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17 views

Perturbation of resolvent

Let $H$ be a Hilbert space.$A$ is densely defined closed operator and $B$ is $A$-bounded:$||Bx||\leq a||Ax||+b||x||$.Let $\lambda\in\rho(A)$,such that ...
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0answers
19 views

perturbation solution of two singular ODEs

I need some help with solving the following system of ODEs: $$\epsilon \frac{dx}{dt}=Ay +ABx(1-y)$$ $$\epsilon\frac{dy}{dt}=Bx(1-y)-y-\epsilon y$$ I'm confused by the fact that both equation are ...
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1answer
34 views

Stuck on perturbation theory for finding a root of polynomial, with rescaling

I have been given the polynomial $$\epsilon x^3+x-2=0,$$ where epsilon is very small and I need to find the roots using perturbation theory. So far I have found the first root, 2, using the direct ...
5
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1answer
77 views

Perturbation theory PDEs

I have the solution of a PDE of the form: $$ \Delta \Psi(r,\theta, \phi) = k \Psi(r,\theta,\phi)$$ on a set $\mathbb{R}^3 \backslash B(0,R)$. Hence, the actual solution is known there! Regarding ...
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0answers
28 views

Method of dominant balance and perturbation theory

We know perturbation theory express the desired solution of differential equations in terms of a formal power series in some "small" perturbation parameters: $y=y_0+\epsilon ^1 y_1+\epsilon ^2 ...
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1answer
34 views

non-homogeneous constant co-efficient 2nd order linear differential equation

I am doing a perturbation theory question and am having trouble with the (seemingly simple!) differential equation method of undetermined coefficients... I have reduced my given system so that now I ...
2
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2answers
46 views

Introductory text on perturbation theory for dynamical systems

I am working on my thesis which is about oscillations and as far as I realise I need to know about perturbation theory and methods in solving differential equations, specifically dynamical systems. A ...
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0answers
18 views

Outer Expansion of $\frac{e^{x/\epsilon}}{x}$

The whole function is $f(x,\epsilon)=\frac{e^{x/\epsilon}}{x}+\frac{\sin(x)}{x}-\coth(x)$ I think I've found the inner expansion by setting $x=\epsilon y$ $F(y,\epsilon)=1+\frac{e^y-1}{\epsilon ...
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0answers
30 views

Strong versus weak coupling expansion to solve hard problems

For the quintic equation $$ x^5 + x = 1 $$ it can be seen that when taking the strong coupling limit to solve $$ x^5 + \epsilon x = 1 $$ perturbatively, summing the terms of all orders in ...
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0answers
25 views

Perturbation of eigenvalues

I am looking at a certain operator, that is an integral operator which is Hilbert-Schmidt on $L^2(X,d\mu)$ to $L^2(X,d\mu)$.I want to see how its eigenvalues or singular values change as its kernel is ...
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1answer
48 views

curves bounding discs

I'm interested in the following question. Please forgive me if my question is lacking in precision. I'm not a mathematician, and need some help getting started: If I have a smooth, simple curve ...
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1answer
200 views

Asymptotic Matching for boundary layer problem

The question asks to find a global approximation to the leading order of $\epsilon$. $\epsilon y'' + xy' + \epsilon y =0$, with boundary conditions $y(0)=1,y(1)=-1$. I assumed it's a boundary layer ...
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2answers
48 views

How can I do a perturbation on this ODE?

I have an ode of the form $$\frac{dy}{dx}=y(x)+\sqrt{y(x)+\epsilon\cdot f(x)}$$ and I would like to do a perturbation up to first order $\epsilon$. My advisor gave me an example for a simpler ODE, ...
2
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1answer
53 views

Derive a perturbation of period $2\pi$, to order $\epsilon$

I have the following problem: In the equation $\ddot{x}+\Omega^2x+\epsilon f(x) = \Gamma \cos t$, $\Omega$ is not close to an odd integer, and $f(x)$ is an odd function of x, with expansion, $$f(a\cos ...
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0answers
13 views

Subordinate operators

Let $A$ be a linear densely defined operator on a Hilbert space $H$ and $L$ is a selfadjoint operator with discrete spectrum such that $\mathcal{D}(L) = \mathcal{D}(L)$ and $$\|Tf\| \leq M ...
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0answers
59 views

Perturbation in Maple for ODE

I am working on doing a perturbation analysis for a model of opsin delivery in the eye. The ODE is VERY complicated ${\frac {d}{dt}}\eta \left( t \right) =1/2\,{\frac {-K_{{2}}kP_{{0}}+ \eta \left( ...
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0answers
45 views

Can't match boundary conditions on a perturbation series solution to a non-linear ODE?

I'm trying to generate a naive perturbation series solution (with all associated secular terms included) to the Rayleigh equation: \begin{equation} \frac{d^2y}{dt^2} + y = \epsilon ...
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0answers
42 views

Does analytical solution of these nonliear ODE's exist?

I have following first order nonlinear ordinary differential and i was wondering if someone can suggest some method by which either i can get an exact solution or approximate and converging ...
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3answers
153 views

Laplace integral - Asymptotic expansion

$$\int_{0}^{\infty} \frac{t^{x}}{{\cosh (t)}}dt$$ I'm trying to use Laplace's method to find the leading asymptotic behavior for x>>1, but I'm having some trouble. Could someone help me? Thanks in ...
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0answers
33 views

Does exact solution for these nonlinear ODE's exist?

I have following first order nonlinear ordinary differential and i was wondering if someone can suggest some method by which either i can get an exact solution or approaximate and converging ...
0
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0answers
16 views

Help please! Perturbation method on ODE with variable gravitational field and air resistance [duplicate]

I'm a little stuck with a problem and I was hoping that you guys could help. It's not quite homework but I'll tag it as such so the answers explain what they're doing. Question: A projectile is fired ...
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0answers
71 views

perturbation question

I'm a little stuck with a problem and I was hoping that you guys could help. Question: A projectile is fired up from the earth with an initial velocity of $v_0$ upwards. Accounting for air resistance, ...
2
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2answers
160 views

Asymptotic expansion of an integral

I came up with a simpler example which illustrates the technical difficulty I have encountered in my work. Consider an integral depending on parameter $\epsilon$: \begin{equation} ...
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1answer
72 views

Analytical fixed point iteration method

I have a system of nonlinear ordinary diffferential equations and i want to use analytical fixed point iteration method. Unfortunately i could only see computational resources but i couldnt find ...
1
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1answer
258 views

Analytical solution of nonlinear ordinary differential equation

I have following first order nonlinear ordinary differential and i was wondering if you can suggest some method by which either i can get an exact solution or approaximate and converging perturbative ...
1
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1answer
91 views

Preservation of Positive-Definiteness from Small Perturbations

Let $A$ with real positive entries be a Hermitian positive definite matrix. I'm wondering if one perturbs $A$, e.g., $\hat{A}=A+\Delta A$, would the matrix still be positive definite? I'm told this is ...
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1answer
68 views

Inverse a matrix $B+\lambda C$, where $\lambda$ is variable.

In my research I need to calculate $\operatorname{Trace}(A^{-1} C)$ where $A$ is given by two large, but sparse, matrices $B$ and $C$ by $A=B+\lambda C$. I need to do this inversion many times, so ...
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3answers
187 views

Why does the Method of Successive Approximations for a Differential Equation work?

Time dependent perturbation theory in quantum mechanics is often derived using the Method of Successive Approximations for a Differential Equation. I have not seen an explanation or a more rigorous ...
2
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1answer
82 views

Effect of change in a column of a matrix in its determinant and eigenvalue

I have matrix $R$ which based on it, another matrix $H$ is computed using $$ H(R) =\frac { (R R^T+ sI)^{1/2} } { trace(R R^T+ sI)^{1/2}} $$ $s$ is a small value. Now, one column of the matrix $R$ ...
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0answers
65 views

Absolute error bound for a perturbed linear equation

Consider a perturbed linear equation \begin{equation} (A+\delta A)(x + \delta x)=(b+\delta b) \end{equation} where $Ax=b$. It is well-known that a relative error bound for the solution $x+\delta x$ ...
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0answers
9 views

Etablishing a bound for derivativces

I am trying to show that $$|v^{(k)}_\epsilon|\leq C(1+\epsilon^{-(k-2)/2})$$where $v_\epsilon=v_0+\epsilon v_1$ and $\epsilon$ is a small parameter. I am given that $|v^{(k)}_0|\leq C$ and ...
0
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0answers
51 views

singular perturbation

I'm looking for literature pertaining to singulary perturbed fourth order parabolic equations. ex:$$u_{t}=-\epsilon u_{xxxx}+f(x,u)u_{xx}+g(x,u,u_x)$$ for sufficiently smooth functions $f,g$. I ...
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1answer
28 views

Asymptotic solutions for inequalities

How do I determine the order (big o) of $\omega$ in $e^{-\omega/\epsilon}\leq10^{-9}$ and $e^{-\omega/\epsilon}\leq\epsilon$, where $\epsilon$ is a small parameter.
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1answer
215 views

A relatively bounded perturbation of a closed operator is a closed operator.

Please I need help with an example I cant figure out and which will hopefully help me to get the theory: Let $X$ be Banach space and $A, B$ general operators. Furthermore $A$ is closed, ...
3
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1answer
81 views

Perturbation of a transcendental equation

I ran into a transcendental equation of the following form: $$116.2e^{-2t}-16t+12570=0$$ and was naively thinking that I could turn this into a perturbation problem by changing the problem into ...
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1answer
114 views

I am struggling to solve this boundary value problem and obtain the leading-order approximations using asymptotic matching

I am trying to solve this boundary value problem and obtain the leading-order approximations using asymptotic matching. But I got my solution wrong and I am stuck along the way. I would really ...
0
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1answer
54 views

How to find the largest possible matching region for asymptotic matching of any differential equation

I am reading Bender and Orszag's book page 336, example 2. I pasted the problem below: I am wondering why they chose $\epsilon^{-1/5}\ll x\ll\epsilon^{-1/4}$ as the overlapping region. Then ...
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1answer
183 views

What exactly is outer/inner region and outer/inner solution?

I have been searching in some literature and Wikipedia about the definition or explanation of outer region, outer solution, inner region and inner solution of boundary layer theory, perturbation ...
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1answer
96 views

Some questions on outer region and outer solution of a boundary problem

There is this problem below that I have some doubts and confusions, I will appreciate if anyone could provide some clarifications and explanations. I am new to the Boundary Layer Theory, this question ...
2
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1answer
96 views

How can I find the nth-order problem?

I was reading Bender and Orszag's 'Advanced Mathematical Methods for Scientists and Engineers', I came up with a problem in Chapter 7 on approximating solution of an initial value problem using ...
2
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1answer
65 views

Regular perturbation for a elliptic equation?

I focus on the following problem $$u_{xx}+u^3=f+\epsilon g,\quad x\in(0,1),\\u(0)=u(1)=0,\tag{1}$$ If we have known $w$ solves the problem $$w_{xx}+w^3=f,\quad x\in(0,1),\\w(0)=w(1)=0,\tag{2}$$ and ...
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1answer
61 views

Exp. Stability of perturbed system with temporally vanishing perturbation

I have a perturbation problem for which I can't find a fitting theorem in Khalil's Nonlinear Systems. Maybe someone can point me in the right direction: Given a nominal system $\dot x(t) = ...
1
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0answers
73 views

optimal control -Taylor expansion - PDE problem

I am trying to follow perturbation analysis in this paper (Optimal control of fluid limits of queuing networks and stochasticity corrections) and I am stuck at one point. For the given control ...
6
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1answer
159 views

Bounding the roots of the sum of two monic polynomials with real coefficients.

Let $P_1(z)$ and $P_2(z)$ be monic polynomials with real coefficients and roots $\{z_1^{(1)},z_1^{(2)},...\}$ and $\{z_2^{(1)},z_2^{(2)},...\}$, respectively. Are there any results relating the ...
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0answers
30 views

Error bound on matrix vector multiplication

I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate. Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i,j}$, $2a_{i,j}$}. ...
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2answers
53 views

Does a constant eigenvalue of a linearly parameter-dependent matrix have a constant eigenvector?

Let $A(\alpha)=A_0+\alpha A_1$, with $A_0,A_1\in\mathbb R^{n\times n}$ and $\alpha\in\mathbb R$, such that there exists a $\lambda\in\mathbb C$ with the property that for all $\alpha$: $\det(\lambda ...
2
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1answer
59 views

Condition of an eigenvector problem #2

This one of the problem, where the only thing I can do is ask for help. Let $A$ be a diagonalisable matrix, $\lambda_i\in\mathbb{R}$ a simple eigenvalue of $A$, and $B$ any matrix. Show that, for ...
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4answers
64 views

Evaluating a summation

I am trying to solve a homework question in which a part involves the evaluation of a summation. The summation is: $$ \sum_{i=0}^n2^{2i+1} $$ and this is my attempt which i am stuck at. Any lead ...
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1answer
317 views

Finding approximate eigenvalues of perturbed matrix

Assume I have some constant matrix $A$ to which I add a perturbation, resulting in $M(\epsilon )=A+\epsilon B$ the perturbed matrix ($B$ is constant as well), and that I can easily find the ...
0
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1answer
27 views

Order of the solution to an IVP

Suppose you have the following IVP $$\dot{y}(t) = d_1 y^2 + d_2 \epsilon^{-1} y$$ with $y(0) = y_0$ and where $d_1$ and $d_2$ are two positive constants independent of $\epsilon$. What can you ...