Solving Dirichlet problem on a Square

[Scootertaj]

1. Problem Statement
Solve by inspection the Dirichlet problem, where \Omega is the unit square 0\leqx\leq1, 0\leq y \leq 1, and where the data is:
f(x,y) = {
x for 0\leqx\leq1, y=0,
1 for x = 1, 0\leq y \leq 1,
x for 0\leqx\leq1, y=1,
0 for x=0, 0\leqy\leq1

Homework Equations

Dirichlet Problem:

\Delta u = 0 in \Omega
u = f on d\Omega

The Attempt at a Solution

I don't know where to really start with this one. What does it mean "solve by inspection the dirichlet problem?" What is u? How does the "data" help, it's the initial conditions right?

 

[hunt_mat]

It's the boundary conditions. How much PDE theory do you know?

 

[Scootertaj]

Almost nothing, sadly. I have had only one lecture in it so far and that was just going over the three main operators in PDEs.
Is there a good website that shows how to solve the Dirichlet problem with boundary conditions or just how to solve a problem with boundary conditions?

 

[hunt_mat]

So if I mention separation of variables, this would draw a blank to you? It sounds like you need to go to a few more lectures before you start looking at the questions.

Have you tried a google search? It would be the first thing I would do.

 

[Scootertaj]

So if I mention separation of variables, this would draw a blank to you? It sounds like you need to go to a few more lectures before you start looking at the questions.

Have you tried a google search? It would be the first thing I would do.

Unfortunately, mainly yes. I do know that we will be covering three main methods: Separation of Variables, Green's Method, and Variational/Numerical methods, but we haven't even discussed what each means. It isn't even until next week we start discussing the separation of variables though.

I did a google search for solving differential equations using boundary conditions and it made sense. Specifically, I used http://tutorial.math.lamar.edu/Classes/DE/BoundaryValueProblem.aspx"
A google search of dirichlet problem did no good.

I suppose I'll just have to keep reading, thank you for the help!

 

[hunt_mat]

How about a search on wikipedia?

 

[Scootertaj]

How about a search on wikipedia?

I came across http://en.wikipedia.org/wiki/Dirichlet_boundary_condition" .
So, what I'm getting from it is I need to find a solution to a PDE based on the boundary conditions given. Is my PDE \Deltau = 0 in the unit square and u = f where f is our data?
I'm just confused on where to start.

 

[hunt_mat]

Basically yes, your f is your boundary data. Like I said you're probablt jumping into the problems a little too early.

 

[LCKurtz]

Yet the original question was to solve that Dirichlet problem by inspection. By examining the values on the boundary of the square, it really isn't much to ask to guess a solution, never mind the lack of advanced techniques available.

 

[Scootertaj]

How would you go about that?

 

[Dick]

How would you go about that?

Is there a simple function in the square that fits your data on the boundary of the square? Does that function solve the Laplace equation 'by inspection'?