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

learn more… | top users | synonyms

0
votes
0answers
6 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 ...
1
vote
0answers
25 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, ...
0
votes
0answers
24 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 ...
7
votes
3answers
229 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 + ...
1
vote
0answers
26 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 ...
0
votes
2answers
29 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
votes
1answer
56 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 ...
1
vote
1answer
42 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 ...
0
votes
0answers
31 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 ...
0
votes
0answers
41 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
votes
0answers
49 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 ...
1
vote
0answers
25 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
votes
1answer
43 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: ...
3
votes
0answers
26 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 ...
1
vote
1answer
29 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 + ...
-1
votes
1answer
25 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 ...
0
votes
1answer
34 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: $Y'+Y=e^{- \tau\epsilon}$ ...
0
votes
1answer
29 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
votes
1answer
64 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 ...
0
votes
0answers
13 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
votes
2answers
48 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 ...
1
vote
0answers
46 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 ...
2
votes
1answer
58 views

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 ...
0
votes
0answers
55 views

Finding null geodesic that intersects a point and a time-like geodesic

I'm trying to find the right way to calculate the point of intersection of the null hypersurface emitted by a point $S$ event in a spacetime manifold with metric $g_{\mu \nu}$, with another time-like ...
0
votes
0answers
30 views

Perturbation of steady solutions to a non-linear ODE

I would appreciate if someone could please help me clarify this problem. Given the system of equations $\frac{dx}{dt}=-4x+y+x^2$, $\frac{dy}{dy}=\frac{3}{2}\alpha-y$, where $\alpha$ is an arbitrary ...
2
votes
1answer
64 views

Method of matched asymptotic expansions with Van Dyke

Consider the boundary value problem $$\varepsilon \frac{d^2y}{dx^2}+(1+x)\frac{dy}{dx}+y=0$$ subject to $y(0)=0$, $y(1)=1$, for $0 \leq x \leq 1$. Use the method of matched asymptotic expansions to ...
2
votes
1answer
82 views

Boundary value problem with rescaling

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

How to reduce the chemical reaction system $A+B \rightleftharpoons^{k_+}_{k_-} C$ with transfer using the quasi steady state assumption

Suppose we have the reaction $$A+B \rightleftharpoons^{k_+}_{k_-} C$$ within some reaction $\Omega\subset\mathbb{R}^3$, we assume this region to be well mixed and we denote the concentration of $A$ as ...
0
votes
0answers
27 views

Method of stationary phase to evaluate an integral.

I need help showing that $$I(x) = \int_{0}^{\infty} \frac{1}{1+t}e^{ix\left(\frac{t^{3}}{3} -\frac{3t^{2}}{2} +2t\right)} dt \thicksim ...
1
vote
2answers
52 views

asymptotic expansion/approximation

Find the small solution of $$y''-y\left ( 1-y^{2} \right )=0 \text{ with } y\left ( 0 \right )=\epsilon \ll 1$$ Making a pun, I decided that $$y^{3}\left ( 0 \right )\ll y\left ( 0 \right )$$ so ...
0
votes
0answers
15 views

perturbated smallest enclosing ball

Suppose we have points $p_1,p_2,..,p_n$. It can be shown that the minimal enclosing circle of these points can be found by the following optimization problem: $$ ...
1
vote
0answers
28 views

Coordinate change expansion

Suppose there is a function $f(\vec{x})$ which can be given as a perturbative expansion of the form $$f(x) = f_0 (x) + f_1 (x) + f_2 (x)+\cdots$$ where $f_n$ represents a function of order some ...
5
votes
1answer
50 views

Uniqueness of Asymptotic Matching

I'm studying the basics of Boundary Layer theory in which one makes a number of asymptotic expansions of the solution in various regions separated by a number of layers and then matches them to have ...
1
vote
1answer
27 views

2nd Order Perturbation Theory General Matrix

I Have a hamilton matrix in perturbation form ($H=H_{o}+H'$): $ H= ...
0
votes
1answer
33 views

Showing series is asymptotic

Consider the integral $$I(x) = e^{-x} \int_1^x \frac{e^t}t \, dt, \, \, x \to \infty$$ (a) By IBP repeatedly develop a series expansion of the following form, $$I(x)=\bigg( \frac1x +\frac1{x^2} ...
1
vote
1answer
30 views

Perturbation theory, why are the assumptions of the method satisfied?

I am a undergrad student interested in math taking quantum mechanics. Yesterday I was introduced to what physicists call perturbation theory, non-degenerate case. According to authors Griffiths, ...
0
votes
2answers
37 views

Relation between perturbed matrix and condition number of the matrix

If A is non‐singular but the perturbed matrix (A+δA) is singular, then show that  $$∥A∥/∥δA∥≤y $$ Where y is condition number of the matrix A. Tried for a solution The relation $$(A+δA)(x+δx)=b $$ ...
1
vote
2answers
55 views

laplace method $\sim \frac12 \sqrt{\frac{\pi}x}$

Use Laplace's method to show that $$I(x)=\int \limits_0^{\infty}\frac{e^{x(2t-t^2)}}{1+t^2}dt \, \, \sim \, \, \frac12 \sqrt{\frac{\pi}x}$$ as $x \rightarrow \infty$. So we make the top limit into ...
0
votes
1answer
72 views

Finding singular root terms by dominant balance method

Consider the problem $$ε^2x^3 + x^2 −4 = 0$$ where $0 < ε <<1$. The two regular roots of this problem are $$x = ±2−2ε^2±5ε^4$$ Find the first two non-zero terms in the asymptotic expansion for ...
0
votes
1answer
34 views

Solving equation with undetermined gauges

Working through Murdock's Pertubations 1.5.5 and 1.6.3 here... We're working to solve $x^3 - 2x^2 + x - \epsilon^2 = 0$. When trying with rescaled coordinates, I'm having trouble choosing ...
3
votes
0answers
56 views

Convergence of spectrum [closed]

Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...
0
votes
2answers
61 views

Method for finding roots of perturbation singular problems

So I am confident with finding roots of regular perturbation problems, e.g. $x^{2}+2\epsilon x-4=0$. However, I am getting confused when I have a singular problem, where $\epsilon$ affects the leading ...
1
vote
2answers
54 views

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 ...
0
votes
2answers
29 views

show that the orbit represented by the function r() is an ellipse

let $r(θ)=a(1-β^2)/(1+β\cos\theta)$ representing the distance from the Sun to a planet. With $0<β<1$, show that the orbit represented by this function $r(θ)$ is an ellipse described by ...
0
votes
0answers
19 views

Stability of least isolated eigenvalue under positive perturbation

This would be a useful theorem. Have you seen it anywhere? $\mathbf{Theorem:}$ Suppose a self-adjoint operator $H_0$ on a Hilbert space has a simple isolated least eigenvalue $0$ with separation ...
5
votes
1answer
60 views

Proof that asymptotic approximation on interval plus monotonicity (decreasing) implies uniformity?

I am having a difficult time verifying the following theorem, and hope that someone can lend me a hand. Holmes, in his book Introduction to Perturbation Methods (Second Edition) states: "Theorem ...
1
vote
1answer
63 views

Perturbation theory - Algebraic equations (Repeated roots)

Obtain a three term approximation (for $E\to 0$) of the roots of the following equation: $$x^2 + (4+E)x + (4-E) = 0$$ I understand what to do in basic cases but I've no idea what to do in this ...
0
votes
0answers
13 views

Linear perturbation of PDE with function of integrated quantity

Background As an introductory example consider the following heat equation: $$ \dot{T} = aT'' + s(T) $$ where $\dot{T}$ is the time derivative of $T$, $T'$ is a spacial derivative of $T$ and $s(T)$ ...
0
votes
1answer
38 views

Construct a single-valued function increasing arbitrarily quickly at a point $x=x_0$?

Title says it all. (How) can we construct a function (a regular function, perhaps?) whose first derivative is arbitrarily large at a point $x = x_0 < \infty$?
4
votes
1answer
49 views

Autonomous exponentially stable steady-state and small non-vanishing perturbations

My question considers if an autonomous system having a exponentially stable steady-state will continue to do so for non-vanishing small perturbations. Consider the system ...