Example of a non-periodic function constructed from two periodic functions.
f:=x->cos(x)+cos(2*Pi*x);
plot(f(x)-2,x=0..200,y=-0.01..0.01);
fsolve(f(x)=2,x);

Fourier's finite rod heat conduction problem has some demonstrations available.

Bunsen burner on one end of the rod (say x=1) heats the rod, which
has wax rings located between x=0 and x=1. As the heat transfer takes
place, the wax rings melt and fall to the lab table.
http://demonstrations.wolfram.com/ExperimentOnHeatConduction/


A finite rod held at zero temp at the ends (x=0 and x=1) is submerged in a 100C water
bath at time t=0. The graphic in the demo shows the progression in time of the heat solution series, using more and more terms of the series. 

http://demonstrations.wolfram.com/HeatTransferAlongARod/