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Can anybody tell me some practical/physical example where we use Dirichlet and Neumann Boundary condition. Is it possible to use both conditions together at the same region?

If we have a cylindrical symmetric array of lenses to focus ions, which regions it comes the Dirichlet BC and Neumann BC? This types of systems can be simulated using SIMION. I know SIMION solved Laplaces equation. But I am confused with the boundary conditions.

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2 Answers

up vote 5 down vote accepted

There is a standard book which contains everything about electrostatics, the Laplace/Poisson equation and boundary conditions: Classical Electrodynamics by J. D. Jackson. Get the book from the library of your choice, read all chapters labeled "Electrostatics", and you will find the answers to all your questions (if you are simulating this, you need to know all this stuff anyway).

The nature of the boundary condition depends on the system you are describing. It means something else if you are calculating heat flow than electrostatics - obviously!

Let's say you have a differential equation (e.g. the Poisson equation $\Delta\varphi(\vec r)=-\frac{\varrho(\vec r)}{\varepsilon_0}$ describing electrostatics, and you solve it for the function $\varphi(\vec r)$. At the boundaries of the region (e.g. a cylinder, a cube, etc.) you have to fix some property of $\varphi(\vec r)$.

  • Neumann boundary condition: You fix $\frac{\partial\varphi(\vec r)}{\partial\vec n}=const$ along the boundary, where $\vec n$ is the normal vector to the surface. It's basically the derivative of $\varphi$ if you go straight away from the surface. It can be a different value for every $\vec r$.
  • Dirichlet boundary condition: You fix $\varphi(\vec r)=const$. It can be a different value for every $\vec r$.

You can only fix one of those two, or the sum (this is called Robin boundary condition).

Physical examples for electrostatics:

  • Neumann boundary condition: The aforementioned derivative is constant if there is a fixed amount of charge on a surface, i.e. $\frac{\partial\varphi(\vec r)}{\partial\vec n}=\sigma(\vec r)$.
  • Dirichlet boundary condition: The electrostatic potential $\varphi(\vec r)$ is fixed if you have a capacitor plate which you connected to a voltage source. E.g. if you have two capacitor plates which are at 0V and 5V, respectively, you would set $\varphi(\vec r)=0$ at the first plate and $\varphi(\vec r)=5$ at the second plate. That way, you can calculate the capacitance.

For heat flow, fixing the field $u$ (=Dirichlet BC) means fixing the temperature. If you have elements in your system that have a fixed temperature, you would use that one.

If you lenses are based on electrostatics, they probably only have Dirichlet boundary conditions, because that's how you describe a capacitor plate. If you have outer boundaries which are not capacitor plates, you should use a Neumann BC = 0 (in this case, it has nothing to do with fixing the charge), because that's the best BC for simulating an "infinite" system.

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+1 for a good answer –  Hal Swyers Feb 9 '13 at 13:37
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Zonk's answer is very good, and I trust that there is an understanding that Dirichlet BC specify the value of a function at a set of points, and the Neumann BC specify the gradient of the function at some set of points.

I will add this additional example as described here, and it covers the importance of boundary conditions in our understanding of T-duality in superstring theory.

In this situation we have the requirement that on an open string, momentum does not leak off the string (e.g. remains conserved and does not leak off into space). This constraint is represented as Neumann boundary conditions:

$$\dfrac{\partial X^\mu}{\partial \sigma}\bigg|_{\sigma=0} = \dfrac{\partial X^\mu}{\partial \sigma}\bigg|_{\sigma=\pi}=0$$ $$\mu = 0,1,\dots,d-1$$

Which basically is a statement that the change in position with respect to the boundaries in all dimension are fixed at zero.

If we want to describe boundary conditions for a T-dual theory where some number $p$ dimensions are left non-compact and the rest are compactified on a torus, we would convert some boundary conditions from Neumann BC to Dirichlet BC and write:

$$\dfrac{\partial X^\mu}{\partial \sigma}\bigg|_{\sigma=0} = \dfrac{\partial X^\mu}{\partial \sigma}\bigg|_{\sigma=\pi}=0$$ $$\mu = 0,1,\dots,p$$ For the non compact dimensions, and:

$$X^\mu(\tau,0) = X^\mu(\tau,\pi)=X^\mu_0$$ $$\mu = p+1,\dots,d-1$$

For the compactified dimensions.

This then is an example of a situation where some variable respect Neumann BC and others respect Dirichlet BC.

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