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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.

Let $X$ be a function space and $T$ an operator on $X$. A eigenfunction of $T$ is a nonzero function $f \in X$ such that the following holds $$ T f = \lambda f $$ where $\lambda$ is a scalar and called the eigenvalue.

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Can we construct Sturm Liouville problems from an orthogonal basis of functions?

Given a sequence of functions orthogonal over some interval, which satisfy Dirichlet boundary conditions at that Interval, can we construct a Sturm Liouville problem that gives these as its eigenfunctions? For example, if we take the sequence of…
Manishearth
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physical meaning of laplace-beltrami eigenfunctions?

The eigenfunctions of Laplace-Beltrami operator are often used as the basis of functions defined on some manifolds. It seems that there is some kind of connection between eigen analysis of Laplace-Beltrami operator and the natural vibration analysis…
Fei Zhu
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Eigenfunction of which differential operator?

I suspect this question is rather naive, but here goes. I have a set of basis functions that I suspect are eigenfunctions of an unknown differential operator (due to some results I've seen in some of my research). I have no idea what differential…
VarunShankar
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Eigenvalue problem-Expand the function to the eigenfunctions of the problem

Having solved the eigenvalue problem $$y''+ λ y=0, 0 \leq x \leq L$$ $$y(0)=y(L)=0$$ which solution is: $$\text{The eigenvalues are: } λ_n=(\frac{n \pi}{L})^2$$ $$\text{ and the eigenfuctions are: }y_n=\sin (\frac{n \pi x}{L})$$ I am asked to expand…
Mary Star
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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 form $u(x)=a_0+a_1x+a_2x^2$. Find the possible…
Ray
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Finding eigenfunctions

I seek to solve an eigenproblem like: $\Delta U+\lambda U=0$ $U|_{\partial \Gamma}=0$ I want to "try" solving it for initial guess $\lambda_{0}$. Suppose that equation for this guess holds within $\varepsilon$ and produces a function $U_{0}$. I can…
Vsevolod A.
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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 1- or 2- dimensional differential equation…
Gibby_McGinge
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What kind of function do I have to find?

To show that the eigenfunctions of an eigenvalue problem don't form a complete set, I have to show that there is a function that satisfies the boundary conditions of the problem, but it cannot be written as a linear combination of the functions that…
Mary Star
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Eigenvalue problem, BVP with periodic conditions

Good Day, I am struggling to connect 2 parts of the lecture. Consider the Eigenvalue Problem with periodic conditions $$y''+ky = 0, y(0) = y(1), y'(0) = y'(1) $$ $$\text{Solution is in the form:} \\ y =A\sin(\sqrt{k}x) + B\cos(\sqrt{k}x) \\ y'…
SuperMage1
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Completeness of eigenfunctions of higher order differential equation

I have a third order linear differential equation, with a free parameter, and boundary conditions that depend on that parameter. I don't think it is possible to obtain an analytic solution, but I would like to know if the eigenfunction, i.e. the…
user2535797
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1-periodic function; fourier; eigenvalues

Let $k\colon\mathbb{R}\to\mathbb{C}$ be a 1-periodic function with $k|[0,1]\in L^2([0,1])$. Define the convolution operator $T$ as $f\mapsto\int\limits_{[0,1]}k(s-t)f(t)\, dt$. Develop $k$ in a Fourierseries and with this find the eigenvalues…
user34632