# Boundary condition Definition and 25 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.

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1. ### The meaning of the electric field variables in the boundary condition equations

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...
2. ### A Wavefunction of the Universe considering all possible boundaries?

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...
3. ### A Boundary conditions of eigenfunctions with Yukawa potential

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...
4. ### Initial condition of Wave functions with Yukawa Potential

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...

10. ### A Replacing a non-harmonic function with a harmonic function

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...
11. ### I How do irrational numbers give incommensurate potential periods?

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...
12. ### Electromagnetism Help-- Magnetostatics Boundary Problem

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...
13. ### (Numerical) Boundary Value Problem for Schrodinger's Equation

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...
14. ### I Boundary Conditions for System of PDEs

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...
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### Partial differential equation boundary

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...
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### Boundary conditions with l=0

The question is basically find the boundary conditions when ##l=0##, for energies minor than 0. Homework Equations $$V(r)=\begin{cases} & 0\text{ r<a_0}\\ &V_0\text{ a_0<r<a_1}\\ & 0\text{ r>a_1}\\ \end{cases}$$ ...
17. D

### Boundary conditions for eigenfunctions in a potential step

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...
18. ### I A question about boundary conditions in Green's functions

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...
19. ### Boundary condition for dielectric sphere

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 ?
20. ### Abaqus - Boundary Conditions Comparison of two models

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...
21. ### I Is dψ/dx zero when x is infinite in QM?

In QM, we all know that the wavefunction ψ is zero when x is infinite. However, Is dψ/dx also zero when x is infinite? And the d2ψ/dx2?
22. ### A Boundary condition question

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 ...
23. ### I Infinite square well solution - periodic boundary conditions

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...
24. ### Green's first identity at the boundary

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...
25. ### Ordinary differential equation with boundary value condition

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}...