Questions tagged [eigenfunctions]
For questions on eigenfunctions, each of a set of independent functions that are the solutions to a given differential equation.
0
votes
0answers
46 views
How to proceed further in this Eigen Boundary value problem
I have the following eigenvalue BVP
$$
\lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \beta_h^2 F = 0
$$ wit BC(s) $F(0)=0,\frac{F''(0)}{F'(0)}=\...
0
votes
0answers
30 views
Converting BVP into standard Eigenfunction Eigenvalue form
I have a eigen boundary value problem
$$
\lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \beta_h^2F=0
$$
$\mu$ is the separation variable or the ...
0
votes
0answers
26 views
Finding pattern in a Eigenvalue BVP [Analytic possiblity] ??
I have the following two third order linear ODEs which have been arrived at after applying separation of variables to a coupled system of three PDEs.
\begin{eqnarray}
\lambda_h F''' - 2 \lambda_h \...
0
votes
0answers
26 views
Eigenvalues of a partial differential equation
Why $\lambda_n=sgn(n)\pi i \sqrt{n^2+\alpha}$?
I have this:
$\varphi_{xx}-(\alpha+\lambda^2)\varphi=0$ and $\varphi(0)=\varphi(1)=0$ then $\varphi(x)=c\sin(\sqrt{-(\alpha+\lambda^2)}x)+d\cos(\sqrt{-...
0
votes
0answers
20 views
Evaluating Eigen values of [Cubic ODE]
I have a DE (resulting from variable separation applied on a PDE with $\mu$ acting as the separation coefficient, all other terms are constant and $>0$)
$\lambda_h F''' - 2 \lambda_h \beta_h F'' + ...
0
votes
1answer
138 views
Eigen values of a Third Order Linear Homogenous ODE
I have two third order linear ODE which have been arrived after applying separation of variables to a system of PDEs
\begin{eqnarray}
\lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \...
0
votes
0answers
25 views
Helmholtz equation with moving boundary in the plane
Let us assume we have a unit disk $D\subset\mathbb{R}^2$ s.t. $\vec{0}\in D$. To obtain the eigenfrequencies and eigenmodes (or eigenvalues and -functions if you like) we must solve $\Delta \psi + \...
2
votes
0answers
27 views
Can this Helmholtz PDE with Robin boundary conditions be solved analytically?
Consider the following Helmholtz problem in the infinite triangle $y>0,\;x>y$ with parameters $Q<0$, $P\ge0$, $P<|Q|$.
$$\left\{\begin{align}
&\psi^{(2,0)}(x,y)+\psi^{(0,2)}(x,y)+E\...
3
votes
0answers
46 views
What's the name of this “shooting”-like method for numerical solving of PDE eigenproblems?
Consider a PDE of the following form:
$$\left(\partial_r^2+\frac5r\partial_r+\frac4{r^2} \hat L\right)\Psi(r,p)+(E- V(p)U(r))\Psi(r,p)=0,\tag1$$
where $\hat L$ is a differential operator ...
2
votes
0answers
44 views
Weird eigensystem of ODE with regular singularity
Consider the eigenvalue problem of the following 2nd-order ODE $$(x/2+a)^2y(x)-xy'(x)-x^2y''(x)=\lambda^2y(x),$$ in which $y\in(-\infty,+\infty)$ and parameter $a>0$. It has a regular singularity $...
0
votes
1answer
41 views
Determine the normalised eigenfunctions for the BVP: $y''+λy=0, y(0)=0, y(1)=0$
Solving it I get:
$y(x)=c_1 \cos(x \sqrt{\lambda}) + c_2 \sin (x \sqrt{\lambda})$
$y(0)= C1 + 0 = 0, C1=0$
$y(1)=0+C2\sin(\sqrt{\lambda})=0$
So, $(\sqrt{\lambda})=n\pi$, $({\lambda})=(n\pi)^2$
So, ...
2
votes
1answer
27 views
Distinct eigenvalues implies $A \in \mathbb{R}^{n \times n}$ is diagonalisable
Theorem:
If an $n \times n$ matrix has n distinct eigenvalues then A is diagonalisable.
Proof:
Let $A \in \mathbb{R}^{n \times n}$. Suppose A is not diagonalisable.
Then, by definition, for a ...
2
votes
0answers
22 views
An upper bound of the first eigenvalue of Laplacian on a Riemannian manifold.
I'm reading the Cheng's thesis ""Eigenvalue Comparison Theorems and Its Geometric Applications," and the author obtains an estimate of eigenvalues of the Laplacian based upon his theorem:
If $M$ is $...
-3
votes
1answer
57 views
Legendre's Equation, sturm liouville - eigenvalues/eigenfunction
Linear Differential Equation,Legendre's Equation, sturm liouville - eigenvalues/eigenfunction
Consider the linear differential operator:
$$ L = \frac{1}{4}(1+x^2)\frac{d^2}{dx^2}+\frac{1}{2}x(1+x^...
2
votes
1answer
45 views
Legendre Eigenvalue problem
I have the eigenvalue problem,
$\frac{d}{dx}\big((1-x^2)\frac{du}{dx}\big)+\lambda u=0$,
on $[-1,1]$ subject to single boundary condiction $u(-1) = u(1)$.
Assume that there is an eigenfunction of the ...
1
vote
2answers
79 views
Eigenvalue of a given operator
If $u_0$ is a positive radial symmetric nontrival solution of
$$
-\frac{1}{2}\frac{d^2u}{dx^2}+\lambda u -u^3=0
$$
Then how to show $-3\lambda$ is a eigenvalue of
$$
Lu=-\frac{1}{2}\frac{d^2u}{dx^2}+...
1
vote
1answer
51 views
Finding eigenfunctions and eigenvalues from a differential equation
Consider the differential equation
$$X''(x)+\lambda X=0$$
on $0 \leq x \leq 1$with boundary conditions
$$X'(0)+X(0)=0 \ \ \ \ \text{and} \ \ \ \ X(1)=0.$$
I have a few problems here that I think ...
0
votes
0answers
37 views
Addition property of Laplace-Beltrami eigenfunctions in symmetric spaces
Consider the eigenvalue equation for the Laplace-Beltrami operator on a manifold with metric $ds^2=|K|^{-1}[d\chi^2+\sin_K^2\chi(d\theta^2+\sin^2\theta\,d\phi^2)]$, where:
$$\sin_K\chi=\left.
\begin{...
1
vote
0answers
77 views
Eigenvalue problem; second order differential equation.
I have arrived at following differential equation
$\psi^{''} + (x^2 - E/x + E^2) \psi =0$,
where $E$ is a constant.
Is it possible to recast this equation as an eigenvalue problem, that is:
$\psi^{'...
1
vote
0answers
16 views
Neumann Laplace eigenfunctions
Let $u_k, u_m$ be two Neumann Laplace eigenfunctions on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, corresponding to eigenvalues $\mu_k, \mu_m$ respectively. ...
2
votes
0answers
29 views
Understanding the product of Normal Random Variable and Eigenfunction
Consider a symmetric function (Mercer Kernel) $K : T \times T \rightarrow \mathbb{R}$ and define an operator $H_k: L^2(T,\nu) \rightarrow L^2(T,\nu)$ where $H_kf(x) = \int_X K(x,y)f(y)d\nu(y)$.
...
1
vote
0answers
18 views
Eigenfunction of Dirichlet Laplacian on smooth domain in $\mathbb{R}^n$
I was reading about eigenfunctions of the Dirichlet Laplacian on bounded domains $\Omega \subset \mathbb{R}^n$. It seems that such eigenfunctions are real analytic in the interior of $\Omega$ and ...
1
vote
1answer
33 views
Find eigenvalues and eigenfunctions for integral operator
I'm trying to find the eigenvalues and eigenfunctions for the integral operator
$Ku=\displaystyle \int_{-1}^1 (1-|x-y|) \,u(y) \, dy$
Since I want to find $\mu,u$ such that $Ku=\mu u$, we get the ...
0
votes
0answers
18 views
2D Elliptic Eigenproblem
Consider the elliptic eigenproblem \begin{align}
\nabla^2\phi&=0 \ \ \ \ \ \ \ \ \text{in $\Omega$}\\
\frac{\partial\phi}{\partial r}&=\lambda\phi \ \ \ \ \ \text{on $\Gamma_1$} \\
\phi&=0\...
3
votes
1answer
22 views
Eigenfunctions Laplacian - Bounding the Fourier Coefficients
Let $\Omega \subset \mathbb{R}^{N}$ be an open set with boundary of class $C^{\infty}$ and let $\{\lambda_{k}\}$ and $\{v_{k}\}$ be the eigenvalues and eigenvectors of -$\Delta$ with Dirichlet ...
1
vote
2answers
62 views
What is the error I am making on getting general solutions for this Sturm-Liouville problem?
Given this Sturm-Liouville problem:
$$X'' + \lambda X = 0$$
There are general solutions (Eigenfunctions) for three cases on $\lambda$:
$$\lambda > 0$$ Has the characteristic equation: $r^2+\lambda ...
0
votes
0answers
19 views
Orthogonal polynomials as eigenfunctions of a second-order difference operator
I am Reading the theorem 6.1.3 of this book https://books.google.com.mx/books?id=RusIDAAAQBAJ&pg=PA146&lpg=PA146&dq=up+to+normalization,+the+charlier,+krawtchouk,+meixner,+and+chebyshev%E2%...
0
votes
1answer
41 views
SO(n) as a manifold
I cannot find some basic information on $SO(n)$ ($n$ general, not just 3) as a manifold: what is the geodesic distance between two matrices, what are the eigenfunctions and eigenvalues of the Laplace-...
1
vote
0answers
32 views
Eigenvalue/function of Laplace Operator for Exterior of Disk - Reference Request
I'd like to know more about,
$$\lambda u = \nabla ^2 u$$
for unbounded domains (particularly the exterior of a disk in $\mathbb{R}^2$ or ball in $\mathbb{R}^3$), but have had a hard time finding ...
3
votes
1answer
70 views
What is the relation between separation of variables and the eigenfunctions and eigenvalues for PDEs?
Studying Fourier Series and its application of solutions for Partial Differential Equations, in particular (historically) for the heat equation, one starts by separating variables. Somehow related to ...
1
vote
0answers
38 views
find the eigen function and eigen value of differential operator
I have an operator, defined in the cylindrical coordinate system with cylindrical symmetry, given by:
$\frac{\partial^2}{\partial r^2}+ \frac{\partial}{r\partial r} $
I would like to find the ...
1
vote
1answer
56 views
How do I compute the eigenfunctions of an operator that contains another operator?
Given the operator $A = (X\frac{d}{dx}+2)$, where $X$ is a linear operator, how can I find the eigenfunction of $A$ corresponding to a zero eigenvalue?
In general, this is just a matter of solving ...
3
votes
1answer
47 views
Need Help Interpreting the Sturm-Liouville Operator
I am given the following "Sturm-Liouville Problem with Operator $\mathcal{L}$ ":
$$\mathcal{L}_{SL}=-\frac{1}{x}\left[\frac{d}{dx}\left(x\frac{d}{dx}\right)-\frac{1}{x}\right]$$
which is defined on ...
1
vote
0answers
27 views
Finding the eigenfunctions of an operator with multiple derivatives
The differential operator is given by
$L= x\cdot \frac{d^2}{dx^2} + \frac{d}{dx} -\frac{a}{x}$
Any advice or strategies to find the eigenfunctions of this operator would be greatly appreciated. I ...
1
vote
0answers
23 views
Eigenfunctions of PDE
I came across the following eigen-function problem
$$
\begin{align}
(a(x,t)\nabla + b(t)\Delta)f =& \lambda f\\
f(x)=&0 (\forall x \in \partial \Omega),\\
f(x)>&0 (\forall x \in \Omega),...
0
votes
1answer
79 views
PDE Problem on Eigenvalues and the Heat Equation
Let $a$ be any fixed real number. Consider the eigenvalue system:
$$
\begin{cases}
X'' + \lambda X = 0, & 0 ≤ x ≤ 1\\[0.1cm]
aX(0) = X(1)\\[0.1cm]
aX'(0) = -X'(1)
\end{cases}.
$$
Prove that if $...
1
vote
2answers
56 views
Expanding the function $f(x) = 1, 0 < x < 1$ in a series of the eigenfunctions?
I have a problem that asks me to expand the function $ f(x) = 1, 0 < x < 1$ in a series of the eigenfunctions of the given problem. For example, one given problem is $y'' + 2y' + ( \lambda + 1)y ...
2
votes
1answer
38 views
Since $\phi_n$ and $\phi_m$ are eigenfunctions, they must satisfy the ODE …
I came across this:
Theorem 1
The eigenfunctions of Sturm-Liouville BVP above satisfy the integral relationship:
$$\int_a^b r(x)\phi_n(x) \phi_m(x) \ dx = 0$$ if $m \not= n$,
where $\...
2
votes
3answers
51 views
Finding eigenfunction to this operator
I have the operator
$$
A : -e^{-2ax}
\frac{\partial}{\partial x}
\left(e^{2ax}\frac{\partial}{\partial x}\right)\\
D_A = \left( v \in C^2[0,L] \quad | \quad v(0) = v(L) = 0 \right)
$$
...
0
votes
2answers
45 views
how to make use of t' in tx = Ax vs general case
I ran into this problem on my differential equations homework set. Previously, the questions were in the form x' = Ax, and they were relatively straight forward and easy to complete.
I was gifted a ...
3
votes
4answers
65 views
Find eigenvalues & eigenvectors for an integral.
Can anyone please explain me how to solve it?
Find the nonzero eigenvalues and the corresponding eigenvectors:
$T:[-1,1]\rightarrow[-1,1]$
$$T((f(x))=\int_{-1}^1(x^2 y + y^2 x) f(y) \, dy$$
2
votes
1answer
67 views
How to find eigenvalues and eigenfunctions of simple-looking differential operator
What process can be used to solve for the eigenvalues and eigenfunctions of the following differential operator?
$$H=A\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+\cos(...
0
votes
0answers
24 views
Fixed points of cubic transformation
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a Lipschitz continuous operator and let fix$(f)$ denote the set of fixed points of $f$.
Define the operator $g = (1-a) f + a(1-b) f^2 + a b f^3$, for ...
2
votes
0answers
40 views
Operator acting on a product of functions
Let $f \in \mathcal{H}$ be a function $f: \mathbb{R} \rightarrow \mathbb{C}$ in a Hilbert Space $\mathcal{H}$.
Now suppose that for some linear operator $T$ that acts on $\mathcal{H}$ and for some $f \...
1
vote
0answers
103 views
Solve this integral equation.
Let $h(x)$ be a known well-behaved function, I have to solve for $\sigma(t)$:
$$
\phi(x) = \int_a^b\log\left[\left(x-t\right)^2 + \left(h(x) - h(t)\right)^2\right]\sigma(t)dt
$$
Where, $b>a>0$, ...
3
votes
0answers
24 views
Cylindrical Operator
Maybe you can help me solve this (simple?) problem I'm too stupid to tackle :-(
I want to find the eigenfunctions to the Operator
$$
\widehat{O} = -\partial_z^2 - \frac{1}{r}\partial_r r\partial_r
$$
...
2
votes
1answer
64 views
Eigenvalues of second order ordinary differential equation
The question is find all the eigenvalue of the following equation
$-\frac{d^2y}{dx^2}+x^2y=\lambda y$
I have found the first function which is $y=e^{\frac{-x^2}{2}}$, however I have no clue on how ...
4
votes
0answers
98 views
Eigenvectors of discrete Laplace matrix for 2D unit square under Neumann boundary condition
Eigenvectors of discrete Laplace matrix for 2D unit square with free boundary is simply
$$
\phi(x,y)= \cos(\frac{\pi}{n} kx) \cos(\frac{\pi}{m} ly)
$$
It is easy to see that its 2nd order derivative ...
2
votes
0answers
25 views
Scale-dependent isoperimetric inequalities (e.g. with heat kernel?)
Suppose $\Omega\subseteq\mathbb R^2$ is a compact, connected planar region with a smooth connected boundary $\partial\Omega$. Take $\lambda_0,\lambda_1,\lambda_2,\ldots$ to be the Laplacian spectrum ...
1
vote
0answers
41 views
Eigenvalue solution of a 2nd order ODE of a geophysical fluid dynamic problem
community. I am working on a project with a professor of mine and he suggested me to numerically solve a geophysical fluid dynamic instability ODE equation of a paper of Boccaleti et al. (2007) (...
