Old Exams and the Laplace Equation

What IS the Laplace equation?

Whenever researching mathematics, there almost never seems to be a simple explanation of what the function actually does. Usually, this is because a function will convert one abstract notion of a small branch of mathematics into another abstract notion of a small branch of mathematics.

It means absolutely nothing for the average person.

To begin, lets breakdown the Wikipedia page. This is usually my default go-to, as it's written by people, not robots (usually).

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.

Great. I'm still none the wiser.

This is often written as Δφ = 0  or  ∇²φ = 0

Okay, Now we're getting somewhere.
Luckily for me, I have a vague idea what the Laplacian operator, or ∇², actually refers to.

I hear you saying; 

"wow, it's really that simple - all it does is relate the divergence of the rate of change in n-dimensions of a scalar value to be non-moving, or 0!"

You're probably thinking one of the following options;

• What does divergence mean in a mathematical context?
• What does that mean?; or
• ...What?.

Well, divergence (in a mathematical concept), at a point (x,y,z) is the measure of a vector flow (i.e. has direction and magnitude) out of a surface surrounding that point1. See  description below (from maxwells-equations.com)

That is, imagine a vector field represents water flow. Then if the divergence is a positive number, this means water is flowing out of the point (like a water spout - this location is considered a source). If the divergence is a negative number, then water is flowing into the point (like a water drain - this location is known as a sink).

Going back to the Laplace equation, and to summarise - the rate-of-change of the divergence, scaled to some scalar operator (i.e. some real number) must equal zero. If it is non-zero, then the system defined by this equation is still dynamic, and does not satisfy the Laplace Equation.

Now you're probably wondering where this is actually used, and why it's incredible useful (it is, trust me.).

The Laplace equation is fundamental behind the vast majority (citation needed) of modelling natural phenomena. Some key examples are fluid-flow, heat-flow, and electrostatics (N.B. Does not include electro-magnetics)

Great.

Now, the title implies there's some relevance to old uni exam papers.

All this sort of thing used to be covered by one of the units here at Curtin, and I recently found one of the exam papers.

The question asks;

"What are the two important boundary conditions used to provide a specific solution for the Laplace Equation".

The key-word here is 'specific solution'. Solutions for Laplace' equation, mathematically, are many and nonsensical². To get a specific solution, we impose constrains on the range of possible answers, such that a single, sensible, answer is available.

To paraphrase the second link: In a heat-equation problem, assuming a rectangular system is heated only from one side, and all other sides must be zero, we can assume the following points;

1. The first condition, T(x, ∞) = 0, rules out all solutions with an oscillating y-axis, because the temperature profile decays monotonously from left, to right. (i.e. the solution cannot physically increase at any other point in the system, as the only source is at T(x, 0))
2. The same boundary condition also contradicts solutions which increase, rather than decrease along the y-axis. i.e. positive exponential. (i.e. again, the solution cannot physically increase at any other point in the system, as the only source is at T(x, 0))
3. The second boundary condition, T(0, y) = 0, requires that the oscillating function along x is zero at x = 0. Therefore, of the remaining two solutions, we can rule out the one with a cosine term. (The cosine allows for non-zero terms at x = 0)
4. We can use the third boundary condition, T(10, y) = 0, to put constraints on the constant k. Inserting the BC into the one remaining general solution, we have e−kysin10k=0

The second link has pictures, and is reasonably well formatted.

The point is, the two boundary conditions used to provide a specific solution are that

1. All boundaries, besides one, are zero.
2. That one boundary is a function which satisfies a Dirichlet (some function), or Neumann (the derivative of some function) type condition.

There are answers for this exam somewhere. I'll update once I find them.

Hopefully I'm right....

Edit;

The two important boundary conditions, according to the unit coordinator of the unit, are the conservation of charge, and the conservation of energy across the boundary. (Referenced the Parasnis book)

¹Link: http://www.maxwells-equations.com/divergence.php
²Link: http://users.aber.ac.uk/ruw/teach/260/laplace2.php