Solving Laplacian Equation Analytically

[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?

 

[vela]

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.

 

[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?

 

[Astronuc]

How about writing the steady-state heat conduction equation.

Then the proposed solution, e.g., Q(x,y) = X(x)Y(y), where Q(x,y) is the function one is trying to determine.

Then right the boundary conditions, Q(0,y) = X(0) Y(y) = . . . .

 

[vela]

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?

You don't choose the specific functions beforehand, if that's what you're confused about. After you separate the variables, you'll have two differential equations. You solve those to find the actual forms of X(x) and Y(y), and then apply the boundary conditions.