Questions tagged [eigenfunctions]
For questions on eigenfunctions, each of a set of independent functions that are the solutions to a given differential equation.
772 questions
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Functional Equation of Eisenstein series using (or not?) symmetry of Laplacian (possible error in Garrett's book)
I am following the proof of the functional equation of the (non-holomorphic) Eisenstein series $E_s$ in Garrett's book "Modern Analysis of Automorphic Forms", Corollary 1.10.5. Let $f=E_{1-s}...
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Why are normal mode frequencies discrete?
Consider a partial differential equation in a box $x \in [0,1]$ in $u \equiv u(t,x)$
$$ u_{t} = L(x, \partial_x, \partial_x^2, \ldots) u$$
with boundary condition $u(1)=0$. One can separate variables ...
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Are periodic functions created by fractional derivatives?
I just finished a long flight and I was thinking about fractional derivatives and I have a question.
I’m not a mathematician, just a nerd who likes to learn about these things, so first of all I beg ...
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Bounds on eigenvalues of Fredholm integral equation of the first kind
Consider the Fredholm integral equation of the first kind \begin{equation} \mu_n^2\psi_n(\tau) = \int_0^T K(\tau,\tau')\psi_n(\tau')\mathrm{d}\tau',~n\in\mathbb{Z}^+\end{equation} where $K(\tau,\tau') ...
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Confusion about hemispherical harmonics
It is known (see this answer on physics.se and this one on mathoverflow) that to obtain a basis of harmonics (i.e. Laplacian eigenfunctions) on a hemisphere $\Omega$ (say, the Northern $z\geq 0$ ...
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Adjoint and spectrum of derivative operator $\hat{D}$
$\hat{D} = \frac{d}{dx}$
Calculate the adjoint in the space of the periodic functions of $L^2[a, b]$:
$$\begin{align} (g, \hat{D}f) = \int_a^b \overline{g(x)}f'(x) &= \left[\int_a^b \overline{g(x)...
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Eigenfunctions of the Fourier Transform operator
I hope you can forgive the Electrical Engineering notational convention (I'll try to use $i^2=-1$ instead of $j$), particularly the use of the unitary continuous-time Fourier Transform. So it's "...
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Numerical algorithms for minimal eigenvalue to a couple of non linear ordinary differential equation
Problem:
I have a problem of this form:
$$
\left( - \frac{\partial_r^2}{2} + h(r,f,g) - p(r,f,g) \right) f(r) = K f(r) \\
\left( - \frac{\partial_r^2}{2} + h(r,f,g) + p(r,f,g) \right) g(r) = K g(r)
$$...
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Solution of a nonsymmetric eigenfunction equation.
Sometimes even very much non-traditional problems lead to (almost) familiar territory. In trying to find exact solutions to erosion models (here, the "streampower erosion model") that I can ...
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Axisymmetric Stokes flow field in cylindrical coordinates in a semi infinite domain
I am trying to solve the following simultaneous second-order PDEs for deriving the 2D and axisymmetric Stokes flow field (in cylindrical coordinates) $\mathbf{u}(r,z) = u_r(r,z)\hat{\mathbf{e}}_r + ...
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How to Discretize a Coupled System of Differential Equations with Finite Differences for a $4\times 4$ Matrix?
I am working on discretizing a system of coupled differential equations using the finite difference method for a $4 \times 4$ matrix. The system involves both derivative terms and functions that ...
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Is the Laplace-Beltrami metric tensor unique for eigenpairs on $S^2$?
Consider that I have found an eigenpair ($f(\theta,\varphi),\lambda$) which solves the Laplace-Beltrami eigen equation on the 3D unit sphere, $S^2$: $\Delta_gf+\lambda f=0$ with the standard metric ...
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Sturm-Liouville theory of PDEs
I'm looking for information about the eigenvalues and eigenfunctions of boundary value problems of two-dimensional PDEs i.e. Sturm-Liouville problems of the form:
$$
\nabla.(p(x,y)\nabla f(x,y))+q(x,y)...
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Showing $Ker(u-\alpha_1 Id)\circ (u-\alpha_2 Id) = Ker(u-\alpha_1 Id)\oplus Ker(u-\alpha_2 Id)$ for derivative operator.
Let $\partial$ denote the derivative operator on a space of infinitely differentiable functions.
I am trying to prove that
$$\ker \left(a(\partial - \alpha_1 \operatorname{id})\circ (\partial - \...
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Equivalent matrix given a constraint on elements
Assume there is a matrix which has elements with a normal distribution. How to find a matrix which every element can only be -1, 0, or 1 such that it has closest eigenvalues to the original matrix. ...
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