Boundary Definition and 1000 Threads

Homework Statement Because q(x,t) = A*exp[-(x-ct)2/σ2] is a function of x-ct, it is a solution to the wave equation (on an infinite domain). (a) What are the initial conditions [a(x) and b(x)] that give rise to this form of q(x,t)? (b) if f(x) is constant, then Eq. (2) shows that solution is...

1) Solve $\begin{aligned} {{u}_{t}}&=K{{u}_{xx}},\text{ }0<x<L,\text{ }t>0, \\ {{u}_{x}}(0,t)&=0,\text{ }{{u}_{x}}(L,t)=0,\text{ for }t>0, \\ u(x,0)&=6+\sin \frac{3\pi x}{L} \end{aligned}$ 2) Transform the problem so that the boundary conditions get homogeneous: $\begin{aligned}...

I like to know if electromagnetic waves pass through a medium (Glass) and if this medium were in contact with two other mediums in its boundary which the first one with same optical dense (Glass) and the second with less optical dense (air). Is there any tendency or priority for wave to pass...

Hi, I am solving the diffusion equation using explicit finite difference to model the diffusion of an analyte through a membrane. I am interested in the concentration of the analyte on the other side vs time elapsed. On one side of the membrane is an initial concentration, which I am...

http://imageupload.org/getfile.php?id=171120&a=a955b8cf905480ad89ee96abfea140da&t=4f27421f&o=36BBD6199C49C981F6CC7562A37DC69B664D4F14FBA06DFCF24E9B24875233233B2BA7CB1980&n=%E6%9C%AA%E5%91%BD%E5%90%8D.png&i=1 I have just been given the question without much support so I can't even get a...

Homework Statement From a previous exercise (https://www.physicsforums.com/showthread.php?t=564520), I obtained u(r,\phi) = \frac{1}{2}A_{0} + \sum_{k = 1}^{\infty} r^{k}(A_{k}cos(k\phi) + B_{k}sin(k\phi)) which is the general form of the solution to Laplace equation in a disk of radius a. I...

Homework Statement An electrostatic point charge of 1 Coulomb (C) placed symmetrically between two infinitely/perfectly conducting parallel plates. These two infinitely large conducting plates are parallel to the yz plane.The region between the two plates is designated as “Region A.” Starting...

So I'm reading through Jackson's Electrodynamics book (page 39, 3rd edition), and they're covering the part about Green's theorem, where you have both \Phi and \frac{\delta \Phi}{\delta n} in the surface integral, so we often use either Dirichlet or Neumann BC's to eliminate one of them. So for...

How do I know which function is the upper boundary curve and which is the lower boundary curve. For example find the area between the curves e^x and x bounded on the sides x=0 and x=1. We can draw it and we know that e^x is the upper curve and x is the lower curve. Thus the area is ∫e^x-∫x...

Hi, I am trying to understand solving boundary valued partial differential equations and it's relation to hyperbolic functions. In one of my problems, there is a PDE and the solution contains the hyperbolic function "cosh". I was just curious if anyone has any information for me to read up on...

I'm trying to solve a third-order nonlinear ordinary differential equation. I couldn't get the answer even using Mathematica. The equation is: u'''(t) + u/2 u''(t) = 0 with conditions u(0)=0, u'(0)=0, u(10)=1. I need to get both analytic solution and numerical solution. For the...

Hi, Can anyone explain the difference between axisymmetric and cyclic symmetry boundary conditions? Isn't it the same i.e. bith cyclic symmetry and axisymmetric?

I hear the statement that global symmetries in the boundary field theory corresponds to gauge symmetries in the bulk. 1) Is this a generic statement that is expected to hold for all holography pairs? (Maldacena states this towards the end of his first lecture at PiTP2010, which was supposed to...

I'm writing a method that checks a given coordinates neighbors to see if they are part of the path. the problem I am having with is the conditions that ihave to make sure i don't go outside of the grid. it doesn't seem that the two conditions for col are being checked. i put print statements...

Hi all, I want to calculate the electrostatic potential for an two-dimensional area with given Dirichlet boundary conditions (say, a square) with a charged ring in it (like a wedding ring, but inifinitely thin) with a given line charge density. I figured out that the problem should be...

Hellllllllllo World 1- http://www.physicsclassroom.com/Class/refrn/u14l3a1.gif When a light ray falls on the boundary between two media , some of the energy is reflected back to the first medium,some energy refracts and some energy is absorbed. how is this exlplained?does it has something...

Homework Statement Prove the following: an accumulation point of a set S is either an interior point of S or a boundary point of S. Homework Equations None The Attempt at a Solution Suppose x is not an interior point. Then you cannot find a neighborhood around x such that N is a...

I keep seeing this symbol, something like \oint_{{\partial}\omega}\omega, I know it is a contour integral and read that {\partial}\omega is called a boundary, but I don't know what it means or why there isn't a differential at the end of it. Can someone please answer these questions and explain...

Hi! I have this two related questions: (1) I was thinking that \mathbb{Q} as a subset of \mathbb{R} is a closed set (all its points are boundary points). But when I think of \mathbb{Q} not like a subset, but like a topological space (with the inherited subspace topology), are all it's...

Homework Statement Let Ω\subsetR2 be a region with boundary \Gamma=\Gamma1\bigcup\Gamma2. On Ω we must solve the PDE -{div}(\frac{h^{3}}{12\mu}{grad} p+\frac{h}{2}{u})+kp=f with h and f functions of the spatial coordinates, \mu and k given constants, u a given constant velocity...

Hi, I want to show that the set of boundary points on a manifold with boundary is well defined, i.e the image of a point on a manifold with boundary can not be both the interior point and boundary point on upper half space. To do this, it is enough to show that R^n can be homeomorphic to...

http://en.wikipedia.org/wiki/D-brane says, "The equations of motion of string theory require that the endpoints of an open string (a string with endpoints) satisfy one of two types of boundary conditions: The Neumann boundary condition, corresponding to free endpoints moving through spacetime at...

Hi! I'm implementing a scheme to solve the following equation \frac{\partial \psi}{\partial t}=-c_{s} \cdot \frac{\partial \phi}{\partial x} \frac{\partial \phi}{\partial t}=-c_{s} \cdot \frac{\partial \psi}{\partial x} c_{s} is just the isothermal velocity of sound. The equations are for a...

How would you show that if X is a non-empty set and A\subseteq X then A and A^c have the same boundary? The definition is x\in \partial A \iff there exists r>0 such that the open ball B(x,r) intersects both A and A^c but this is precisely the statement that x\in \partial A^c!

I was looking at this webpage: http://www.ap.smu.ca/demos/index.php?option=com_content&view=article&id=120&Itemid=85 I was wondering, when n2(imag)=0 what would be the merits of using a metal surface and a glass/air boundary (ie internal reflection in a prism) as a mirror surface? Also...

Homework Statement A current I is flowing along the y-axis and a spherical surface with radius 1 m has its center at origin, as in the figure left. A closed contour C is chosen as in the figure, which is a boundary between two semi-sphere surfaces S1 and S2. Based on the...

Hi guys! I'm to find the solution to \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} Subject to an initial condition u(x,0) = u_0(x) = a \exp(- \frac{x^2}{2c^2}) And Neumann boundary conditions \frac{\partial u}{\partial x} (-1,t) = \frac{\partial...

I am working on a hypersonic vehicle project and would like to calculate the boundary layer. The vehicle that I am studying is the Apollo re-entry capsule. I'm assuming a 2-D flow. Here's what I have done so far. Calculate the local surface inclination angle at any point on the blunt body...

Homework Statement Given w'' - w = f(x) w'(0) = 1 w'(1) = 0 Homework Equations Find the Green's Function The Attempt at a Solution The solution to the homogeneous equation is known as: w(x) = A*exp(-x) + B*exp(x) For G's function we have: u(x) = A1*exp(-x) +...

A harmonic function in a region is zero on an open portion of the boundary, and its normal derivative is also zero on the same part, and it is continuously differentiable on the boundary. I have to show that the function is zero everywhere, but I have no idea how. I have tried this for hours...

Hello all: I'm a newbie, trying to write/use code for solving a 2D advection-diffusion problem. I'm not sure how many boundary conditions I should have for the property that is being transported. In my problem, I have diffusion switched off (advection only). The property being...

Not actually a homework question, this is a question from a past exam paper (second year EM and optics): Homework Statement Use a Gaussian surface and Amperian loop to derive the electrostatic boundary conditions for a polarization field P at an interface between media 1 and 2 with...

Hi. I'm trying to solve the heat equation with the initial boundary conditions: u(0,t)=f_1(t) u(x_1,t)=f_2(t) u(x,0)=f(x) 0<x<x_1 t>0 And the heat equation: \frac{\partial u}{\partial t}-k\frac{\partial^2 u}{\partial x^2}=0 So when I make separation of variables I get: \nu=X(x)T(t)...

Hello all: I would very much appreciate advice on setting up a problem. Apologies in advance... This is probably a silly question--I'm more of a chemist than an engineer/math person! I have written a code for calculating changes in concentration/mass within a domain over time, as new...

Lets go straight to the point. I need to find a way to calculate the displacement thickness of a turbulent boyndary layer. The laminar part has been simulated with Thwaite's method but I need to go from there. I'v heard of "Head's-model" but can't find the solution for it. Anyone that can...

Homework Statement Hi, this is the first time I post a thread in this forum. I am not sure if I could post this question here since it is not a homework problem. I have trouble understanding two boundary condition between nonconductor and conductor from Maxwell's equations in dynamic case...

I'm asked to determine if for the solution y=c_{1}e^{x}cos(x)+c_{2}e^{x}sin(x) for: y"-2y'+2y=0 whether a member of the family can be found that satisfies the boundary conditions: y(0)=1, y'(\pi)=0 Not quite sure what to do here. The examples in my book give boundary conditions for the same...

Homework Statement See figure attached for problem statement, as well the solution. Homework Equations The Attempt at a Solution I'm confused as to how he is writing these equations from the boundary conditions. What I understand as the boundary condition for D is, \hat{n}...

Homework Statement No problem, I just have a confusion about a certain concept. Homework Equations The Attempt at a Solution I'm confused as to how they draw the result, \oint_{C} \vec{E} \cdot \vec{dl} = E_{1t}\Delta l - E_{2t}\Delta l = 0 You don't really need to do the...

Homework Statement Prove: If x is an isolated point of a set S, then x is an element of bd S. I believe this could be reworded as, if x is an isolated point of S, then x is also a boundary point of S. Homework Equations None The Attempt at a Solution My book says that an...

Is there a term to describe something like a boundary condition but which can be applied within the domain, not just on the boundaries? For example in a heat transfer problem you might specify a constant rate of heat generation in some region. Is that still called a boundary condition...

EDIT: The subscripts in this question should all be superscripts! Homework Statement I'm trying to solve a temperature problem involving the diffusion equation, which has led me to the expression: X(x) = Cekx+De-kx Where U(x,y) = X(x)Y(y) and I am ignoring any expressions where...

Homework Statement Let X = R2 with the Euclidean metric and let S = {(x1, x2) : x1^2+x2^2 <1}.Prove that Closure of S ={(x1,x2):x1^2+x2^2<= 1} and that the Boundary of S= { (x1, x2) : x1^2 +x2 ^2=1 } . Homework Equations The Attempt at a Solution I was able to prove all my...

Homework Statement Let S = {(x,y): x^{2}+y^{2}<1}. Prove that \overline{S} is (that formula for the unit circle) \leq 1 and the boundary to be x^{2}+y^{2}=1. Homework Equations Boundary of S is denoted as the intersection of the closure of S and the closure of S complement. p \epsilon...

Hi! I am currently studying homology theory and am using Vicks book "Homology Theory, An introduction to algebraic topology". When I was reading I found a definition that troubles me, I simply cannot get my head around it. Vick defines that: if PHI is a singular p-simplex we define di(PHI), a...

Homework Statement Let m be the Lebesgue measure on \mathbb R^d , and define the open sets O_n = \{ x \in \mathbb R^d : d(x,E) < \frac1n \} where d(A,B) = \inf\{ |x-y| : x \in A, y \in B \} 1) Find a closed and unbounded set E such that \lim_{n\to\infty} m(O_n) \neq m(E) . 2) Find an...

Homework Statement The deflection y of a non-uniform beam of length equal to 1, simply supported at both ends and with uniformly distributed load q, is governed by the equation (E*I(x)*y'')'' + k*y=q y(0)=0, y''(0)=0, y(1)=0, y''(0)=0 I(x)=A[1-0.5(1-x)2]2, 0<=x<=1 where E =Young’s...

For fixed-fixed BC's, i have to arrest x and y displ. For simply supported case, i have to arrest y displ. Then my doubt is, while applying force at the end of the column, how the displacement will happen for fixed-fixed column.

Is it possible to impose boundary conditions on the other 2d lattices like a rhombic lattice? a hexagonal lattice? an oblique lattice? How does one typically index such lattices?

Say you have a free particle, non relativistic, and you want to calculate the density of states (number of states with energy E-E+dE). In doing that, textbooks apply periodic boundary conditions (PBC) in a box of length L, and they get L to infinity, and in this way the states become countable...