I'm using Maple these days to generate C code. More precisely, I use Maple to calculate high order derivatives. For example, I know that :
dt( u(x,t) ) = a(x) * dx( u(x,t) ) + b(x) * dx( p(x,t) )
dt( p(x,t) ) = c(x) * dx( p(x,t) ) + d(x) * dx( u(x,t) )
and I ask Maple to calculate the 5th time-derivative of u using these 2 properties. But in fact, I use 8 variables instead of 2 and PDEs are far more complicated.
Finally, I get thanks to Maple expressions of time-derivatives for my 8 variables.