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Questions tagged [eigenfunctions]

For questions on eigenfunctions, each of a set of independent functions that are the solutions to a given differential equation.

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How to Choose Basis Solutions for Eigenvalue Problem

I have a real, fourth order linear operator $L$ and want to solve the eigenvalue problem \begin{equation*} Lv = \lambda v, \end{equation*} where $\lambda \in \mathbb{C}$. I further want to impose ...
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For a compact Riemannian manifold $M$, $L^2(M)$ is spanned by the eigenfunctions of the Laplacian.

In some paper I read the following statement: For a compact Riemannian manifold $M$ and the corresponding Laplace-Beltrami operator $\Delta$ on $M$ we have, that $$L^2(M) = \widehat{\bigoplus_{\...
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Orthogonal Degenerate Eigenfunctions

Regarding Fourier Series, it is easy when talking about non-degenerate eigenfunctions to prove they are orthogonal using Green's identity. However I'd like to know if it is possible to prove that ...
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2answers
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Sturm-Liouville, find normalized eigenfunction

I want to find the eigenvalues and normalized eigenfunctions of the problem $$-y'' = \lambda y, y'(0) = y(1) = 0. $$ By solving $r^2 + \lambda = 0$ I found the general solution $y(x) = c_1\cos(\sqrt{\...
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How to turn ODE into an Eigenvalue Problem

My Question: What is the general formula/method for turning an ODE into an eigenvalue problem? My book turns this equation from an example $$\frac{d^2u}{dx^2}+ u=e^x, u(0)=u(\pi)=0$$ into $$\frac{...
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Solving the BVP $y''+2y'+(\lambda+1)y=0$ for $y(0)=y(\pi)=0$

Convert the differential equation $$y''+2y'+(\lambda+1)y=0$$ to Sturm-Liouville form, and obtain the solutions satisfying the boundary conditions $$y(0)=y(\pi)=0.$$ Using the integrating factor $\mu(...
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Relation eigenfunction of linearized PDE and solution of the original PDE

Consider $$ \frac{\partial^2}{\partial x^2}u+\mu \sin(u) = 0 \\ u(0) = 0 = u(1) $$ The linearized version is for small $u$ $$ \frac{\partial^2}{\partial x^2}u+\mu u = 0 $$ This gives for the general ...
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What functions can be represented as a series of eigenfunctions

Consider the differential equation: $y'' = \lambda y$ with the boundary conditions $y(0) = y(2\pi) = 0$. This equation has eigenfunctions $\mu_n(x) = \sin(\frac{nx}{2})$ with the corresponding ...
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Why is a matrix multiplied by an eigenvector not parallel to that eigenvector?

If $\lambda$ is a non-zero eigenvalue with a corresponding eigenvector $v$, then $A v$ is parallel to $v$. This statement is false. Why is that? Would it be parallel to $\lambda v$?
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Proving that the eigenvalues of the Airy problem are positive

I am solving an exercise concerning the Airy eigenvalue problem $$ -y''+xy =\lambda x, \quad y(0)=y(1)=0, \quad (*) $$ which (among other things) asks me to prove that all eigenvalues are positive. I ...
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How can I find the following one dimensional heat conduction solution?

$$\frac{\partial T}{\partial t}=\alpha\frac{\partial^2 T}{\partial^2 t}$$ with an initial condition and boundary conditions $$T(x,0)=T_0$$ $$T(L,t)=T_0$$ $$-k\left.\frac{\partial T}{\partial x}\right|...
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Checking Eigenvalues Of an ODE

When checking the eigenvalues of an ODE that you separate from a PDE like: $\displaystyle \frac{d^2\phi}{dx^2} = -\lambda \phi$ $\phi(0)=0$ $\phi(L)=0$ Why do you separate the problem into cases ...
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Heuristic understanding of the eigenvalue equality between two Neumann quantum graph

Lemma: Let $G$ be a quantum graph (not necessarily connected) with two vertices $v_{1}$ and $v_{2}$ with the Neumann conditions imposed on it. Modifying the graph G by merging the two vertices into ...
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Finding basis of the null space of eigenvector equation

Question I'm trying to do part (ii) which asks for the basis for the null spaces of the eigenvector equation with the two respective eigenvalues. What's the easiest way of doing so? I tried to find ...
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Fredholm Equation with Exponential Sum Kernel

I'm trying to solve the following integral equation to find the function $f(x)$ \begin{equation} f(x) = K(x) - \int_0^\infty K(x-t)f(t)dt \end{equation} where \begin{equation} K(x) = \sum_{i=1}^N ...
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Can this problem be reduced to a Sturm-Liouville form?

From a system of three coupled PDEs \begin{eqnarray} \frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\ \frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c ...
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Help in building final solution after solving the separated Eigenvalue problems

I (with help from a MSE user) used the following substitution to seperate variables in a second order linear PDE $$\theta_w = e^{-\beta_hx}F'(x)e^{-\beta_cy}G'(y)$$ The following two ODEs (...
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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)}=\...
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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 ...
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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 \...
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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{-...
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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'' + ...
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1answer
157 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 \...
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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 + \...
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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\...
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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 ...
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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 $...
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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, ...
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1answer
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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 ...
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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 $...
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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^...
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1answer
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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 ...
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2answers
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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}+...
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1answer
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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 ...
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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{...
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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^{'...
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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. ...
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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 $H_kf(x) = \int_X K(x,y)f(y)d\nu(y)$. ...
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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 ...
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1answer
59 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 ...
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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\...
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1answer
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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 ...
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2answers
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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 ...
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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%...
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1answer
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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-...
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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 ...
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1answer
92 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 ...
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41 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 ...
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1answer
59 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 ...
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1answer
49 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 ...