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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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Intuitive understanding of Maximin Principle

From the the book page $324$, does someone could explain to me the Theorem $2$. Maximin principle? I have a bit of difficulties to well understand how works this theorem. A simple example would be ...
4
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1answer
925 views

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is that,...
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1answer
162 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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3answers
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Eigenvalue of a linear transformation substituting $t+1$ for $t$ in polynomials.

Let $V$ be the linear space of all real polynomial $p(x)$ of degree $\leq n$. If $p \in V$, define $q=T(p)$ to mean that $q(t)=p(t+1)$ for all real $t$. Prove that $T$ has only the eigenvalue $1$. ...
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1answer
660 views

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 ...
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2answers
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Sturm-Liouville Problem for Bessel Functions

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 \...
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1answer
524 views

What can be said about the eigenvalues of the Laplace operator in $H^k(\mathbb{T}^2)$

Consider the Laplace operator $$\Delta: H^{k+2}(\mathbb{T}^2) \to H^k(\mathbb{T}^2)$$ where $\mathbb{T^2}$ is the two-dimensional torus (which is a compact manifold without boundary), so that $$ H^k(\...
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1answer
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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-...
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1answer
790 views

Eigenvalue problem $y'' + \lambda y = 0,$ $y'(0) = 0$, $y(1) = 0$

Find the eigenvalues of $$y'' + \lambda y = 0, \; y'(0) = 0, y(1) = 0$$ For $\lambda >0$, $$y(x) = c_1 \cos(\sqrt{\lambda} x) + c_2 \sin(\sqrt{\lambda}x)$$ We get that $y'(0) = 0 \implies c_2 =...
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1answer
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Eigenvalue problem for the Laplacian on the unit ball [closed]

I want to find out what are the eigenvalues and eigenfunctions of the eigenvalue problem for the Laplacian on the unit ball in $\mathbb R^3$, with the Dirichlet boundary conditions.
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1answer
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Proof for Sturm Liouville eigenfunction expantion pointwise convergence theorem

In "Elementary Partial Differential Equation" by Berg and McGregor, the following theorem is given without proof: Let $f(x)$ be piecewise smooth on the interval $[a,b]$ and let $\{\varphi_n(x)\}$ ...
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0answers
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Spherical Harmonics & Beltrami operator

I don't know if I can ask this question here, but there's a question on MO for which I have a good interest. The problem is I don't think I have competencies to do it. On the page The spherical ...
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1answer
121 views

Write generalized Fourier series for $f(x) = 1$ in terms of the eigenfunctions from a Sturm-Liouville Problem

I solved the following Sturm-Liouville Problem: $\begin{matrix} w^{\prime \prime}(x) = \mu w(x), \\ w^{\prime}(0) = > w(1) = 0 \end{matrix}$ and found that the eigenvalues were $\displaystyle \...
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0answers
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Can this problem be solved with eigenfunction expansion? Thought it could, then things got weird.

I am being asked to determine whether the following problem $\begin{align} u_{xyy}(x,y) + u_{xxy}(x,y)=0, && 0<x,\, y<1 \\ u(x,0)=u(x,1)=0, && 0<x<1 \\ u(0,y)=f(y),\,u(1,...
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0answers
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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 ...
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1answer
236 views

Eigenfunction expansion

Use the appropriate engenfunction expansion to represent the best solution. $$u''=f(x), u'(0)=\alpha, u'(1)=\beta$$ I use the function $$\phi''+\lambda\phi=0$$ to get the eigenfunction is $$\phi=A\...
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2answers
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Why $\sum_nA_n(X_n,X_m)=A_m(X_m,X_m)$?

Why in the following proof $$\sum_nA_n(X_n,X_m)=A_m(X_m,X_m)$$ ? The author says it's because orthogonality but orthogonality means $(f,g)=\int_a^bfgdx=0$. So how come orthogonality helps to prove it ...
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1answer
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Need help putting $-Y^{\prime\prime} - \mu Y^{\prime} = 0$ into Sturm-Liouville form

This question is related to the question I asked in this post. I'm trying to find an integrating factor or some way to turn $-Y^{\prime\prime}(y) - \mu Y^{\prime}(y) = 0$ into Sturm-Liouville form. ...
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0answers
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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 ...
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1answer
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How do I find values of $a,b,c,$ and $d$ of a $2\times 2$ matrix given these eigenvalues?

You are told that a matrix $A= \begin{bmatrix} a & b \\ c & d \end{bmatrix} \in \mathbb{R}^{2 \times 2}$ has eigenvalue $\lambda_1 = 1$ and $\lambda_2 = 8$. Calculate the value of $a^2+7ad-5bc+...
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Can you help with the Method Of Eigenfunction Expansion of a Non-Homogeneous PDE problem?

Here is the Problem: Solve $\frac{\partial T(x,t)}{\partial t} = \frac{\partial^{2} T(x,t)}{\partial x^{2}} +2xe^{-t} $ with the following boundary conditions $T(0,t)=10, and \frac{\partial T}{\...
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2answers
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Finding the eigenvalues and eigenvectors with each eigenvalue, solving the general solution with initial conditions.

Consider the system $x'_1 = x_1 + 2x_2$ and $x'_2 = 3x_1 + 2x_2$ If we write in matrix from as $X' = AX$, then a) $X =$ b) $X' =$ c) $A =$ d) Find the eigenvalues of A. e) Find eigenvectors ...