This is probably a lot to go through over a forum. Can you show us the code you have so far?
When it comes to numerically solving PDEs, most often you use NDSolve. NDSolve uses the method of lines for PDEs which works well for some kinds of equations and not well for others. What numerical method were you planning on writing?
Please see the documentation on how equations are written out and how NDSolve is used with PDEs.
If you are new to programming in Mathematica, it's probably best to get a good understanding of the language in general before tackling a problem like this. The Mathematica virtual book provides an introduction into the basics of programming with the Wolfram Language in Mathematica: