Fall 2003 

As an example of a system with many variables, consider a drum modeled by a fine net. The heights at each interior node needs the average the heights of the 4 neighboring nodes. The height at the boundary is fixed. With n^{2} nodes in the interior, we have to solve a system A x=b of n^{2} equations. The problem solved here is a discrete version of the Dirichlet problem, which we will see later in class and solve explicitly. 
x_{11}=a_{21}+a_{12}+x_{21}+x_{12} x_{12}=x_{11}+x_{13}+x_{22}+x_{22} x_{21}=x_{31}+x_{11}+x_{22}+a_{43} x_{22}=x_{12}+x_{21}+a_{43}+a_{34} 
Fixing the heights at the boundary and the requirement that inside, the heights are the averages of the neighboring heights fixes the heights everywhere. 
Numerical solution of a situation with n=24. The computer was asked to solve the system of linear equations. 
The problem has other interpretations. If we had given the temperature distribution at the boundary, then the solution would give the temperature distribution inside the region. We could replace temperature also with voltage and replace each connection between nodes by a resistor. The solution would give then the voltage distribution inside the resistor network. 
Numerical solution with n=200. The computer had to solve a system of 40'000 unknowns. 