What is Boundary conditions: Definition and 415 Discussions
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.
Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems involves the eigenfunctions of a differential operator.
To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devoted to proving that boundary value problems arising from scientific and engineering applications are in fact well-posed.
Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's principle.
Could someone plase hep me with normal coordinate substitutions with periodic boundary conditions, I can't see where the 1/N cancels in the attached file
Thanks
Doug
Hi all,
This pertains to a pretty common method of simulating semiconductor devices, but unfortunately I've looked through tons of sources that have been unable to answer my question:
I'm currently working on a 1D device simulator in MATLAB that uses a Newton-Raphson iteration to solve...
Hi friends,
I'm new to CST microwave studio. Just finish constructed a structure of an L-probe patch antenna (from IEEE paper) and just run the simulation by transient time solver, the curve of the return loss(S11) against frequency that i get is different from what showing on the IEEE paper...
Hello
I am trying to build a 3D Poisson solver using method of moments. I need to find out the Green's function for the system. My system is a rectangular box and boundary conditions are as follows:
On all surfaces BC is neumann.
Only on the upper and lower surface, the middle 1/3 region...
I'm reposting this because there was a problem with the title/LaTeX last time.
Homework Statement
Solve the wave equation (1) on the region 0<x<2 subject to the boundary conditions (2) and the initial condition (3) by separation of variables.Homework Equations
(1) \frac{\partial^2...
Though this question arose in quantum mechanics, i think it should be posted here.
Consider a particle in a well with infinite walls:
$i i \frac{\partial \Psi}{\partial t} = -\frac12 \frac{\partial^2 \Psi}{ \partial x^2},\:0<x<a$
but the wall start to squeeze :devil:
$\Psi(x=0,t) \equiv...
Homework Statement
Homework Equations
N/A
The Attempt at a Solution
Really stuck with this, i can't work out how to apply the boundary conditions to generate the simultaneous equations to find the specific solution. Can't find any similar examples either.
Help appreciated.
Homework Statement
Homework Equations
N/A
The Attempt at a Solution
The problem and attempt are as above, I'm not sure where to go from here though. I'm not sure what to do with the boundary condition of dx/dt=-2 and t=0.
Any help appreciated.
Homework Statement
Hi all.
I have the following expression, which relates the incoming amplitude with the reflected amplitude at a point x = L in a coaxial cable:
A_{\text{reflected}} = \frac{R-Z_0}{R+Z_0}A_{\text{incoming}}.
Here R is the resistance at the point x = L and Z0 is the...
Please can anyone tell me how to set this up?
I know how to do the required settings in the Physics/Period Conditions.
However, to fully implement it, I'm also required to choose boundary conditions in the 2D incompressible navier stokes solver (e.g. wall, inlet, outflow, open boundary...
Homework Statement
One half of the region between the plates of a spherical capacitor of inner and outer radii a and b is filled with a linear isotropic dielectric of permittivity \epsilon_1 and the other half has permittivity \epsilon_2, as shown in the figure. If the inner plate has total...
Hi, I have been given a differential equation to use in order to solve for the Hydrogen wavefunction in the ground state using Euler's method.
d^2u_nl/dr^2 -(l(l+1)/r^2)*u_nl + 2k*(E_nl-V(r))*u_nl = 0
V(r) = -a/r where a = 1/137.04
I have been given initial conditions u_nl(0) = 0 an...
Homework Statement
Solve
\left(\frac{\partial^{2}}{\partial z^{2}}- \frac{1}{\kappa} \frac{\partial}{\partial t}\right) T=0
with boundary conditions
(i) T(0,t)=T_{0}+T_{1} Sin[\omega t]
(ii) T is finite as z\rightarrow\inftyHomework Equations
Separation of Variables gives:
T =...
I am trying to solve an ode of the form u"(x)=a(x) where a(x) is some known function and the domain is from -inf to +inf. I am required to use Green's function.
The boundary condition is u(0) = Integral[u(x),{x,0,1}] = 0
My Green's function has the form
G(x,y) = A(y)*x+B(y) x<y
G(x,y)...
Homework Statement
Calculate potential function and the electric field for the region between two concentric cylinders, where V ( inner cylinder) = 0 at r = 0.015 m and V(outer cylinder) = 100 for r = 0.025
Homework Equations
\Delta (square ) V = 0
The Attempt at a Solution
so...
Hi.
You know that B_{1n} = B_{2n} as one of the boundary condition when
magnetic field is go across from material 1 to material 2, n means direction perpendicular to
the boundary surface.
I wander this theorem is right in non-uniform field which is function of space variable r...
Hello everyone,
if an electric field in a medium with permittivity eps_1 moves at an angle into a material with eps_2, there is refraction. The field in the second medium changes angle.
we can see that the tangential component is not changed but the one normal to the surface of separation...
Firstly I really feel so lucky to find this forum. Since I don't have a strong physics background but now dealing with many problems directly related to physics.
I'm now doing some simulation in comsol and need to solve some PDEs. I'm using this PDE coefficient form in comsol. The equations...
Homework Statement
Hi all.
I am given the following differential equation:
X'' - k*X=0.
I am told that k = -m^2, so the general solution is given by:
X = a*cos(m*x)+b*sin(m*x),
where a and b are constants. I am also given boundary conditions:
1) X(-Pi) = X(Pi)
2) X'(-Pi) =...
Homework Statement
Solve Laplace's equation inside the rectangle 0 \le x \le L, 0 \le y \le H with the following boundary conditions
u(0,y) = g(y)\text{, } u(L,y) = 0\text{, } u_y(x,0) = 0\text{, and } u(x,H) = 0
Homework Equations
The Attempt at a Solution
I know that with...
Free Electron Model: Why periodic boundary conditions and what is "L"?
Right, hello!
The quantum free electron model for electrons in solids (in One dimension) says we need to use periodic boundary conditions such that if Y(x) is the wavefunction, then Y(x) = Y(x+L).
Where L seems to be...
Mods: I wasn't sure if I should put this here or in DE's, if you think it would be better there, feel free to move it.
Everyone:
If I have a bunch of particles, which I'm modelling as a continuum, and I want to put periodic boundary conditions which reflect (completely specular reflection)...
Hey folks,
I'm trying to find the Green function for the equation
-\partial_\mu \partial^\mu \phi = K
where K is some source term. Its a 2D problem with the wave confined to a rectangular cavity where the cavity is located at z = 0 and z=a.
This tells me that G|_0= G|_a=0
I've pretty...
Hi there, I am solving laplace equation in a code i made in C. I am using Finite Difference Method in order to simulate the current pattern. I used the stream function to do that...
Hey people can u please tell me what will be the boundary conditions for a circular plate with a central hole clamped at the circumference... Plate is axis symmetric and is under uniform load..
What are the general boundary conditions for nonviscous, incompressible fluid flow? I am trying to find the velocity of fluid at the surface of a sphere with the incident fluid having uniform velocity. I am surprised to find in the solution that the radial velocity at the surface does not...
I can't understand this conditions, and in general every boundary conditions for problems like this. they states "the choice of boundary conditions can be determined by mathematical convenience (!?) ... for if the metal is sufficiently large, we should expect its bulk properties not to be...
Hello friends,
Thanks in advance for your answers,
I am using genreal form of PDE to solve system of PDEs. I am dealing with cyllindrical co-ordinates under axisymmetric case. I am not able to understand how to implement boundary conditions such as,
DEL. Gamma=F is system of PDEs
where...
Hi
I'm trying to solve a magnetostatic problem and I'm not sure which boundary conditions must be applied to the magnetic vector potential (A) on magnetostatic problems?
Thanks in advance.
Hey all,
I'm wondering if someone can help me understand how to apply the boundary conditions to the diffusion equation in one dimension. Diffusion equation is:
\frac{\partial u}{\partial t}=D*\frac{(\partial)^{2}u}{\partial x^{2}}
The initial condition is:
u(x,0)=0
And the boundary...
Hello hello,
I cannot for the life of me wrap my head around the idea of a boundary condition. I understand the idea (at least I think I do) of solving a differential equation with given initial conditions. But is solving for a magnetic field or electric field while enforcing...
I am studying acoustic wave reflection. The boundary conditions of acoustics are continuity of pressure and normal particle velocity. Can anyone tell me if these boundary conditions are completely independent? (Since the pressure and particle velocity are in phase, I would believe they are not...
Hey guys, just need some hints with this doosey
Homework Statement
We have
(x^2 y')' + ax^2y = 0 where a the eigenvalue (a sturm-lioville problem) (sp?)
with y'(0)=y(1) = 0 and we get the hint to substitute f = y/x.
The Attempt at a Solution
Ok so i get the general solution being a sum of...
Homework Statement
A particle of mass m is confined to move in one dimension. its wavefunction is periodic with period L\gg 1 - i.e. periodic boundary conditions are imposed.
a)Determine the eigenfunctions and eigenvalues of momentum. Normalise the eigenfunctions on the interval [0,L)...
If I have a finite boundary, say of length L. Is it possible to demonstrate that if I were to allow all possible CONTINUOUS values of a wave to exist (with unit amplitude) then deconstrutive interference destroys all waves except those with wavelength:
k=\frac{n\pi}{L}
Where n =0,1...
One thing that's always bothered me about Bloch's theorem is the periodic boundary conditions which are imposed on the system. Clearly, when dealing with an actual solid, the more natural choice would be to impose zero at the boundaries. I know that periodic conditions make the math easier, but...
I wasn't completely sure where to put this (programming or Diff.E.'s), so if there's a better place, maybe the mentors could move it for me.
I'm doing some numerical simulations involving the (2-D) wave equation, and was wondering if anyone could tell me (or give a reference to a paper which...
solve the next differential equation:
y´´- a*y= \delta (x-d)
with the boundary conditions:
\left.\frac{\partial y}{\partial x} \right|_ {x=0} = 0
lim _{x\rightarrow\infty} y = 0
I get the homogeneous solution: y_H = C_1 exp (\sqrt{a}x) + C_2 exp (-\sqrt{a}x)
and then to...
Hi,
I'm trying to find an analytical solution of Laplace's equation:
\phi_{xx} + \phi_{tt} = 0
with the tricky boundary conditions:
1. \phi(x=0,|t|>\tau)= 0
2. \phi(x\neq0, |t|>>\tau)=0
3. \phi_{x}(x=0, |t|<\tau)=-1
4. \phi_{t}(x, |t|>>\tau)=0
I have the following ansatz(I...
Homework Statement
What is the stationary (steady state) solution to the following reaction diffusion equation:
\frac{\partial C}{\partial t}= \nabla^2C - kC
Subject to the boundary conditions C(x, y=0) = 1, C(x = 0, y) = C(x = L, y) (IE, periodic boundary conditions along the...
Hi to all community of Physic's help from Florence,
looking at born-von karman BC I'm a bit confused. I put this condition when i assume periodicity of wave function where the period is the spatial dimension of my system. I found that BC first in solid state physic, then I've noticed that...
Hi,
Can anyone please tell me how to go about solving this system of coupled ODEs.?
1) (-)(lambda) + vH''' = -2HH' +(H')^2 - G^2
2) vG'' = 2H'G - 2G'H
lambda and v are constants.
And the boundary conditions given are
H(0) = H(d) = 0
H'(0) = omega * ( c1 * H''(0) + c2 * H'''(0) )...
Hi
I am new to solid state. I just read about fermi gas in a cube. For some reason the author used periodic boundary conditions? Why didn't they choose finite well potential where the height of the well is related to the work function?
I need help figuring out the solution to this diff.eq.
y(x) = x + (1/2)*∫(from u=-1 to 1)[ ( 1-| x – u | ) y(u) du] , x є [ -1, 1]
I have to show that:
y``(x) + y(x) = 0 , x є [ -1, 1]
subject to:
y(1) + y(-1) = 0
y`(1) + y`(-1) = 2
Thanks for any help you can give.
Homework Statement
The edges of a square sheet of thermally conducting material are at x=0, x=L, y= -L/2 and y=L/2
The temperature of these edges are controlled to be:
T = T0 at x = 0 and x = L
T = T0 + T1sin(pi*x/L) at y = -L/2 and y = L/2
where T0 and T1 are constants...
how do you prove/show that there really is a vector space defined by certain boundary conditions?
unfortunatly this part of pde's was glossed over in my professor's lecture notes and I don't recall him talking about it in class.
I have done most of a question except for the most important part, putting in the boundary conditions, I can't really interpret them.
The question is:
I managed to solve this, with -c^2 as a separation constant, and I got:
T(x,t) = X(x)F(t) = (A_{1} \cos{\frac{cx}{\sqrt{\kappa}}} + A_{2}...