# What is a boundary condition?

While I've been doing my university homework, I've come across a question asking for the boundary condition for the electric flux density of a charged conductor. However, I seem to be stuck as I realise I don't really know the meaning of a 'boundary condition'. I've always taken it to be the limits in a definite integral, but somehow I don't feel that is the case. Searching up online gives me explanations with big words which I can't catch onto, such as the it being "a condition that is required to be satisfied at all or part of the boundary of a region in which a set of differential conditions is to be solved."

Can anyone offer a more dumbed down explanation of what boundary conditions are?

• @DavidWhite: Why not turn that into an answer? – Kyle Kanos Feb 5 '17 at 19:49
• @KyleKanos, I considered my comment to be a bit too simplistic to serve as an official answer, but I took your advice and posted it. Thanks for the suggestion. – David White Feb 5 '17 at 20:38

In physics, you will deal with differential equations which relate functions and their derivatives; a solution to such an equation is an unknown function, a priori.

A boundary condition is a type of condition or requirement we place on this function. I'll present an example; the mathematical details are not so important here, as long as you grasp the idea of a boundary condition.

Illustrative Example: Injecting into a tube

Suppose we have a tube of some length $L$ and it is sealed at both ends. Now imagine we take a needle and inject a substance; how this diffuses is described by the diffusion equation for $c(x,t)$, the concentration of the substance:

$$\frac{\partial c}{\partial t} = D \frac{\partial^2 c}{\partial x^2}$$

where $D$ is a constant. The equation itself is not important! Now, so far we can write down a general solution describing how it will disperse, but we need to specialise to our specific case to get a physically applicable answer.

Remember the tubes were sealed at both ends; this means the flow of the substance at either end of the tube has to be zero. It turns out this is proportional to the first derivative, $\partial_x c$ of the concentration. So, we'd like this to be zero at both ends:

$$\frac{\partial c}{\partial x} \bigg\rvert_{x = 0} = 0, \quad \frac{\partial c}{\partial x} \bigg\rvert_{x = L} = 0.$$

(To complete the problem, we'd also describe what the concentration is like initially; in this case it is of the form, $c(x,0) \sim \delta(x)$ since we inject it all at a point.)

The big picture is: boundary conditions apply constraints on solutions to equations that are motivated by the physical problem being considered. In this example, we are imposing the fact that the tube which is really the domain of the function, is sealed at both ends.

A boundary condition is a known value that must be true for the problem that you are working. Example: if I am sitting at a stop light and the light turns green, I absolutely know that my vehicle starts from rest as I accelerate through the (now) green light. This is a boundary condition for a physics problem involving distance, velocity, and acceleration vs. time for the automobile that I am driving. Also note - boundary conditions are usually used to evaluate constants of integration when you are performing an indefinite integral.

• This answer does not really distinguish a boundary condition from an "initial condition". – user1583209 Feb 5 '17 at 20:50

As David White already stated: A boundary condition is an equation that gives you constraints of your problem you want to solve.

• So your vehicle being at rest at time $t=0$ is a boundary condition: $v(t=0)=0$

• Or a grounded conducting sphere has the boundary condition of vanishing potential (grounded potential) in its volume and on the surface $\Phi(r<R)=0$

• Or and oscillating, on both ended fixed string has also two boundary conditions: It can't move on each end: e.g. $v(x=0)=0$, $v(x=L)=0$.

Also see this question for more quantitative description

• You could add why they are needed for solving a problem. – user1583209 Feb 5 '17 at 20:22
• Your vehicle example sounds more like initial conditions since you specify position (traffic light) and velocity, both at t=0. – user1583209 Feb 5 '17 at 20:33
• Mathematically speaking there's no difference between initial and boundary conditions. Your domain simply includes a temporal dimension, and is bounded by a temporal boundary. – Pirx Feb 5 '17 at 22:20