Boundary Definition and 900 Threads

Hey! :o I have a question.. (Wasntme) When we have the following boundary value problem: $$u_{xx}+u_{yy}=0, 0<x<a, 0<y<b (1)$$ $$u_x(0,y)=u_x(a,y)=0, 0<y<b$$ $$u(x,0)=x, u_y(x,b)=0, 0<x<a$$using the method of separation of variables, the solution would be of the form $u(x,y)=X(x) \cdot Y(y)$...

Hey! :o I have to solve the following boundary value problem: $$u_{xx}+=u_{yy}=0, 0<x<a, 0<y<b$$ $$u_x(0,y)=u_x(a,y)=0, 0<y<b$$ $$u(x,0)=x, u_y(x,b)=0, 0<x<a$$ By using the method of separation of variables, the solution is of the form $u(x,y)=X(x)Y(y)$. By substituting this it the problem we...

Hi, Given a holomorphic function u(x,y) defined in the half plane ( x\in (-\infty,\infty), y\in (-\infty,0)), with boundary value u(x,0) = f(x) , the solution to this equation (known as the Poisson integral formula) is u(x,y) = \int_{-\infty}^{\infty} \frac{y\ f(t) }{(t-x)^2 +y^2} \...

For an imperfect conductor, when there is current, an electric field is set up inside the wire along the direction of the current flow, and is parallel to the wire. If this is true, then what I don't understand is boundary condition tells me the tangential E-field is always continuous, if...

I don't really understand boundary conditions and I've been trying to research it for ages now but to no real avail. I understand what boundary conditions are, I think. You need them along with the initial conditions of a wire/string in order to describe the shape of motion of the string. I...

Hi all, I was reviewing some old material on the representation of orientable surfaces in terms of disks and bands , in page 2 of: http://www.maths.ed.ac.uk/~jcollins/Knot_Theory.pdf Please tell me if I am correct here. Assume there is a horizontal line dividing the surface into an...

Homework Statement A tight string lies along the positive x-axis when unperturbed. Its displacement from the x-axis is denoted by y(x, t). It is attached to a boundary at x = 0. The condition at the boundary is y+\alpha \frac{\partial y}{\partial x} =0 where \alpha is a constant. Write the...

Homework Statement I have been given a complex function I have been given a complex function \widetilde{U}(x,t)=X(x)e(i\omega t) Where X(x) may be complex I have also been told that it obeys the heat equation...

Homework Statement A unit sphere at the origin contains no free charge or conductors in its interior or on its boundary. It is, however, embedded in a dielectric medium. The dielectric is linear, but the permitivity varies by angle about the origin. It is constant along any radial direction...

Homework Statement Hello, I have to demonstrate that multiplying a differential equation: -d/dx[a(x)*d/dx{u(x)}]=f(x), 0<x<1 subject to u(0)=0 and u(1)=0. by some function v(x) and integrating over an interval [0,1], I get a new equation that can be used in an optimisation problem, that...

Hi all, Generally boundary condition (Dirichlet and Neumann) are applied on the Load Vector, in FEM formulation. The equation i solved, is Generalized eigenvalue equation for Scalar Helmholtz equation in homogeneous wave guide with perfectly conducting wall ( Kψ =λMψ ), and found, doesn't...

Homework Statement Stuck on two similar problems: "State the normal stress boundary condition at an interface x_3-h(x_1,x_2,t)=0between an invisicid incompressible fluid and a vacuum. You may assume that the interface has a constant tension." The second question in the same but the fluid is...

This is my first post so I hope this in the right place. I am fairly sure this is quite a straight forward question but I having trouble working out the details of it. "State the boundary conditions for an inviscid fluid at an impermeable fixed boundary x_3-h(x_1,x_3)=0 where we do...

I've been trying to prove that the closed unit ball is a manifold with boudnary, using the stereographic projection but I cannot seem to be able to get any progress. Can anyone give me a hint on how to prove it? Thanks in advance :)

Hi guys, so I'm stuck on quite an interesting problem, and have been for a few days now. If anybody can take the time to have a look at it that would be the most incredible thing ever, because I have reached a point where I am at a loss. Solve the following 4th order differential equation...

hi all... I have written codes for 2d fdtd tfsf with berenger's pml absorbing boundary. But I have serious leakage at front and right boundary. At the first place I think the problem is because the pml, but pml is working perfectly in the left and bottom boundary. I need an advice whether the...

So I know individually how these form. Unfortunately I haven't found any sources that describe more detailed questions that pop up in my mind. Could someone help me answer a couple of questions? 1. So if a thermal boundary layer forms in a 'plug flow' model i.e. when there is no momentum...

Hi, it is easy solving these PDEs with the idealized homogeneous BCs they throw out in class, but I am having some difficulty solving the transient problem posed in the images below. I have tried working through it, but I don't have confidence in the result. I overlook the solution when the...

Homework Statement Find the volume of an object bounded by x2 + y2 ≤ 1, x2 + z2 ≤ 1 and y2 + z2 ≤ 1. Homework Equations The Attempt at a Solution This stuff is very new to me (multiple integrals to find volume) so I am not entirely familiar with it. My first thought was to put...

Homework Statement Consider the Laplace’s equation, ∆u(r,θ) = 0, inside the quarter-circle of radius 2 (0 ≤ θ < π, 0 ≤ r ≤ 2), where the boundary θ is insulated, and u(r,\theta/2)=0 Show that the insulated boundary condition can mathematically be expressed as \frac{\partial u}{\partial...

Homework Statement Determine the interior, the boundary and the closure of the set {z ε: Re(z2>1} Is the interior of the set path-connected? Homework Equations Re(z)=(z+z*)/2 The Attempt at a Solution Alright so z2=(x+iy)(x+iy)=x2+2ixy-y2 so Re(x2+2ixy-y2)= x2-y2 >1 So would...

Homework Statement Find the deflection at x=L/4 and x=L/2 for the beam Homework Equations See attached pic. The Attempt at a Solution So I have the solution derived in class. Only 0<x<L/2 is derived because since the load on the beam is at L/2, the equation is valid for the...

I'm working on a student design project building a multirotor UAV to host a sensor array. The airframe supports arm beams with motors producing thrust at the end, a battery, a flight controller, payload, ESC's and needs to be custom made so that it is of a size that can support large blades and...

Hello, I am looking to solve the 3D equation in spherical coordinates \nabla \cdot \vec{J} = 0 using the Ohm's law \vec{J} = \sigma \cdot (\vec{E} + \vec{U} \times \vec{B}) where \sigma is a given 3x3 nonsymmetric conductivity matrix and U,B are given vector fields. I desire the...

This is part of a solution to find the average of power consumption. In the solution, the boundary taken is from -T/2 to T/2. Can we integrate from 0 to T instead?

Hi, I have to use a wide 5 point stencil to solve a problem to fourth order accuracy. In particular, the one I'm using is: u'' = -f(x + 2h) + 16f(x + h) - 30f(x) + 16f(x - h) - f(x - 2h) / 12h2 or when discretized u'' = -Uj-2 + 16Uj-1 -30Uj + 16Uj+1 -Uj+2 / 12h2 In addition to...

I'm trying to implement a numerical code for the diffusion equation with moving boundaries. I have no problems with the numerical implementation, but with the transformation of coordinates. In spherical coordinates, the diffusion equation is \frac{\partial c}{\partial t} = D...

Hi, I'm trying to show that if ## M^n ## is orientable and connected, with boundary (say with just one boundary component), then its top homology is zero. Sorry, I have not done much differential topology/geometry in a while. I'm trying to avoid using Mayer-Vietoris, by using this argument...

Hello! I have a nifty set of problems (or rather one problem, gradually building itself to be a great problem) that I like to collectively call "The final problem" as it is the last thing I need before I can take the exam in Numerical Methods.Information There is given a Laplace equation...

In Griffith's section about electrostatic boundary conditions, he says that given a surface with charge density \sigma , and take a wafer-thin Gaussian pillbox extending over the top and bottom of the surface, Gauss's law states that: \oint_{S} \mathbf{E} \cdot d \mathbf{a} =...

hi all, i have wrote codes for 2d fdtd in different permittivity (epsilon). in this code, cell size is 200 x 200, start with eps=1 from center, and different permittivity started at boundary (i1,i2) = (25,75) = (j1,j2), epsilon = 2. the problem is, when the wave propagates and approach...

Hi, please could someone help clarify the reason why a mechanical wave is inverted at a boundary as I'm really stuck! Some sources I have read seem to suggest it can be explained by Newton's 3rd law whilst others suggest its to do with conservation of momentum. Newton's 3rd law - consider...

Can anyone explain me why the thermal boundary layer develops faster for viscous fluids? I would just say it would develop more slowly because due to high viscosities there are low reynoldsnumbers and thus less turbulence or mixing. This causes a slow homogenization of temperature (assume a...

Homework Statement now I have a PDE $$u_{xx}+u_{yy}=0,for 0<x,y<1$$ $$u(x,0)=x,u(0,y)=y^2,u(x,1)=0,u(1,y)=y$$ Then I want to know whether there are some method to make the PDE become homogeneous boundary condition. $$i.e. u|_{\partialΩ}=0$$

Homework Statement Consider a charged particle, of mass m and charge q, confined in a device called a Penning Trap. In this device, there is a quadrupole electric field described in cartesian coordinates by the potential Phi[x,y,z] = U0 (2z^2 - x^2 - y^2) / (r0^2 + 2z0^2) Where U0 is...

In linearized gravity we can one sets $$(1) \ \ g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$$ where h is taken to be a small perturbation about the flat space metric. One common decomposition of h is to write the spatial part as $$ h_{i j} = 2 s_{ij} - 2\psi \delta_{ij} \ h_{0i} \equiv...

Homework Statement Just need some quick confirmation. For a beam which has a load applied to it, will its free end always have a shear force, bending moment and curvature of zero?

Homework Statement A flat plate moves in water (20°C) in the direction of the plate at a speed of 1 m/s. What is the boundary layer thickness 0.1 meter downstream of the plate?Homework Equations Reynolds number: ##Re_x=\frac{xU_∞}{\nu}## Boundary layer thickness for laminar flow...

Homework Statement Determining two sets of boundary conditions for a double integral problem in the polar coordinate system. Is the below correct? Homework Equations The Attempt at a Solution There are two sets of boundary conditions that you can use to solve this problem in the polar...

I have two questions: [SIZE="5"](1)As the tittle, if u(a,\theta,t)=0, is \frac{\partial{u}}{\partial {t}}=\frac{\partial^2{u}}{\partial {r}^2}+\frac{1}{r}\frac{\partial{u}}{\partial {r}}+\frac{1}{r^2}\frac{\partial^2{u}}{\partial {\theta}^2} and \frac{\partial^2{u}}{\partial...

My first analysis/topology text defined the boundary of a set S as the set of all points whose neighborhoods had some point in the set S and some point outside the set S. It also defined the closure of a set S the union of S and its boundary. Using this, we can prove that the closure of S is...

What is the answer of this differential equation. ((d^2) r)/((ds)^2) +(m/(r^2)) -(nr/3)=0 the boundary conditions (i) r=a when s=0 and (ii) dr/ds =0 when r=b. m and n are constants.

Homework Statement Given a non-linear decision boundary line: (1 + X1)^2 + (2 − X2)^2 = 4 Argue that while the decision boundary is not linear in terms of X1 and X2, it is linear in terms of X1,X1^2 , X2, and X2^2 . The Attempt at a Solution I'm honestly not sure. I realize the...

Homework Statement Ignore the text in German. You just need to see the picture. 2 conductors both with potential 0 are given. \alpha is the angle between the conductors. (r, \varphi) are polar coordinates pointing to a point in the plane. Homework Equations What we need to do is...

In the textbook I am working with, an isolated point of A is defined to be a point X in A such that there exists a neighborhood (open ε-ball) centered on X containing no point in A other than X itself. A boundary point of A (which need not be in A) is defined as a point X in A such that...

So when we calculate divergence (especially referring to the gauss divergence theorem), why aren't the components of the vector field parallel to the boundary considered? I mean even of, say fluid, is traveling parallel to the boundary when it comes out, fluid is exiting, or diverging out...

Homework Statement solve the heat equation over the interval [0,1] with the following initial data and mixed boundary conditions.Homework Equations \partial _{t}u=2\partial _{x}^{2}u u(0,t)=0, \frac{\partial u}{\partial x}(1,t)=0 with B.C u(x,0)=f(x) where f is piecewise with values: 0...

Hi, I'd love a to have a more graphical understanding of how a plane wave interacts with a boundary. I know the maths that describes it, Fresnel's equations etc, and how Brewster's angle is derived and stuff. I'm rather confused with the dipole concept. From what I understand, when a plane...

What are the boundary conditions of the universe?

Wheres the limit between quantum mechanics and classical mechanics. I mean,when can I expect quantum behavior on a system, is it depends on the system size?Tempature? Something else...and if so what are the numbera for those limits. As we know in nature everything is continuous, so, the...