Tagged Questions
1
vote
0answers
31 views
First eigenvalue of the given linear operator
I have the following question:
Let us denote $H_2^N: = \{u\in (H^2(0,1))^2: u'(0) = u'(1) = 0\}$.
Let an operator $L:H_2^N \to (L^2(0,1))^2$ be given by
$Lu = -Du'' + Cu$, where $D$ is a positive ...
1
vote
1answer
69 views
Considering the linear system Y'=AY
What would be an equation that I can use when I compute the eigenpairs for the coefficient matrix A.
1
vote
1answer
48 views
Differential Equation: Complex Eigenvalue
For the following system
$$ x'=\left( \begin{array}{ccc}
\frac{-1}{2} & 1 \\
-1 & \frac{-1}{2} \end{array} \right)x $$
To find a fundamental set of solutions, we assume that $$ x = Ee^{rt}$$
...
2
votes
1answer
56 views
complex eigenvectors with non zero real parts
I'm wondering about how to deal with complex numbers in eigenvectors that have non zero real parts, as in my eigenvector is $\bigl[\begin{smallmatrix}1-2i\\-1\end{smallmatrix}\bigr]$ that is supposed ...
1
vote
1answer
34 views
Are the eigenvalues limited?
While studying elliptic operators, I encountered the following problem, which I'm having problems to prove or give a counter-example:
Let $\Omega$ be an open subset of $\mathbb{R}^m$, and suppose ...
0
votes
1answer
83 views
Show the stable age structure
Considering the population process described by
where $γ$ is the dominant eigenvalue of $L$ $l$ denotes the survival function of the Leslie matrices and $L$ is the Leslie matrix below
We are trying ...
1
vote
0answers
64 views
Solving a Sturm-Liouville differential equation variationally
This is a problem from Haim Brezis' functional analysis book (Exercise 8.41). I solved parts of it, but am stuck on some parts/want confirmation on the method. The problem is as follows:
Let $q ...
1
vote
1answer
50 views
Boundary-value problem in differential equations
Consider the problem:
$$u^{(4)} + \lambda u = 0, \ \ \ 0<x<\pi; \ \ \ u(0) = u(\pi) = u''(0) = u''(\pi) =0$$
Find the eigenvalues.
How should one proceed about this problem? I am complete ...
2
votes
3answers
127 views
Spectrum of eigenvalues and eigenfunctions
Our O.D.Es professor had the "amazing" idea of heavily introducing advanced linear algebra material (which is not an official prerequisite for the course) along with boundary value problems. Not being ...
0
votes
0answers
10 views
Qualifying Parameters
you have two parameters, 1) rates of trees per land size, ranging from 30%-100%, and 2) rates of birds per land size, ranging from 5%-30%
goal is that you're trying to find out which is overall ...
0
votes
2answers
89 views
Eigenvalue of some Sturm–Liouville problem
I have one simple question.
How I suppose to show that $\lambda =0$ is an eigenvalue of some problem.
Does it mean that I must have non-trivial solution for $\lambda=0 $?
Thanks!
UPD:I mean by ...
0
votes
1answer
68 views
Solving wave eigenmodes in a cylinder
I am trying to find the eigenfrequencies of waves in a cylinder, or put into equations:
$$\frac{1}{c^2}\frac{\partial^2}{\partial t^2}u = \Delta u$$ with $u = u(t,r,\theta,z)$. Giving, in cylindrical ...
2
votes
0answers
67 views
Positive eigenvalues in differential-algebraic equations not appearing in time-domain simulation
I am solving a system of equations derived from power system applications. It consists of index-1 differential and algebraic equations in the form:
$$\dot{x}=f(x,y) \\ 0=g(x,y)$$
To get the ...
1
vote
1answer
57 views
eigen value of the gradient operator
Eigen value of the following differential equation
$$\nabla \phi (\vec r) = a \vec {k} \phi(\vec{r})$$
is
$$ \phi(\vec{r}) = e^{a \vec{k}.\vec{r}}$$
How can i derive this result?
0
votes
1answer
108 views
Finding the eigenvalues for a $3\times 3$ matrix
With the matrix $A$ given by $$\left( \begin{array}{ccc}
0 & -1 & 0 \\
0 & 1 & a \\
1 & 0 & 1 \end{array} \right)$$
the solution to the initial value problem $x'=Ax$, $x(0) = ...
1
vote
1answer
117 views
Repeated Root Eigenvalues
The question is:
Solve the initial value problem:
$$\begin{align*}
\frac{dx_1}{dt}&=40x_1-6x_2+18x_3,\\
\frac{dx_2}{dt}&=-6x_1+45x_2+12x_3,\\
\frac{dx_3}{dt}&=18x_1+12x_2+13x_3,\\
...
1
vote
2answers
69 views
Eigenvalues and IVPs
So I have this question:
Solve the initial value problem:
$$\begin{align*}
\frac{dx_1}{dt}&=3x_3-2x_4,\\
\frac{dx_2}{dt}&=-2x_3+3x_4,\\
\frac{dx_3}{dt}&=3x_1-2x_2,\\
...
0
votes
1answer
50 views
General solution and finding eigenvalues
Find the general solution:
$\hat y_1'= y_2$
$\hat y_2' = 3y_1 +2y_3$
$\hat y_3' = -y_2$
If I put it in matrix form and get the eigenvalues I got
...
0
votes
1answer
148 views
Sturm-Liouville problem: eigenvalues
I have a Sturm-Liouville problem
$$
y'' + \lambda^2 y = 0, \\
y'(0) + \alpha_1 y(0) = 0, \\
y'(L) + \alpha_2 y(L) = 0,
$$
where $\alpha_1 \alpha_2 \neq 0$. I found that eigenvalues are ...
1
vote
3answers
70 views
Eigenvalue products
Prove that if $detA > 1$ then $A$ has at least one eigenvalue with
$|\lambda |> 1$.
The answer says:
If all $|\lambda_j | \le 1$ then so is their product $1 \ge |\lambda_1 ...\lambda_n| ...
2
votes
1answer
156 views
Complete matrix
Which of the following are complete eigenvalues (by complete,
corresponding eigenspace has the same dimension as its multiplicity)
for the indicated matrix? What is the dimension of the ...
1
vote
0answers
69 views
Partial Differential Equation Eigenvalue of zero question
In the event that I'm solving a partial differential equation through separation of variables, if I end up with an eigenvalue of zero, what do I do with the corresponding eigenfunction?
That is to ...
0
votes
0answers
59 views
Polya's fake Zeta function $ \zeta ^{*} (1/2+iz) $ and differential operator
i have heard that the Polya's fake function's zeros $ \zeta ^{*} (1/2+iz)=0 $ are realted to the Eigenvalues of the operator
$$ - \frac{d^{2}}{dx^{2}} + e^{2x} $$
with boundary conditions $ ...
1
vote
2answers
179 views
A differential equation of Buckling Rod.
I tried to solve a differential equation, but unfortunately got stuck at some point.
The problem is to solve the differentail equation of hard clamped on both ends rod.
And the force compresses the ...
0
votes
0answers
85 views
Eigenvalue in Sturm-Liouville problem: why isn't this one valid?
I am trying to solve the following S-L problem:
$\Phi_{xx}-2\Phi_x+\lambda\Phi=0$
$\Phi_x(-1)=\Phi_x(1)=0$
where $\Phi_{xx}$ is the second derivative of $\Phi$. Professor's solution is ...
0
votes
0answers
152 views
an eigenvalue problem of a differential equation
Let $L = - (\frac{d}{d \theta})^2 $. Consider the eigenvalue problem $L \phi = \lambda \phi \; (a < \theta < b )$. If $\phi(a) = \phi(b) = 0$, can I derive $\lambda_n$ formula ? Here $\lambda_n$ ...
0
votes
1answer
29 views
stuck on a differential equation
let be the differential equation
$ x^{2} y''(x)+ y(x)(a^{2}+k^{2} _{n})=0 $
the boundary conditions are $ \int_{0}^{\infty}dx |y(x)|^{2} < \infty $ and $ y(0) $ must be finite (regular solutions ...
1
vote
1answer
42 views
Small perturbations
Background:
Let $x_1,\ldots,x_n$ be the variables satisfying the equations of motion $\ddot{x_i}=f_i(x_1,\ldots,x_n)$ for $i=1,\ldots,n$
We introduce a small perturbation such that $x_i(t)=x_i^0 ...
1
vote
2answers
475 views
Solving an eigenvalue problem
Eigenvalue problem:
$y''+ \lambda y = 0$ subject to $y'(0) = 0$ and $y(1) + y'(1) = 0$
The question is:
show that the eigenvalues are given by $\lambda = \mu^2$ with $\mu$ any root of $ \mu ...
1
vote
1answer
232 views
Differential equations: Connection between repeated roots of characteristic equation and generalized eigenvectors
My question is about homogeneous linear equations with constant coefficients: $ay''+by'+cy=0$.
When you solve this equation via a characteristic equation (see e.g. here) problems arise when you ...
0
votes
1answer
204 views
Interpreting numerically-estimated eigenvalues of a Jacobian matrix
For small systems of differential equations, it is easy to solve for the eigenvalues of the Jacobian. From there, it is then easy to interpret what those eigenvalues mean, and to see how changing ...
0
votes
1answer
156 views
about first order differential equation
Suppose I have a first order differential operator in matrix format:-
$$Dx = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 \cdots \\
-1 & 0 & ...
4
votes
3answers
797 views
Finding eigenvalues by inspection?
I need to solve the following problem,
In this problem, the eigenvalues of the coefficient matrix can be
found by inspection and factoring. Apply the eigenvalue method to find
a general ...
1
vote
1answer
204 views
Boundary value problem and eigenvalues
This boundary value problem has (countably) infinitely many eigenvalues $\mu_n\in \mathbb R$:
$$y^{(4)}-\mu y=0$$ with boundary conditons: $y(0)=y(1)=y'(0)=y'(1)=0$
What happens to the ...
1
vote
0answers
47 views
Further simplifiable?
I have an equation $(xy')'+kxy=xf(x)$ where $k$ is not an eigenvalue; $y(x)$ and $f(x)$ are subjected to boundary conditions (i) bounded as $x\to 0$; (ii) $y(1)=0=f(1)$
I want to get the ...
2
votes
0answers
235 views
Differential equation, eigenvalues and eigenfunctions
How does one find all the permissible values of $b$ for $-{d\over dx}(-e^{ax}y')-ae^{ax}y=be^{ax}y$ with boundary conditions $y(0)=y(1)=0$? I assume we have a discrete set of $\{b_n\}$ where they ...
1
vote
0answers
59 views
Orderable eigenvalues
I want to show that there are countably infinite number of eigenvalues (that can put in ascending order-- with a minimum value) to the 2nd order ODE $x''+\lambda x = 0$ subjected to boundary ...
2
votes
1answer
122 views
Eigenvalues of diff-system(can't understand)
In this paper the authors have the dynamical system
$$\begin{align}
T_f \dot{y}_f & = -y_f + (1-\alpha(v))\varphi(z,d) \\
T_r \dot{y}_r & = -y_r + \alpha(v) \varphi(z,d) \\
\dot{z} ...
4
votes
2answers
435 views
integral transforms: why do roots in frequency domain correspond to eigenvalues in time domain (and how does it help solve differential equations)?
In Wikipedia you can read about integral transforms, esp. the Laplace transform which maps a differential equation in the time domain into a polynomial equation in the complex frequency domain:
...
9
votes
1answer
727 views
How to compute the first eigenvalue of Laplace operator in an ellipse?
Let $\mathcal{E}$ be an ellipse in the $\mathbb{R}^2$ plane with center in $o=(0,0)$, given focal distance $c\geq 0$ and given area $A>0$.
It is a fact that the eigenvalue problem for the Laplace ...
