Boundary conditions Definition and 364 Threads

Hi everyone! I am studying spectral methods to solve PDEs having in mind to solve a heat equation in 2D, but now i am struggling with the time evolution with boundary conditions even in 1D. For example, $$ u_t=k u_{xx}, $$ $$ u(t,-1)=\alpha, $$ $$ u(t,1)=\beta, $$ $$ u(0,x)=f(x), $$ $$...

Hi! I want to use Euler's equations to model a 2 dimensional, incompressible, non-viscous fluid under steady flow (essentially the simplest case I can think of). I'm trying to use the finite difference method and convert the differential equations into matrices to be solved using MATLAB. I set...

I'm solving the heat equation on a ring of radius ##R##. The ring is parameterised by ##s##, the arc-length from the 3 o'clock position. Using separation of variables I have found the general solution to be: $$U(s,t) = S(s)T(t) = (A\cos(\lambda s)+B\sin(\lambda s))*\exp(-\lambda^2 kt)$$...

Most undergrad textbook simply say that it is intuitive that boundary conditions should not play a role if the box is very large. Other textbooks suggest that this should be taken for granted since the number of particles at the surface are orders of magnitude smaller that the number of bulk...

Hi I have a project regarding micromechanics of composites. I'm starting my analysis on the Fiber Matrix RVE. Right now I'm trying to find the natural frequency of the unit cell. The Unit cell has some unique geometry which I will keep on changing to see how natural frequency changes. I have...

I will try to ask the question, saving as much calculations as possible, so as not to weigh down those who want to try to help me. Starting from the general electromagnetic problem in empty space, taken as a domain a volume V delimited by a closed surface S, Elliot (1) shows how the field (i.e...

I am seeing the heat conduction differential equation, and I was wondering about a boundary condition when the equation is of transient (unsteady) nature. When analyzing boundary conditions at the surface of say, a sphere, the temperature does not depend on time. For example, if you have...

Hi all, Kirchhoff's equation for this simple circuit is equivalent to \dot I=\frac{V}{L} Where V=V_0 \sin(\omega t). Integrating both sides should give I(t) = -\frac{V_0}{L\omega} \cos(\omega t)+c where c is an arbitrary constant (current). Here, most of the derivations I've found simply drop...

Hello guys. I am studying the heat equation in polar coordinates $$ u_t=k(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}) $$ via separation of variables. $$u(r,\theta,t)=T(t)R(r)\Theta(\theta)$$ which gives the ODEs $$T''+k \lambda^2 T=0$$ $$r^2R''+rR+(\lambda^2 r^2-\mu^2)R=0$$...

The book of Balanis solves the field patterns from the potential functions. Let say for TE modes, it is: F_z(\rho, \phi, z) = A_{mn} J_m(\beta_{\rho}\rho) [C_2 \cos(m\phi) + D_2 \sin(m\phi)] e^{-j\beta_z z} There is no mention of how to solve for the constant A_{mn} . Then, from a paper...

Hi everyone! This is the first time I'm posting on any forum and I'm still rather unsure of how to format so I'm sorry if it seems wonky. I'll try my best to keep the important stuff consistent! I am working on infinite square well problems, and in the example problem: V(x) = 0 if: 0 ≤ x ≤ a...

I am operating via finite differences. Say for example, I have this pipe that contains a fluid. I have the boundary condition at x = x1: k is the effective thermal conductivity of the fluid, T is the temperature of the fluid at any point x, hw is the wall heat transfer coefficient, and Tw is...

I have the following PDE I wish to solve: \frac{\partial u}{\partial t}=D\frac{\partial^{2}u}{\partial x^{2}} With the following boundary conditions: \frac{\partial u}{\partial x}(t,1)+u(t,1)=f(t),\quad u(t,0)=0 Now, I wish to do this via the Crank-Nicholson method and I would naively...

Consider the following linear first-order PDE, Find the solution φ(x,y) by choosing a suitable boundary condition for the case f(x,y)=y and g(x,y)=x. --------------------------------------------------------------------------- The equation above is the PDE I have to solve and I denoted the...

PS: This is not an assignment, this is more of a brain exercise. I intend to apply a general derivative boundary condition f(x,y). While I know that the boxed formulation is correct, I have no idea how to acquire the same formulation if I come from the general natural boundary condition...

Premise: everything that follows is done in the frequency domain. Boundary conditions If there are superficial currents (electric and magnetic) impressed on the boundary between two media, we have these discontinuities for the tangential components of the fields...

I'm not quite sure if my problem is considered a calculus problem or a statistics problem, but I believe it to be a statistics related problem. Below is a screenshot of what I'm dealing with. For a) I expressed f(t) in terms of parameters p and u, and I got: $$f(t)=\frac{-u \cdot a + u \cdot...

Hi everyone, in the attached file I tried to find the transmitted and the reflected coefficients. I ran into trouble applying the boundary conditions to the linear components of the electric field. Check the outlined boxes and see if they make sense. Thanks

Homework Statement [/B] I have a metal disc adhesively bonded at its edges to a piezoelectric ring. The piezoelectric ring vibrates radially which leads to the plate vibrating transversely. I am looking to work out the resonant frequency of the metal disc which I believe will depend on the...

Homework Statement A conductor sphere of radius R without charge is floating half-submerged in a liquid with dielectric constant ##\epsilon_{liquid}=\epsilon## and density ##\rho_l##. The upper air can be considered to have a dielectric constant ##\epsilon_{air}=1##. Now an infinitesimal...

Homework Statement I am trying to fill in the gaps of a calculation (computing the deflection potential ##\psi##) in this paper: http://adsabs.harvard.edu/abs/1994A%26A...284..285K We have the Poisson equation: ##\frac{1}{x}\frac{\partial}{\partial x} \left( x \frac{\partial \psi}{\partial...

Hi everyone! I have to solve a problem using Poisson's equation. There are two parallel infinite conductor planes in vacuum. The distance between them is d and they are both kept at a potential V=0. Between them there is a uniform volume density charge \rho_0>0 infinite along the directions...

In the textbook (attached image) it says that the boundary condition is V=0 at r=R. This creates a correlation that ##B_l=-A_l R^{2l+1}## but the potential at any boundary is continuous so when we take this account, we get. ##B_l=A_l R^{2l+1}## These two clearly contradict each other. I'd...

Hello, If I have a homgeneous linear differential equation like this one (or any other eq): $$y''(x)-y'(x)=0$$ And they give me these Dirichlet boundary conditions: $$y(0)=y(1)=0$$ Can I transform them into a mixed boundary conditions?: $$y(0)=y'(1)=0$$ I tried solving the equation, derivating...

We are using griffith's 4 edition in my electromagnetic course atm. and there's something I just don't understand about boundary conditions. It says that if we have a surface charge, and we put a pillbox on it, in such a way that half of it extends under the surface charge, and the other half...

How would you solve for the Amplitude(A) and Phase Constant(ø) of a spring undergoing simple harmonic motion given the following boundary conditions: (x1,t1)=(0.01, 0) (x2,t2)=(0.04, 5) f=13Hz x values are given in relation to the equilibrium point. Equation of Motion for a spring undergoing...

I wish to numerically compute solutions of the 1D heat equation using the Crank-Nicholson scheme: The equation is: \partial_{t}u=\partial^{2}_{x}u I use the discretisation: u_{i+1,j}-u_{i,j}=s(u_{i+1,j+1}-2u_{i+1,j}+u_{i+1,j-1})+s(u_{i+1,j+1}-2u_{i+1,j}+u_{i+1,j-1}) Where s=\delta...

Griffith's writes in chapter 7 electrodynamics that D1.a - D2.a = sigma. a. But minus sine comes when we evaluate the dot product first. How does the minus sign occur without evaluating the dot product?

I am trying to decipher if an error occurred in a calculation given in this paper. It is understandable that if two compressible fluids of different uniform densities have a common interface (e.g. Figure 1), then to be in equilibrium and supported against gravity, there must be a pressure...

Hi, my classmate asks me an interesting question: For a finite 4D volume in spacetime, its boundary is a 3D close surface. If the 4D volume is a 4D rectangular, the boundary consists of eight 3D surfaces. The boundary condition is specified on these eight 3D surface. Please explain the physical...

At the interface between: 1) conductor/conductor 2) conductor/semiconductor (or dielectric) 3) semiconductor/semiconductor (or dielectric/dielectric) What quantity should be continuous? Is it the electrochemical potential, only the chemical potential or is it the electric potential? Since they...

Homework Statement A particle is represented by the following wave function: ψ(x)=0 x<-L/2 =C(2x/L+1) -L/2<x<0 =C(-2x/L+1) 0<x<+L/2 =0 x>+L/2 use the normalization condition to find C Homework Equations ψ(x) must be...

-If we have string of length L that has fixed ends, then we can easily find frequencies with which this string can oscillate: We just need to solve wave equation: ∂2y/∂x2=1/c2*∂2∂t2 (c is determined by strings properties (linear density and tension), with Dirichlet boundary conditions...

I am solving the Laplace equation in 3D: \nabla^{2}V=0 I am considering azumuthal symmetry, so using the usual co-ordinates V=V(r,\theta). Now suppose I have two boundary conditions for [V, which are: V(R(t)+\varepsilon f(t,\theta),\theta)=1,\quad V\rightarrow 0\quad\textrm{as}\quad...

Hi at all, I'm tring to solve Schrodinger equation in spherically symmetry with these bondary conditions: ##\lim_{r \rightarrow 0} u(r)\ltimes r^{l+1}## ##\lim_{r \rightarrow 0} u'(r)\ltimes (l+1)r^{l}## For eigenvalues, the text I'm following says that I have to consider that the...

Hello PF community, I am currently self-studying electrodynamics from Griffiths textbook, and I'm at a point where the book discusses electrostatic boundary conditions. If someone can please check if my reasoning is right. So, as I am approaching an infinite, uniformly charged plane (let the...

I am unsure how to choose the boundary conditions for a system of PDEs or for a single PDE for that matter. The situation I am stuck with involves a system of 4 PDEs describing plasma in a cylinder. The dependent variables being used are Vr, Vt, Vz, ni, and the independent variables are Rr...

Homework Statement [/B] Determine the Green's functions for the two-point boundary value problem u''(x) = f(x) on 0 < x < 1 with a Neumann boundary condition at x = 0 and a Dirichlet condition at x = 1, i.e, find the function G(x; x) solving u''(x) = delta(x - xbar) (the Dirac delta...

There are few thing I'm not sure of and be happy for clarifications. In general: at steady state, what are the electric-field,potential, and current boundary conditions between a conductor and a dielectric medium? more specific: a) When dealing with a perfect conductor there exist a surface...

Homework Statement I'm trying to find the boundary conditions for the following problem: A plate with length 2L is placed on supports at x = L/2 and x = - L/2. The plate is deforming elastically under its own weight (maximum displacement bowing up at x = 0). Both ends of the plate are free...

I am studying online course notes from University of Waterloo on 'Analytical mathematics in geology' in which the author describes a 'modified Fourier transform' which can be used to incorporate 3rd kind of boundary conditions. The formula is ## \Gamma \small[ f(x) \small] = \bar{f}(a) =...

Hello, I am attempting to solve the 1 d heat equation using separation of variables. 1d heat equation: ##\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}## I used the standard separation of variables to get a solution. Without including boundary conditions right now...

Homework Statement Find E1, E3, and ps2 Homework Equations boundary conditions The Attempt at a Solution (these are class notes)[/B] I understand how to find E1, but I am a bit confused about the reasoning behind finding E3... Why do we leave the 2x(hat) for E3...? I though that only...

Hello, My question is very simple but I do not have a lot of experience with simulation. I want to write some code to simulate a lattice with boundary conditions and then I will perform calculations with the Hubbard model to find different kinds of properties of interest. I would like to know...

Homework Statement A particle with mass m and spin 1/2, it is subject in a spherical potencial step with height ##V_0##. What is the boundary conditions for this eigenfunctions? Find the degeneracy level for the energy, when it is ##E<V_0## Homework Equations Radial equation \begin{equation}...

I've been playing around with Maxima and it's ctensor library for tensor manipulation. I decided to have a crack at deriving Schwarzschild's solution for the interior of a constant-density sphere. I've managed to derive a static, spherically symmetric solution, but am struggling a bit with the...

Q) A conducting sphere of radius R floats half submerged in a liquid dielectric medium of permittivity e1. The region above the liquid is a gas of permittivity e2. The total free charge on the sphere is Q. Find a radial inverse-square electric field satisfying all boundary conditions and...

This question is based on page 71 of Thomas Hartman's notes on Quantum Gravity and Black Holes (http://www.hartmanhep.net/topics2015/gravity-lectures.pdf). The Euclidean Schwarzschild black hole $$ds^{2} = \left(1-\frac{2M}{r}\right)d\tau^{2} + \frac{dr^{2}}{1-\frac{2M}{r}} +...

Hi, I need to solve Laplace equation ##\nabla ^2 \Phi(z,r)=0## in cylindrical coordinates in the domain ##r_1<r<r_2##, ##0<z<L##. The boundary conditions are: ## \left\{ \begin{aligned} &\Phi(0,r)=V_B \\ &\Phi(L,r)=V_P \\ & -{C^{'}}_{ox} \Phi(x,r_2)=C_0 \frac{\partial \Phi(x,r)}{\partial...

(Mentor note: moved here from noon homework thread hence no template) I was studying vibration of a one-dimensional monatomic chain and the textbook used periodic boundary condition (PBC). I wanted to justify the use of PBC, so I came up with this: atoms deep inside the crystal sees an...