Boundary Definition and 1000 Threads

Homework Statement I'm getting through a paper and have a few things I can't wrap my head around. 1. In defining the boundary conditions for a membrane (a function of vector 'r'), the author claims that for a small displacement (u) and a boundary movement (f), the boundary condition can be...

It's obvious that if Maxwell equations are fulfilled by some E(x,y,z,t) and B(x,y,z,t), they are also fullfiled by E(x,y,z,t)+ E_0 and B(x,y,z,t)+B_0, where E_0 and B_0 are constants. This freedom has physical significance as it changes the Lorentz force which act on a charge. It...

Consider the BVP y''+4y=f(x) (0\leqx\leq1) y(0)=0 y'(1)=0 Find the Green's function (two-sided) for this problem. Working: So firstly, I let y(x)=Asin2x+Bcos2x Then using the boundary conditions, Asin(2.0)+Bcos(2.0)=0 => B=0 y'(x)=2Acos(2x)-2Asin(2x) y'(0)=2A=0...

I have this, probably quite simple, problem. In the RNS superstring, when varying the action, we obtain in general a term \int d\tau [X'_{\mu}\delta X^{\mu}|_{\sigma=\pi}-X'_{\mu}\delta X^{\mu}|_{\sigma=0} + (\psi_+ \delta \psi_+ - \psi_- \delta \psi_-)|_{\sigma=\pi}-(\psi_+ \delta \psi_+ -...

Hi, I just can't understand the basics with BZ. How do I find the shortest distance to the BZ boundary, how do I compare the electron energy between the last electron in the 1st BZ with the first electron in the 2nd BZ? I think I need a visual how to calculate these things, does anyone...

Hello all,Suppose C\subseteq \mathbb{R}^{n}, if x \in \text{bd}\;C where \text{bd} denotes the boundary, a sequence \{x_{k}\} can be found such that x_{k} \notin \text{cl}\;C and \lim_{k\rightarrow \infty}x_{k} = x. The existence of such sequence is guaranteed by the definition of boundary...

I'm trying to understand how to set up the problem of a 1D wire that is insulated at one end and has a heat source at the other. I know the heat law, from my textbook: du/dt = B d2u/dx2 + q(x,t) 0 < x < L, t > 0 Where q(x,t) is the source of heat. The problem is, I want the heat...

I'm solving a 2D forced convection problem by using finite difference method. It involves solving the vorticity-stream function equations. I met some problem when I'm trying to solve the vorticity equation, which can be stated as follows: How to deal with the vorticity boundary condition? after...

Hallo Every one, Homework Statement y(x,t)=sin(x)cos(ct)+(1/c)cos(x)sin(ct) Boundary Condition: y(0,t)=y(2pi,t)=(1/c)sin(ct) fot t>0 Initial Condition : y(x,0)=sin(x),( partial y / Partial t ) (x,0) = cos(x) for 0<x<2pi show that y(x,t)=sin(x)cos(ct)+(1/c)cos(x)sin(ct)...

Homework Statement y''+\lambday=0 y'(0)=0 y'(pi)=0 Homework Equations The Attempt at a Solution What's puzzling me is the case when we check if the eigenvalue is zero. y''=0 y'=C1 y=C1x+C2 Now when I check the first boundary value I get C1=0 now How do I check the second one ? with the...

Homework Statement y'' + ßy = 0, y'(0)=0, y'(L)=0 Homework Equations Meh The Attempt at a Solution I so already did the ß>1 and ß<1; I'm stuck on the ß=0. It seems easy enough. y'' = 0 -----> y' = A -----> 0=A, 0=A (from the two initial conditions) ------> No...

Let's consider two media with magnetic permeability \mu_1, \mu_2 . What's the condition for magnetostatic vector potential \vec{A} on the boundary. Is it true that its tangent component should be continuous. Thanks for replay.

Is it true that there always exists Green's function for Dirichlet boundary problem. I mean a function G(r,r') which fullfils the following conditions: div (\epsilon grad G(r,r')) =- \delta(r,r') inside volume V and G(r,r') is 0 on boundary of V. If V is whole space there exists obvious...

Homework Statement So using the D'Alembert solution, I know the solution of the wave equation is of the form: y(x,t) = f(x-ct) + g(x+ct) I'm told that at t=0 the displacement of an infinitely long string is defined as y(x,t) = sin (pi x/a) in the range -a<= x <= a and y =0...

show for two positive numbers x,y>0 that limit for n->infinity : \sqrt[n]{x^n+y^n} = max {x,y} i don't know how to make a upper boundary(lower boundary is >0 i suppose) something like assume for instance x bigger than y and than make a boundary with it, but how?

Hello everybody, I've been puzzling over something (quite simple I assume). Take S^1. Now consider the action of a Z_2 which takes x to -x, where x is a natural coordinate on the cylinder ( -1< x <1). Now we mod out by this action. The new space is an orbifold: smooth except at x=0. It...

1. I have a rod of length 4,cross section 1 and thermal conductivity 1.Nothing is mentioned about the end at the origin x=0, but at the opposite end x=4, the rod is radiating heat energy at twice the difference between the temperature of that end and the air temperature of 23 celcius. Find the...

If space is expanding then at at some point we must reach the point where it's expansion is faster than C. Do we know where that is? Is the expansion uniform or is it dependent on something like Dark Matter clusters? Is the expansion accelerating or decelerating at two fixed points? In...

Apparently Fluent has this thing called an "overflow" boundary condition. I tried searching the help files and found absolutely nothing about it. Does anyone know what this is and why someone would want to use it?

Homework Statement The steady state temperature distribution, T(x,y), in a flat metal sheet obeys the partial differential equation: \frac{\partial^2{T}}{\partial{x}^2}+{\frac{\partial^2{T}}{\partial{y}^2}}=0 Separate the variables and find T everywhere on a square flat plate of sides S with...

Homework Statement A string is attached to a ring of mass m which is free to move up and down a frictionless pole. The string is subject to tension T and its mass per unit length is \rho. The displacement of the string from its equilibrium position along the x -axis is y(x,t). The boundary...

Homework Statement Find the solution to the two-point boundary value problem u'' + 4u' + exu = sin(8x) with u(-1) = u(1) = 0. Homework Equations The Attempt at a Solution I haven't taken an ODE course in years but I need to verify that my numerical solution to the ODE is accurate to the...

So here's the problem: I'm asked to find the solutions to the 1-D Wave equation u_{tt} = u_{xx} subject to u(x,0) = g(x), u_t(x,0) = h(x) but also u_t(0,t) = A*u_x(0,t) and discuss why A = -1 does not allow valid solutions. I can't figure it out at all. The solutions to...

I want to solve: y(x)''-(\frac{m\pi}{a})^2y(x)=0 With boundary condition y(0)=y(a)=0. First part is very easy using constant coef. which give: y(x)=c_1 cosh(\frac{m\pi}{a}x) + c_2 sinh(\frac{m\pi}{a}x) y(0)=0 \;\Rightarrow\; c_1=0 \;\Rightarrow\; y(x) = c_2 \; sinh(\frac{m\pi}{a}...

Homework Statement Consider the Heat Equation： du/dt=k(d2u/dx2), where d is a partial and d2 is the second partial. The B.C.'s are u_x(0,t)=u(0,t) and u_x(L,t)=u(L,t), where u_x is the partial of u with respect to x. The I.C is u(x,0)=f(x) Now, consider the Boundary Value Problem...

Hello all, I am looking for a reference (text or peer-reviewed journal article) for the derivation of the boundary layer thickness over a short square cylinder as a function of distance from the leading edge for the simplest case of steady, uniform and laminar axial flow of an inviscid...

Homework Statement The problem is write this d\Psi/dy(d^2\Psi/dxdy)-d\Psi/dx(d^2\Psi/dy^2=-\nu(d^3\Psi/dy^3) in the form of -ff''=f''' where \Psi(x,y)=-sqrt(V*\nu*x)f(\eta) f(\eta)=integral(from 0 to \eta)(\Pi')*(\overline{\eta})*d(\overline{\eta}) where \overline{\eta} is a dummy...

Homework Statement A hot fluid is flowing through a thick-walled cylindrical metal tube at a constant temperature of 450C. The cylinder wall has an inner radius of 1 cm and an outer radius of 2 cm and the surrounding temperature is 20C. The temperature distribution u(r) in the metal is defined...

Hello, I am looking for a reference which has solutions for the laminar flow boundary layer for the following scenario: circular cylinder, L>>d, length in direction of flow, with flat circular cap uniform laminar flow inviscid, incompressible fluid In other words, I would like the...

I was reading a website that said the boundary of a set's boundary is equal to the first boundary. Visually, this makes sense for subsets of R1 and R2 because the first boundary will not have an interior (no ball about the points will fall into the boundary). However, the reading went on to...

For a string with one endpoint attached to a wall and the other to an oscillator (so that it is under boundary conditions), what is the character of waves that are not at a harmonic frequency?

Homework Statement A high temperature, gas cooled nuclear reactor consists of a composite cylindrical wall for which a thorium fuel element (Ka=57 W/m*K) is encased in graphite (Kb= 3 W/m*K) and gaseous helium flows through an annular coolant channel. Consider conditions for which the helium...

Hi there, I recently read that the equations of motion for an classical open string naturally give rise to two boundary conditions, namely Dirichlet and Neumann boundary conditions. (i) Could someone explain to me what do these boundary conditions physically mean, in particular for open...

given a boundary Hermitian eigenvalue problem L= -\frac{d}{dx}\left[p(x)\frac{dy}{ dx}\right]+q(x)y=\lambda w(x)y with y=y(x) , in one dimension, can we always find two operators D_{1} = \frac{d}{dx}+f(x) and D_{2} = -\frac{d}{dx}+U(x) so L= D_{1} D_{2} , with Adj( D_{1}) =...

Homework Statement du/dt = (d^2)u/d(x^2), t>0, 0<x<1 u(0,t) = 0 = u(1,t) , t>0 u(x,0) = P(x), 0<x<1 P(x) = {0 , if abs(x-1/2) >epsilon/2 {u/epsilon, if abs(x-1/2) <= epsilon/2 i need to find u(1/2,1/pi^2) Homework Equations i have u(x,t) = SUM{ 2/(n*pi)...

Homework Statement The problem can be found http://whites.sdsmt.edu/classes/ee382/homework/382Homework4.pdf" . It is the first one. Note: The subscript x = 0 is supposed to be y = 0 (the teacher typed it in wrong). Homework Equations \vec{\boldsymbol{D}}_{2t} =...

Hello all, This is to do with forced longitudinal vibration of a rod (bar). It's basically a problem to do with the linearised plane wave equation (1d). The rod is fixed firmly at one end, and excited at the other by a harmonic force. The wave equation (with constant rho/E instead of...

Homework Statement d20/de2+1=0 and the boundry condition is -d0/de(evaluated at e=+/- 1)=+/-H0(evaluated at +/-1). The final result yields 0(e)=(1/2)(1-e2)+1/H. What i don't understand is how to use this boundary condition and where the 1/H comes from. The Attempt at a Solution...

Homework Statement Find the potential outside of a long grounded conducting cylindrical rod of radius R placed perpendicular to a uniform electric field E0. Homework Equations V(s,\phi) = a_{0}+b_0{}ln(s) + \sum(A_n{}cos(n\phi)+B_n{}sin(n\phi))*(C_n{}s^n{}+D_n{}s^{-n}) The sum being...

Hey all, I am try to model a problem in which one fluid (of known properties) sits on top of another fluid (with different properties) and there is an imposed slip boundary condition at the interface. I'm wondering if COMSOL has a Slip boundary condition that can be readily used or if...

Hi, I am in Purcell's E&M book at the section explaining why the field is zero inside a hollow conductor of any shape. The proof given is that the potential function inside the conductor must obey Laplace's equation, and that the boundary of the region (in this case a rectangular metal box) is...

What does it mean to say something "extends smoothly" to a boundary in C (or R^2)? I'm studying Cauchy's integral formula, and one of the assumptions of the theorem is that a function be analytic on a domain D and extend smoothly to the boundary of D. What does that mean, exactly?

Homework Statement Let W\subset S \subset \mathbb{R}^n. Show that the following are equivalent: (i) W is relatively closed in S, (ii) W = \bar{W}\cap S and (iii) (\partial W)\cap S \subset W. Homework Equations The only thing we have to work with is the definitions of open and closed sets...

Hi. I have got two questions... 1. How can we determine the pressure at the boundary of two gas phases (say, compressed carbon dioxide released from a can and the atmosphere) which are originally separated? If we apply the ideal gas equation on each of these two phases, different pressures...

Zhonghui Liu of the University of Hong Kong has made the most comprehensive deep-sea core research to date http://www.sciencemag.org/cgi/content/abstract/323/5918/1187?ck=nck. Global SST's fell by an average of 4.5 to 6 degrees F at the E-O boundary, with temps near the South Pole and North Pole...

Hello all, I have a small problem with COMSOL. I am trying to use the particle tracing feature, more precisely I want to use the boundary coordinates to specify where are the points from which to trace the electrons. From what I understand in the User's Giude, the Boundary Coordinates feature...

Hi guys, i need some help on this topic. Basically, i need to calculate the angle of reflection of a oblique shock on a solid boundary. Here is the description of the question i have: an incident supersonic flow reaches a compression corner, with a deflection angle of 15 degrees...

Homework Statement Given a BVP: \Delta(u)+u=1 in \Omega u=0 on \partial\Omega using linear piecewise functions, calculate the corresponding local stiffness matrix on the reference triangle : {(x,y); 0<=x<=1, 0<=y<=1-x}. The domain is a square with one point in the middle (at...

What methods are there to figure out if our universe has a boundary or not? And if it does are we living on the boundary (like in the case of our Earth that we live on a sphere and not inside the earth)?

A partial differential equation requires boundary conditions. Consider a 2-dimensional problem, where the variables are 'x' and 'y'. The boundary is the line x=0 and you are given all sorts of information about the function on that line. If you are given just the values of the function on the...