For questions on eigenfunctions.

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3
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0answers
18 views

Heat equation with mixed boundary conditions

I am trying to solve the following problem $$\left\{\begin{matrix}\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},& 0<x<1,t>0,\\ u(0,t)=\frac{\partial u}{\partial ...
1
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0answers
30 views

rayleigh quotient of eigenvalue problem (sturm liouville theory and partial differential equations)

I am reading "A First Course in Partial Differential Equations with Complex Variables and Transform Methods" (Weinberger, p. 168). if we have the eigenvalue problem $$ (pu')'- qu + \lambda \rho u = 0 ...
0
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0answers
20 views

Neumann Series for integral equation with inhomogeneous term zero

Consider the method described in the following article: http://mathworld.wolfram.com/IntegralEquationNeumannSeries.html In this notation, what happens when $ f(x)=0 $? All the terms seem to be zero ...
0
votes
1answer
44 views

Can the zero vector be within the eigenspace

I have a matrix that looks like this: $$ \begin{pmatrix} 1 & 2 & 4 \\ 2 & 4 & 2 \\ 4 & 1 & 1 \end{pmatrix} $$ Now the calculated eigenvalues are: $-3$, $2$ and ...
1
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0answers
44 views

Eigenfunctions of integral operator

I am faced with the problem of calculating the eigenfunctions for an operator of the form: $(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy $ Does anyone know for which functions (or types of ...
0
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0answers
17 views

For what operators are $sin(ax)$ and $cos(bx)$ eigenfunctions?

So it is clear that the operator $\frac{d^{2}}{dx^{2}}$ is an eigenfunction of $sin(ax)$ and $cos(ax)$. For what other operators are sins and cosines eigenfunctions?
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0answers
17 views

How to identify a process via its Karhunen-Loeve expansion?

Suppose that you are given the following Karhunen-Loève expansion of a real-valued continuous Gaussian stochastic process, $x$. $$x(t) = \sum_{k=1}^{\infty}z_{k}\cdot \frac{\sqrt{2}\sin((k-0.5)\pi ...
2
votes
1answer
68 views

Nodal Lines of the Eigenvalue problem $\Delta u=\lambda u$

I have really enjoyed performing the method of separation of variables to identify the eigenfunctions and nodal lines (the set of points for which each eigenfunctions vanishes) of the 2-D wave ...
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0answers
16 views

Bounds on eigenfunctions of integraloperator

Let $K: [0,1]\times[0,1] \to \mathbb{R}$ be a symmetric positive definit and continuous function. It is known my Mercer's theorem that $$ [T_K \varphi](x) =\int_0^1 K(x,s) \varphi(s)\, ds $$ is ...
1
vote
1answer
28 views

Solving Laplace equation in a square with one insulated border

I keep getting stuck on this problem, so if someone could point out where my method is flawed and how I should approach this problem, that would be extremely useful. We're considering the square ...
0
votes
1answer
25 views

Initial Value Problem Eigenfunctions

Given an operator and boundary conditions, there often exist eigenfunctions which allow for Fourier summation solutions. Is there a similar way to solve initial value problems, for example, $$ ...
0
votes
1answer
21 views

Eigenfunctions for the symmetric kernel of an integral equation

The solution of the symmetric integral equation below: $$g(s) = f(s) + \lambda \int_{-1}^{1} (st +s^2t^2)g(t)dt \tag{$*$}$$ with separable kernels method is $$g(s) = f(s) + \lambda \int_{-1}^{1} ...
1
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0answers
28 views

Simple random walk on the $N$-cycle

I am considering the following example: In my lecture notes we noted that "the functions $(\phi_j)_j$ form a basis". I think they refer to the space $\mathbb{C}^G$ where $G$ is the above ...
3
votes
1answer
66 views

Least positive eigenvalue of the BVP $y''-\lambda y'+\frac{2\lambda-1}{x}y=0$, $y(0) = y(1/2) = 0$

Find the first positive eigenvalue $\lambda$ of the boundary value problem over $x\in [0,\frac{1}{2}]$. $$y''-\lambda y'+\frac{2\lambda-1}{x}y=0, \quad y(0)=y(\tfrac{1}{2})=0.$$ My approach: I ...
2
votes
0answers
20 views

What are the eigenfunctions of the D'Alembert operator on pseudo-Riemannian manifolds?

Consider the operator $\Box=g^{\mu\nu}\nabla_\mu\nabla_\nu$ acting on a function space $\mathbf{F}(M)$, given by the set of functions $\phi:M\to\mathbb{R}$ whose values go to zero at infinity (at the ...
4
votes
1answer
28 views

How do I find an $A_0$ and $A_n$ which satisfy the initial conditions of this heat equation?

Let's say I have the heat equation $\frac {\partial u}{\partial t} = k\frac {\partial^2 u}{\partial x^2}$, $0 \lt x \lt L$, $t \gt 0$, subject to the boundary conditions $$\begin{cases} \frac ...
0
votes
1answer
34 views

Why is an eigenfunction of the Laplacian determined by its nodal set?

I came across the following statement in a paper, The hyperbolic Laplacian is a real smooth operator, and the system of nodal lines determines $f$ up to a constant multiple. Here, $f$ is an ...
0
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0answers
34 views

Solve PDE using eigenfunction expansion and solve Green's Function

The question is as follows: solve using the method of eigenfunction expansion $$c\rho \frac{\partial u}{\partial t} = \frac{\partial u}{\partial x}\left(K_0\frac{\partial u}{\partial x} \right) + ...
2
votes
1answer
31 views

Sturm Liouville problem for $x^2φ''+ xφ' + λφ = 0$

Consider the eigenvalue problem $$x^2φ''+ xφ' + λφ = 0, \quad 1<x<2, \quad φ(1) = 0, φ(2) = 0$$ (a) Write the problem in Sturm-Liouvile form, identifying $p, q,$ and $σ$. (b) Is the problem ...
0
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0answers
32 views

Proof involving eigenvectors/values of a linear map and polynomials.

Let $V$ be a vector space over a field $k$, and let $T:V\rightarrow V$ be linear, and let $f\in k[x]$. Suppose that $\lambda\in k$ is an eigenvalue of $T$ and let $v\in V$ be a corresponding ...
4
votes
2answers
73 views

Solutions of Sturm-Liouville Problem Dependent on Eigenvalue as Parameter

Let $\;y_1\left(x,\,\lambda\right),\: y_2\left(x,\,\lambda\right)\;$ be respective solutions of eigenvalue problem \begin{align}y\,'' &= \lambda \,y, & ...
4
votes
1answer
15 views

Differential Equations Complex Eigenvalue functions

Show that a function of the form $x(t) = K_1 cos\beta t + K_2 sin\beta t$ Can be written as $x(t) = Kcos(Bt-\phi)$ Where $K = \sqrt (K_1^2 + K_2^2)$ I know that linear systems with complex ...
3
votes
1answer
41 views

Poisson equation - Eigenvalue expansion method; only trivial solutions

I have a problem with the following question (it can be found in the book Partial Differential Equations by Asmar as well if any of you have that; exercise 3.9.6). So I need to solve this equation ...
2
votes
1answer
152 views

Why are eigenfunctions of Laplace-Beltrami operators the minimizer of $\int_\mathcal{M}\| \nabla f(x)\|^2$?

Given a smooth $m$-dimensional manifold $\mathcal{M}$ embedded in $\Re^k$. Suppose we have a map $f : M \to \Re .$ Now, these are my questions: Specific question: i): Why does the $f$ that ...
3
votes
1answer
72 views

An “unusual” PDE eigenvalue problem

Question: (Strauss Partial Differential Equations: An Introduction, Ch. 4.3, Ex. 12) "Consider the unusual eigenvalue problem" $$-v_{xx}=\lambda v \\ v_x(0)=v_x(\ell)=\frac{v(\ell)-v(0)}{\ell}$$ for ...
0
votes
1answer
32 views

Find eigenvector and eigenvalues X1 X2

\begin{vmatrix} 7/2 & 2\\ 2 & 3\end{vmatrix} The characteristic equation is |A - λ I | = 0 $(\frac{7}{2}-λ)(3- λ)-4=0 $ $ λ_1= \frac{13-\sqrt{65}}{4} $ $λ_2 = \frac{13+\sqrt{65}}{4} $ so ...
0
votes
1answer
13 views

Value of the Eigenfunction at a point

I'm reading "Laplace-Beltrami Eigenfunctions for Deformation Invariant Shape Representation" http://www.cs.jhu.edu/~misha/Fall07/Papers/Rustamov07.pdf At a certain point the author states "where ...
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0answers
10 views

Courant-fischer minimax worked example

I am trying to understand Courant's minimax theorem, and am hoping someone can provide me with a simple worked example showing how to find an approximation to, say, the second or third eigenvalue of a ...
0
votes
1answer
67 views

Find the eigenvalues and eigenfunctions for $y"+λy=0, y(0) = y(1)$ and $y'(0) = y'(1)$

So I am in the section of our book about Strum-Liouville problems and all of the previous questions have had the boundary values equal to a constant (such as $y(10) = 0\,\,\,\,or\,\,\ y(L) = 0)$. But ...
0
votes
1answer
141 views

Prove eigenfunctions corresponding to different eigenvalues are orthogonal

I'm considering the eigenvalue problem $L(\phi) = -\lambda \sigma(x) \phi$, subject to a given set of homogeneous boundary conditions. Assume $\sigma(x) \gt 0$ on an interval $[a,b]$. Suppose that ...
0
votes
0answers
25 views

Finding eigenvalue and eigenfunction of the boundary value trouble with finding characteristic polynomial

I've done some searching around here and can't seem to find a similar problem. I've been given the following: y′′+ 2λy' + 16y = 0; λ ≥ 0 with boundary conditions y(0)=0,y(1)=1 and I'm asked to find ...
0
votes
0answers
37 views

Dirichlet eigenfunction cannot be extended to a continuous function on the closure

I need to show that there exist a bounded domain $ \Omega \subset \mathbb{R}^2 $, and a Dirichlet eigenfunction $u$ on $ \Omega$ such that u cannot be extended to a continuous function on $ ...
1
vote
1answer
31 views

eigenenergies when Hamiltonian is $\hat{H}^2$ − $\hat{H}$

If the eigenenergies of the Hamiltonian $\hat{H}$ are $E_n$ and the eigenfunctions are $\psi_n(r)$ , what are the eigenvalues and eigenfunctions of the operator $\hat{H}^2$ − $\hat{H}$ ? Attempted ...
3
votes
2answers
287 views

Bessel Functions Sturm-Liouville problem

I have been given this recently in PDE class involving the solutions to the Bessel fucntion in Sturm-Liouville form, asking for Eigenvalues and Eigenfunctions: $ (xy')'+\lambda x y = 0 \space ...
0
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0answers
25 views

General solution for homogeneous differential equation?

I have a differential equation $$\frac{d^4y}{dx^4}+(ia-2\beta^2)\frac{d^2y}{dx^2}+(\beta^4-ia\beta^2)y=0$$ The characteristic equation is $$r^4+(ia-2\beta^2)r^2+(\beta^4-ia\beta^2)=0$$ and the ...
1
vote
0answers
391 views

What is it meant by a normalized eigenfunction and how do you find it?

I am doing a course called "Partial Differential Equations" and I was told to "find the normalised eigenfunction for": $$y''+ \lambda y=0,\;0<x<1$$ subjected to: $$y'(0)=0,\;y(1)=0$$ Ok, I know ...
0
votes
0answers
36 views

Integral of Laplacian eigenfunctions squared

The Laplacian densely defined in $L^2(\mathbb{R}^3)$ has eigenfunctions $f_k(x)$ that are defined as generalized functions. I need to define the integral of the square of these eigenfunctions in a ...
2
votes
0answers
44 views

Mixed Dirichlet-Neumann eigenvalue problem

Let $\Omega\subset\Bbb R^2$ be a bounded $C^2$ domain. Let $\partial\Omega=\partial\Omega_1\cup\partial\Omega_2$. Does anyone know about the existence of eigenvalues and eigenfunctions for the ...
1
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0answers
66 views

Orthonormal basis of $L^2$ and it's impact on the solution to the heat equation.

If we consider the homogeneous Dirichlet eigenvalue problem on a bounded domain $\Omega\subset\Bbb R^n$ - (one part of my question is if we can assume $\Omega$ to be a Lipschitz domain and still ...
1
vote
1answer
35 views

Which 2D domain with fixed area has the lowest laplacian eigenvalue?

I know that a disc has the lowest laplacian eigenvalue among domains with fixed area. But how do I prove it?
2
votes
1answer
38 views

First Eigenfunction of Simple Equation

Consider the interval $[-a,a]$ and the following problem: $$\phi'' + \lambda\phi=0$$ $$ \phi(\pm a) = 0. $$ The obvious sequence of orthogonal eigenfunctions seems to be $\sin(\frac{\pi n}{a}x)$ ...
0
votes
1answer
41 views

existance of a solution to quadratic form equation

Let $\lambda$ is an unknown scalar and; $Q=Q_1 - \lambda*Q_2$ where $Q_1, Q_2$ are $NxN$ positive symetric matrices, $B=B_1 - \lambda*B_2$, where $B_1, B_2$ are $Nx1$ vectors, $m=m_1 - ...
3
votes
0answers
72 views

$L^\infty$-bounds on eigenfunctions of Laplace-Beltrami opeator

Let $w_k$ be the eigenfunctions of the Laplace-Beltrami operator on a compact manifold $M$ without boundary. We assume that $\{w_k\}$ are orthonormal, thus $\|w_k \|_{L^2} = 1$. We know $w_k$ are ...
1
vote
0answers
28 views

eigenvalue and eigenfunctions of a differential equation

Consider the following eigenvalue problem \begin{equation} \beta xf'(x)+\alpha f'(x)+\alpha xf(x)=Ef(x) \end{equation} I would like to solve for $f(x)$ and $E$. I know one set of solutions. ...
2
votes
5answers
55 views

“eigenfunction” of a transformation

Fourier transform of a gaussian is another gaussian. Fourier/Laplace transforms of $\frac{1}{\sqrt t}$ is something like $\frac{1}{\sqrt \omega}$. I realize that we can't call these eigenfunctions ...
2
votes
0answers
55 views

Eigenvalue problem $y''+py=0$, $y(-2)=0$, $y(2)=0$

The problem states to find the non-negative solutions to the eigenvalue problem given by $y''+py=0$ where p is a parameter which may be varied. Solving this differential equation for the general ...
0
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0answers
119 views

Help finding the critical values of α where the qualitative nature of the phase portrait for the system changes?

I was asked to solved for the eigenvalues in terms of α for 2X2 matrix and so i did and my answer was marked as correct. Then I was asked to solve for this: The roots are complex when? There is a ...
0
votes
1answer
44 views

Given a set of non orthogonal functions. Find another set of functions that are orthogonal to the first set.

Let's say I have a set of complex functions $\{\phi_1,\phi_2,...\}$ defined on $0\leq x \leq1$, and they are not orthogonal i.e. $\int_0^1\phi_m^*(x) \phi_n(x)dx \neq \delta_{mn}$. Is it possible to ...
1
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0answers
54 views

Find eigenfunctions of the integral operator with kernel $\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny)$

Find the eigenvalue and eigenfunctions of the integral operator $Ku=\int_0^\pi k(x,y)u(y)dy$. $k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$. This is how I ...
0
votes
0answers
28 views

Applying different boundary conditions in a quasi Helmholtz problem.

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, Consider the two-dimensional eigenvalue problem $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...