Boundary Definition and 1000 Threads

Homework Statement L[y] = \frac{d^2y}{dx^2} Show that the Green's function for the boundary value problem with y(-1) = 0 and y(1) = 0 is given by G(x,y) = \frac{1}{2}(1-x)(1+y) for -1\leq y \leq x \leq 1\ G(x,y) = \frac{1}{2}(1+x)(1-y) for -1\leq x \leq y \leq...

Let's say I have a set of nonlinear differential equations of the form. x' = f(x,y) \\ y' = g(x,y) Where f and g contain some parameter 'a' that is constrained to within certain values. Let's say I know x(0), y(0) and x(T), y(T) where T isn't a set value. What methods can I use to...

Reading through Spivak's Calculus on Manifolds and some basic books in Analysis I notice that the divergence theorem is derived for surfaces or manifolds with boundary. I am trying to understand the case where I can apply the divergence theorem on a surface without boundary.

Homework Statement Solve the diffusion equation: u_{xx}-\alpha^2 u_{t}=0 With the boundary and initial conditions: u(0,t)=u_{0} u(L,t)=u_{L} u(x,0=\phi(x) The Attempt at a Solution I want to solve using separation of variables... I start by assuming a solution of the form...

Hi I cannot find an equation for a boundary layer in a pipe flow (laminar). I am looking for an equivalent of the equation δ(x)=4.91x/(√Re) that works for a flow between plates (x is the distance downstream). The thing is- I am looking for BL thickness for still undeveloped flow. I would be...

$$ u_{tt} + 3u_t = u_{xx}\Rightarrow \varphi\psi'' + 3\varphi\psi' = \varphi''\psi. $$ $$ u(0,t) = u(\pi,t) = 0 $$ $$ u(x,0) = 0\quad\text{and}\quad u_t(x,0) = 10 $$ \[\varphi(x) = A\cos kx + B\sin kx\\\] \begin{alignat*}{3} \psi(t) & = & C\exp\left(-\frac{3t}{2}\right)\exp\left[t\frac{\sqrt{9...

I'm having trouble finding out how to set up Neumann (or, rather, "Robin") boundary conditions for a diffusion-type PDE. More specifically, I have a scalar function f(\boldsymbol{x}, t) where \boldsymbol{x} is n-dimensional vector space with some boundary region defined by A(\boldsymbol{x})=0...

Homework Statement The steady state temperature distribution T(x,y) in a flat metal sheet obeys the partial differential equation: \displaystyle \frac{\partial^2 T}{\partial x^2}+ \frac{\partial^2 T}{\partial y^2} = 0 Seperate the variables in this equation just like in the...

Hi all, Am reading a few papers for a Uni case study about structures in Turbulent Boundary Layers over a Flat Plate (water), particularly low-speed streaks. I'm confused over what mechanism causes the span wise variations in velocity that seems to cause low speed streaks. Would...

I'm having a tremendously hard time understanding the connection between macro and micro scale electrostatics and how (if?) they're described EQS boundary conditions. I understand that in a medium with mobile ions, an applied current or field will lead to the establishment of an electric double...

μ^{2}\frac{d^{2}u}{dx^{2}}+ae^{u}=0 Boundary conditions: u(-L)=u(L)=u_{0} Solve by multiplying by \frac{du}{dx} and integrating in x I know you have to use substitution, but I keep going in circles.

One of the boundary conditions for a homogeneous uniform waveguide is \frac{\partial H_z}{\partial n}=0. What does this mean physically?

Homework Statement y'' +λy=0 y(1)+y'(1)=0 Show that y=Acos(αx)+Bsin(αx) satisfies the endpoint conditions if and only if B=0 and α is a positive root of the equation tan(z)=1/z. These roots (a_{n})^{∞}_{1} are the abscissas of the points of intersection of the curves y=tan(x) and...

Laplace axisymmetric $u(a,\theta) = f(\theta)$ and $u(b,\theta) = 0$ where $a<\theta<b$. The general soln is $$ u(r,\theta) = \sum_{n=0}^{\infty}A_n r^n P_n(\cos\theta) + B_n\frac{1}{r^{n+1}}P_n(\cos\theta) $$ I am supposed to obtain $$ u(r,\theta) = \sum_{n =...

BOUNDARY CONDITION PROBLEM I have came up with matrix for numerical solution for a problem where chemical is introduced to channel domain, concentration equation: δc/δt=D*((δ^2c)/(δx^2))-kc assuming boundary conditions for c(x,t) as : c(0,t)=1, c(a,t)=0. Where a is channel's length...

Homework Statement BOUNDARY CONDITION PROBLEM I have came up with matrix for numerical solution for a problem where chemical is introduced to channel domain, concentration equation: δc/δt=D*((δ^2c)/(δx^2))-kc assuming boundary conditions for c(x,t) as : c(0,t)=1, c(a,t)=0. Where a is...

Homework Statement Show that the boundary-value problem $$u_{tt}=u_{xx}\qquad u(x,0)=2f(x)\qquad u_t(x,0)=2g(x)$$ has the solution $$u(x,t)=f(x+t)+f(x-t)+G(x+t)-G(x-t)$$ where ##G## is an antiderivative/indefinite integral of ##g##. Here, we assume that ##-\infty<x<\infty## and ##t\geq 0##...

PROBLEM: I am asked to consider a parallel polarized planar wave with frequency ω is normally incident on a dielectric boundary. The incident time average power flux P_i = 100 w/m^2. The first medium is free space and the second has vacuum permeability but ε=4ε_0. We are also given that the...

Homework Statement Using the definition of linearity to determine whether or not ech case is a linear homegeneous boundary condition: i.) Uxx(0,y)=Ux(0,y)U(0,y) ii.)Uy(x,0)=Ux(5,y) Homework Equations The Attempt at a Solution I know Uxx(0,y)=Ux(0,y)U(0,y) is not linear...

I am give probability distribution function f(x)=(e(-x/1000))/1000 of the time to failure of an electronic component in a copier The question is to determine the number of hours at which 10% of all components have failed. My solution: 1) PDF was integrated to obtain: f(x)= e(-x/1000)...

Hi, I was wondering if you can apply shooting method to a 4nd differential eq. two point value boundary problem, specifically I want to use this method to solve Euler-Bernoulli eq. EI y(4)(x)=f(x), y(0)=0,y'(0)=0,y(L)=0,y'(L)=0. Normally, if you have a 2nd order two point value...

Homework Statement Consider a flat plate moving at V = 105 ft/s in an atmosphere with P = 2100 psf and T = 75F, with a viscosity of vis = 3.7373e-7 slug/ft-s. The plate has a 2 foot chord and a 3.5 foot span. a. Find the drag on the plate for: i. A boundary layer that is laminar over the...

I'm currently having a bit of an intuitive problem understanding the Prandtl number effect on boundary layers and I'm hoping that someone can explain it better than what I've read in some heat transfer books. According to various HT books, a low Prandtl number means that heat diffuses quickly...

Let "a" be a non zero vector in R^n and define S = { x in R^n s.t. "a" · "x" = 0}. Determine S^int , bkundary of S, and closure of S. Prove your answer is correct Attempt: Ok I am more sk having trouble proving that the respective points belong to its condition. Such as thr...

Hi As we know, we have two kinds of Electromagnetic Boundary Conditions for interfaces in an electromagnetic problem.one is imposing the continuity of Bz and Hr and the other is applying the continuity of A(Magnetic Vector Potential) and the discontinuity of its derivative with respect to the...

we know that the full life of radioactive sample is infinity. and if we take one radioactive atom in the sample then it's full is measurable and short. it changes when when it emits sub atomic particles. which contradict each other. we need to know more. the dimension of radioactive constant...

Homework Statement Compute the eigenvalues/functions of the given regular S-L problem f''(x)+λf(x)=0 0<x<π f(0)=0 f'(π)=0 2. The attempt at a solution First off, why is π not included in the given boundary if it tells you f'(x) at π? Now for my attempt: assuming λ=0...

$$ \left.(\phi_n\phi_m' - \phi_m\phi_n')\right|_0^L + (\lambda_m^2 - \lambda_n^2)\int_0^L\phi_n\phi_m dx = 0 $$ where $\phi_{n,m}$ and $\lambda_{n,m}$ represent distinct modal eigenfunctions which satisfy mixed boundary conditions at $x = 0,L$ of the form \begin{alignat*}{3} a\phi(0) + b\phi'(0)...

Homework Statement Let A, C \subseteq ℝn with boundaries B(A) and B(C) respectively. Prove or disprove : B(AUC) O B(A)UB(C) and B(A\capC) O B(A)\capB(C) Where O represents each of these symbols : \subseteq, \supseteq, = Homework Equations I know that double inclusion is going to cut the...

I am really confused with the concept of Neumann Boundary conditions. For the simple PDE ut=uxx for the domain from 0<=x<=1 I'm trying to use a ghost point (maintain a second order scheme) for the Neumann Boundary condition ux(0,t) = 0. I understand that I can setup a scheme to...

Homework Statement I have a general wave equation on the half line utt-c2uxx=0 u(x,0)=α(x) ut(x,0)=β(x) and the boundary condition; ut(0,t)=cηux where α is α extended as an odd function to the real line (and same for β) I have to find the d'alembert solution for x>=0; and show that in...

I hear about the balloon analogy, and that there is no need to say that the universe has a boundary, but is that the only reason or would it be problematic to assume that space-time has a volume and a boundary?

Hi all, I'm doing what should be a pretty simple problem, but some theory is giving me trouble. Basically, in this problem I have a conducting sphere, surrounded by a thick insulating layer, and then vacuum outside that. I'm attempting to solve for the potential in the insulating layer by...

Hi, Say I have this pde: u_t=\alpha u_{xx} u(0,t)=\sin{x}+\sin{2x} u(L,t)=0 I know the solution for the pde below is v(x,t): v_t=\alpha v_{xx} v(0,t)=\sin{x} v(L,t)=0 And I know the solution for the pde below is w(x,t) w_t=\alpha w_{xx} w(0,t)=\sin{2x} w(L,t)=0 Would...

Here is the problem: If M is a manifold with boundary, then find a retraction r:U→∂M where U is a neighborhood of ∂M. I realize the Collar Neighborhood Theorem essentially provides the desired map, but I am actually using this result to prove the aforementioned theorem. My thought on how to...

Im wondering how potential can act as a boundary for electrons in a 1-D time independent infinite well?

Hi, I am looking at a problem where I have two electrically conducting fluids where charge accrues on the interface, I know that one of the equations that I have to use comes straight from the usual boundary conditions for the normal component of the electric field, the other one apparent comes...

Homework Statement A one-dimensional wave function associated with a localized particle can be written as \varphi (x) = \begin{cases} 1- \frac{x^2}{8}, & \text{if } 0<x<4, \\ C_1 - \frac{C_2}{x^2}, & \text{if} \,x \geq 4. \end{cases} Determine C_1 and C_2 for which this wave...

Hi all, I am currently studying civil hydraulics in my civil engineering course and we are going through estimating critical shear stresses for sediments. I am confused about the difference between boundary shear stress and particle shear stress. In terms of estimating critical shear stress...

in electromagnetics , considering boundary conditions of dielectric and perfect conductor , inside conductor E = 0. So, there should not be any time varying magnetic field. But in many books i have seen that inside conductor normal component of B is 0 because there is no time varying magnetic...

Hello , i am trying to implement this algorithm for 2d grid. 1) i am not sure if my calculations are correct. 2 ) i don't understand how to return my final calculation ( how will i insert to the matrix i want (the 's' in this example) the new coordinates (xup,xdow,yup,ydown)). I mean ...

Good afternoon, I am a PhD student in motions of damaged ships. I am trying to find a solution of Laplace equation inside a box with a set of boundary conditions such that: ∇2\phi=0 \phix=1 when x=-A and x=A \phiy=0 when y=-B and y=B \phiz=0 when z=Ztop and z=Zbot I have tried...

Is there any boundary of age for any jobs such as International labs, an academic carieer in university ? The reason why i ask you is that I'm at 29 and I still major in MSc physics or physics enginnering. To you, Is it a logical ? So,i have been lated to apply any job in U.S or Europe ...

The electric field in a cubical cavity of side length L with perfectly conducting walls is E_x = E_1 cos(n_1 x \pi/L) sin(n_2 y \pi/L) sin(n_3 z \pi/L) sin(\omega t) E_y = E_2 sin(n_1 x \pi/L) cos(n_2 y \pi/L) sin(n_3 z \pi/L) sin(\omega t) E_z = E_3 sin(n_1 x \pi/L) sin(n_2 y \pi/L)...

I am trying to solve four coupled equations. Three of them are first order differential equations and the fourth is a algebraic one. The equations look something like this: V_{l}(r) = f_{1}(r)W'_{l}(r) (1) h''_{l} + f_{2}(r)h'_{l} + f_{3}(r)h_{l}(r) = U_{l}(r) (2) f_{4}(r)U'_{l} +...

Hi!This is a quite sophisticated problem, but it’s interesting and challenging! Consider the following case: Let’s say we have a 3-dimensional disk with a radius r_{2} and a thickness d (so it actually is a cylinder with a quite short height compared to radius). We’re interested in solving...

i was studying incidence of EM wave at a plane dielectric boundary and encountered equations in the attachment . I just want to know if n2 > n1 then electric field amplitude at the boundary increases . So from where does this extra value comes ? n1 and n2 are intrinsic impedances of 1st and...

Homework Statement At the boundary between water (n=1.33) and flint glass (n=1.66), incoming light at ~49 degrees from the normal is refracted. Of course, I can use Snell's law to calculate the angle of refraction. However, my question is whether any of the light at this boundary is also...

hi can anyone teach me how to put Mur's ABC in my fortran code for 1d fdtd maxwell's equation as below !1d fdtd Simulation in free space subroutine fd1d01(f0,miu,delta,S,E0) implicit none double precision :: f0 !frequency double precision...

Not really a specific problem, but just a general question: Does anyone have any good references (preferably online) for solving E&M problems with this method? I'm using Griffith's Electrodynamics book for my class and I'm trying to get ready for a final. This is the only part I'm having...