Questions tagged [boundary-conditions]
This tag is for questions regarding to the boundary conditions (b.c.) which expresses the behaviour of a function on the boundary (border) of its area of definition. The choice of the b.c. is fundamental for the resolution of the computational problem: a bad imposition of b.c. may lead to the divergence of the solution or to the convergence to a wrong solution.
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Physical meaning of Cahn-Hilliard boundary conditions
Consider the 1D Cahn-Hilliard equation for a two-component mixture, on an interval $x\in[a,b]$:
$\frac{dc}{dt} = -\frac{d}{dx}j(x)$ where the flux $j(x) = -D\frac{d}{dx}\left(c^3 - c - \gamma\frac{d^...
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Difference between boundary conditions in thermodynamic limit
Consider a model for a spin chain. I somehow am able to find a general formula for the expectation value of some observable in both periodic and open boundary conditions. ie.,
under PBC, I have
$\...
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From what equations is magnetic field uniquely determined for a given current distribution?
The Maxwell equations for magnetostatics in the absence of time varying electric field state that -
$$
\mathbf{\overrightarrow{\nabla}} \cdot \mathbf{\overrightarrow{B}} = 0
$$
$$
\mathbf{\...
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Minimum or Stationary Value of a Mixed Boundary Problem
Take the volume integral of the dissipated DC current in a finite volume $\mathcal V$ of conductivity $\sigma$ and stationary potential distribution $\phi$ while assuming charge conservation $\nabla \...
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Robin conditions from action principle
Consider the Lagrangian density
$$L(\tilde{\phi}, \nabla \tilde{\phi}, \tilde{g}) = \tilde{g}^{\mu \nu} \nabla_{\mu} \tilde{\phi} \nabla_{\nu} \tilde{\phi} + \xi \tilde{R} \tilde{\phi}^2$$
with $\...
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1
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Possible boundary conditions in derivation of Euler-Lagrange equations
Given a Lagrange density
$$\mathcal{L} = g^{ij} \phi_{,i} \phi_{,j} - V(\phi)\tag{1}$$
I have read (e.g. here) that the boundary term that occurs through variation of the action
$$ \delta I = \int_V ...
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0
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46
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Bethe diffraction: surface charge density of an ellipsoid
I'm having a hard time following one part of Bethe "Theory of diffraction by small holes" paper, which can be found here: https://web.stanford.edu/class/ee349/Handouts/Bethe_PR1944.pdf
At ...
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Classical open string in Polchinski -- consistency of Neumann boundary conditions with gauge choice
In Section 1.3 of String Theory, Volume 1, Polchinski derives the open string spectrum from the Polyakov action with Neumann boundary conditions, by first considering the classical open string in ...
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Understanding certain boundary conditions of functionals of the form $\int_{p_0}^{p_1}f(x,y)\sqrt{1+y'^2}dx$
A question I had whilst reading section 15 of Fomin's "Calculus of Variations" (great book btw!!)
The General Question:
Among all smooth curves whose end points $p_0$,$p_1$ lie between two ...
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2
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Variation of the Lagrangian expressed as a time derivative of a function
In chapter 4.5 of Jakob Schwichtenberg's Physics from Symmetry, he expresses the variation of the Lagrangian $L = L\left ( q, \dot{q}, t \right )$ with respect to the generalized coordinate $q$ as
$$\...
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Boundary-condition-changing Operators for Free Boson BCFT with Dirichlet Boundary Conditions (or more general BCFTs)?
Is there any literature about boundary-condition-changing (b.c.c.) operators for the Free Boson with Dirichlet Boundary Conditions? The b.c.c. operators I'm interested in would replace boundary ...
0
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1
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Eigenstates of the Laplacian and boundary conditions
Consider the following setting. I have a box $\Omega = [0,L]^{d} \subset \mathbb{R}^{d}$, for some $L> 0$. In physics, this is usually the case in statistical mechanics or some problems in quantum ...
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Electromagnetic Field in a 3D Cavity with Lossy Boundary
I would like to find the electric and magnetic fields inside a cubic cavity with a lossy boundary (i.e. NOT a perfect conductor). I assume that the interior of the cavity is filled with a homogeneous ...
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Static pressure vs ambient pressure
If in a real scenario, a flat surface with a flush perpendicular closed duct of small diameter is exposed to a tangential fluid flow(laminar and naturally with the presence of boundary layer effect), ...
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It's possible to have different potentials (boundary conditions) in the surface of a cylindrical conductor?
Edit I realized that my problem is not clearly stated.
In general, I can solve the Laplace equation for boundary conditions $V(r,\phi, z=0) = f(r,\phi)$ (bottom of the cylinder), $V(r,\phi, z=L) = g(r,...
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