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.
Hey, I have a really short question about electrostatics.
The boundary conditions are :
\mathbf{E}^{\perp }_{above} - \mathbf{E}^{\perp}_{below} = -\frac{\sigma}{\varepsilon_{0}}\mathbf{\hat{n}} ,
\mathbf{E}^{\parallel }_{above} = \mathbf{E}^{\parallel}_{below}.
My question is what is...
The Hawking-Hartle no boundary condition is well known. The authors considered a many worlds/histories model considering a sum over all compact euclidean metrics.
But are there any models or theories that consider a sum over all possible metrics or boundaries?
And finally, if all possible...
Hello, I was going to solve numerically the eigenfunctions and eigenvalues problem of the schrödinger equation with Yukawa Potential. I thought that the Boundary condition of the eigenfunctions could be the same as in the case of Coulomb potential. Am I wrong? In that case, do you know some...
Hello, I was going to solve with a calculator the eigenvalues problem of the Schrödinger equation with Yukawa potential and I was thinking that the boundary conditions on the eigenfunctions could be the same as in the case of Coulomb potential because for r -> 0 the exponential term goes to 1...
Hi everyone!
I am studying spectral methods to solve PDEs having in mind to solve a heat equation in 2D, but now i am struggling with the time evolution with boundary conditions even in 1D. For example,
$$
u_t=k u_{xx},
$$
$$
u(t,-1)=\alpha,
$$
$$
u(t,1)=\beta,
$$
$$
u(0,x)=f(x),
$$
$$...
Hi! I want to use Euler's equations to model a 2 dimensional, incompressible, non-viscous fluid under steady flow (essentially the simplest case I can think of). I'm trying to use the finite difference method and convert the differential equations into matrices to be solved using MATLAB. I set...
Hello guys.
I am studying the heat equation in polar coordinates
$$
u_t=k(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta})
$$
via separation of variables.
$$u(r,\theta,t)=T(t)R(r)\Theta(\theta)$$
which gives the ODEs
$$T''+k \lambda^2 T=0$$
$$r^2R''+rR+(\lambda^2 r^2-\mu^2)R=0$$...
Summary: Questions about the Multiverse hypothesis and the 'No boundary' conditions approach in cosmology
I have some questions about James Hartle and Stephen Hawking's 'No-boundary' proposal:
- In their approach multiple histories would exist. These histories could yield universes with...
the image is given here along with some numerical information:
Now I know that the formula for the electric field in a capacitor is given as:
$$E = \frac{V}{d}$$
which I can use to obtain the three following fomulas:
$$E_1 = \frac{V_1}{d}$$
$$E_2 = \frac{V_2}{d}$$
$$E_3 =...
I am solving a problem of the boundary condition of Dirichlet type, in order to solve the problem, the functions within the differential equations suppose to be harmonic.
I have a function f(x,y,z) (the function attached) which is not harmonic. I must find an equivalent function g(x,y,z) which...
I am trying to understand Aubry-Andre model. It has the following form
$$H=∑_n c^†_nc_{n+1}+H.C.+V∑_n cos(2πβn)c^†_nc_n$$
This reference (at the 3rd page) says that if ##\beta## is irrational (rational) then the period of potential is quasi-periodic incommensurate (periodic commensurate) with...
Homework Statement
Two magnetic materials are separated by a planar boundary. The first magnetic material has a relative permeability μr2=2; the second material has a relative permeability μr2=3. A magnetic field of magnitude B1= 4 T exists within the first material. The boundary is...
Homework Statement
Suppose we have the standard rectangular potential barrier in 1D, with
$$
V =
\left\{
\!
\begin{aligned}
0 & \,\text{ if } x<0, x>d\\
V_0 & \,\text{ if } x>0,x<d\\
\end{aligned}
\right.
$$
The standard approach to solve for tunneling through the barrier is to match the...
I am unsure how to choose the boundary conditions for a system of PDEs or for a single PDE for that matter.
The situation I am stuck with involves a system of 4 PDEs describing plasma in a cylinder. The dependent variables being used are Vr, Vt, Vz, ni, and the independent variables are Rr...
Homework Statement
I have to calculate the stationary field inside a room.
Homework Equations
The Attempt at a Solution
I used the diffusion equation to calculate the temperature, which is
T(x,y)=(Eeknx+Fe-knx)cos(kny),
k=(n*pi/a), a is the length of the room.
Now i have to satisfy boundary...
The question is basically find the boundary conditions when ##l=0##, for energies minor than 0.
Homework Equations
$$V(r)=\begin{cases}
& 0\text{ $r<a_0$}\\
&V_0\text{ $a_0<r<a_1$}\\
& 0\text{ $r>a_1$}\\
\end{cases}
$$
$$...
Dario SLC
Thread
bound states
boundarycondition
schrodinger equation
1. Homework Statement
A particle with mass m and spin 1/2, it is subject in a spherical potencial step with height ##V_0##.
How is the general form for the eigenfunctions?
What is the boundary conditions for this eigenfunctions?
Find the degeneracy level for the energy, when it is ##E<V_0##
2...
I have a couple homework questions, and I'm getting caught up in boundary applications. For the first one, I have y'' - 4y' + 3y = f(x) and I need to find the Green's function.
I also have the boundary conditions y(x)=y'(0)=0. Is this possible? Wouldn't y(x)=0 be of the form of a solution...
Is the potential across the boundary continuous for a dielectric sphere embedded in a dielectric material, so that the potential inside the sphere can be set equal to the potential outside of it at r=R ?
Hi everyone,
This is my first time posting here
I am looking to get some help with Abaqus,
I wish to compare two models and find the residual stress which causes a original model to deform to the other. The deflection between the two models can be calculated by other software.
I plan to do...
Hello everyone,
The boundary condition :
P=0, z=ζ
is very common when studying irrotational flows. When cast with the Bernoulli equation, it gives rise to the famous dynamic boundary conditionn, which is much more convenient :
∂tφ+½(∇φ)2+gζ=0, z=ζ
But what happens if the motion is rotational ...
If we have an infinite square well, I can follow the usual solution in Griffiths but I now want to impose periodic boundary conditions. I have
\psi(x) = A\sin(kx) + B\cos(kx)
with boundary conditions \psi(x) = \psi(x+L)
In the fixed boundary case, we had \psi(0) = 0 which meant B=0 and...
As required by the Green's identity, the integrated function has to be smooth and continuous in the integration region Ω.
How about if the function is just discontinuous at the boundary? Actually, this function is an electric field. So its tangential component is naturally continuous, but the...
Homework Statement
Consider the boundary value problem
\begin{equation}
u''(t)=-4u+3sin(t),u(0)=1,u(2)=2sin(4)+sin(2)+cos(4)
\end{equation}
Homework Equations
Derive the linear system that arise when discretizating this problem using
\begin{equation}
u''(t)=\frac{u(t-h)-2u(t)+u(t+h)}{h^2}...