... 0answers 38 views ### Leading Order \epsilon \frac{\mathrm{d}^2y }{\mathrm{d} x^2} + 12x^{\frac{1}{3} }\frac{\mathrm{d} y}{\mathrm{d} x}+y= 0  I am required to find the leading order outer and inner solutions and then the constants by asymptotic matching. I have shown there exists a boundary layer at x=0 and hence have use the condition y(... 0answers 31 views ### What is the fundamental difference between matched asymptotic expansion and multiple scale analysis? I was wondering about the fundamental difference between the matched asymptotic expansion and the method of multiple scales. They both work extremely well for singularly perturbed problems. Do they ... 0answers 30 views ### solution of small perturbation in fluid dynamics Suppose we have a 2D flow in a narrow gap as shown in the picture The flow is governed by Navier-Stokes equation \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} = - \frac{\... 1answer 37 views ### When is a balance assumption consistent? From Asymptotic analysis and perturbation theory by Paulsen: Find the behavior of the function defined implicitly by$$x^2+xy-y^3=0$$as x\to\infty. [...] The ﬁnal case to try is to ... 1answer 85 views ### Why is \varepsilon x^5 \sim -x? I'm trying to understand what's going on in this lecture on perturbation (the link brings you to 1h 08m 12s). The original problem is to find the real root of$$x^5+x=1.$$We have inserted \... 2answers 41 views ### Algebraic approximation to differential equation Suppose I have a first order differential equation of the form: \frac{dx}{dt} = \frac{1}{\tau}f(x,t) where f(x,t) is a nonlinear function. In the limit where the time constant \tau is small, ... 0answers 19 views ### Writing down the equations for the first-order perturbation for the wave height. Plane waves satisfy Laplace's equation \nabla^2\phi = 0 for the velocity potential \phi. On the surface of the wave y=h(x,t) the boundary conditions are given as \frac{\partial \phi}{\partial t}... 1answer 44 views ### Pendulum loss/gain of time per day given : \ddot{\phi}+\frac{g}{l}\sin{\phi}=0 and max displacement 5^{\angle} Here is what i am given: The oscillations of a pendulum are described by the equation:$$\ddot{\phi}+\frac{g}{l}\sin{\phi}=0$$where \phi is the angle between the pendulum and the vertical axis, l... 0answers 43 views ### Uniform approximation. Two boundary layers? Find uniform approximation up to order O(\epsilon):$$ \begin{cases} \epsilon y''+\epsilon y' - y^2=-1-x^2 \\ y(0)=2 \\ y(1)=2 \end{cases} $$At \epsilon=0 solutions \pm \sqrt{1+x^2} don't ... 1answer 40 views ### Singular Perturbation Theory problem for slightly unstable pde I'm learning about singular perturbation theory for solving an approximate solution to a pde and I'm a little confused on how to apply it. The problem I'm trying to work on is \frac{\partial u}{\... 1answer 20 views ### Determing stretching variable in inner expansion of boundary layer problem I am studying perturbation theory, and I have a problem when reading the book "Introduction to Perturbation Methods" by M.H. Holmes. This is about boundary layer. We know when seeking inner expansion, ... 1answer 18 views ### If an operator A on a Hilbert space has compact resolvent, is Ker(\lambda-A) finite dimensional? If an operator A on a Hilbert space has compact resolvent, is Ker(\lambda-A) finite dimensional, for any \lambda in A's spectrum? P.S: What I know now is that the spectrum of A is discrete. 2answers 25 views ### Roots of a perturbed equation I'm looking to show that the equation$$\displaystyle \psi(\delta) := e^{\alpha g(\delta)} - \delta$$has a real root for \alpha sufficiently small that converges to \delta = 1 as \alpha \... 0answers 13 views ### Renormalization Group I am studying singular perturbation technique right now. Can anyone suggest introductory books on singular perturbation using renormalization group method? I have several books on perturbation theory ... 0answers 45 views ### Using the perturbation method to solve an initial value problem. I am in a differential equation class and I am doing a project involving the perturbation method and this certain question is puzzling me. The question states.... Calculate the first order ... 2answers 65 views ### Singular perturbation problem (ODE) I have found the following singular perturbation problem, \epsilon u_{xx} + |u_x|u_x + u = 0, x>0; with initial conditions, u(0) = \epsilon^2, u_x(0) = 0, where 0 < \epsilon \ll 1. ... 1answer 41 views ### Which singular perturbation method should be used for this system? Consider the system$$ \varepsilon \dfrac{dx}{dt} = -(x^3 - ax + b) \dfrac{db}{dt} = x - x_a$$where \varepsilon \ll 1. Applying regular perturbation methods isn't suitable because when \... 1answer 45 views ### Find the first two terms in the perturbation expansion of the solution I want to find the \mathcal{O}(1) and \mathcal{O}(\epsilon) terms in the pedestrian expansion y = y_0 + \epsilon y_1 + \epsilon ^2 y_2 + \dots, where y satisfies the following second order ODE:... 1answer 53 views ### Inner solution of singular perturbation problem Consider singular perturbation problem$$\epsilon \left[\frac{d}{dx}\left(h^3p\frac{dp}{dx}\right)\right]=\frac{d}{dx}(hp)p(0)=p(1)=1$$where h(x) is a positive smooth function with h(0)\ne h(... 1answer 27 views ### Lagerstrom-Cole equation Consider this boundary value problem$$\epsilon u''+uu'-u=0,\quad u(0)=A\in\mathbb{R},\quad u(1)=3.$$This differential equation is known as Lagerstrom-Cole equation. I trying to construct asymptotic ... 0answers 26 views ### convergence of nonlinear PDE with parameter We have a nonlinear PDE: L(u^{\epsilon}, \epsilon)=f. Here u^\epsilon is the unknown function, \epsilon is a parameter. When \epsilon>0, this is a hyperbolic PDE, and when \epsilon=0, ... 0answers 21 views ### WKB problem with 4 turning points? I was recently given a problem that asked to find the solvability conditions for$$\epsilon^2y''=(W(x)-E)y;\quad y\rightarrow0\text{ as }|x|\rightarrow0$$where W was some piecewise linear, W"-... 1answer 29 views ### Asymptotic Inner and Outer Expansion for a Function In the question above, I understand that to compute the outer layer you take x = O(1). Thus this means in the asymptotic expansion the first term disappears since it is so small. However, there is ... 1answer 63 views ### Boundary layers: approximately satisfying BC I am working on a boundary layer problem for a second order linear ODE. A simpler problem which I think still illustrates the issue I am having is$$\varepsilon y''-y'+y=0,y(0)=0,y(1)=1$$where \... 2answers 63 views ### Why this equation does not develop a boundary layer? The equation I am talking about is$$ \epsilon y''(x)+y(x)+1=0,y(0)=0,y(1)=1 $$The +1 is not essential as y(x) can be decomposed into 1 + y_1, but is kept here for a more direct comparison ... 1answer 32 views ### Problem with constants in Solving first order differential equation with perturbation$$y'+\lambda \ y^4 +y=0$$Where \lambda is very small and y(1)=-0.5. I've tried to solve it by Substituting: y=y_0+\lambda y_1 +\lambda^2 y_2+\cdots but I had problems with the integration ... 1answer 40 views ### Find leading order matched asymptotic expansion for \epsilon y'' + y y' - y = 0; y(0) = -1, y(1) = 0 My attempt: The outer problem for leading order term:$$y_0 y_0' - y_0 = 0$$This has solution: y_0(x) = 0 or y_0(x) = x + c. I notice that y(x) < 0 on this interval, so I assume the ... 2answers 62 views ### Solving a non linear differential equation using Perturbation$$yy'+x+\lambda \frac{x}{y}=0$$Where \lambda is very small and y(0)=R. How can we solve it with Substituting: y=y_0+\lambda y_1 +\lambda^2 y_2+\cdots Also, can we solve it without this ... 0answers 34 views ### Perturbation method that could capture the global behavior? Today I encountered this regular perturbation problem (from a habit of doing random math problems on boring classes):$$y''(x)=-\epsilon y(x)^2-\sin (x),y(0)=0,y'(0)=1 A regular perturbation ...
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 ...