Solving Laplacian Equation Analytically

1. Jan 1, 2010

Harmony

I wish to solve a 2D steady state heat equation analytically. The boundary is a square. The top side is maintained at 100 C, while the other sides are maintained at 0 C.

The differential equation governing the temperature distribution will be the laplacian equation. To solve the equation analytically, I suppose we can guess that the solution is the product of the two independent variable. But how can I proceed from there?

2. Jan 1, 2010

vela

Staff Emeritus
Plug your proposed solution in and then separate variables. You'll find one side depends only on one variable, and the other side depends on the other. The only way the equation can work is if both sides equal some constant.

3. Jan 1, 2010

Harmony

Yes, I read about that. But the problem is, there are so many possible combinations (different products of function y and function x), and how should i use the boundary condition to find the one I need?

4. Jan 1, 2010

Astronuc

Staff Emeritus