PDE's express a function of at least 2 variables so that the value of the function has some sort of relationship to values at nearby points. 3 common methods of solution are Finite Element, Finite Volume & Finite Difference methods. Try a google search for these names. This page (will) shows how a simple PDE can be solved numerically. FD - http://mathworld.wolfram.com/FiniteDifference.html FE - despite being the most widely used PDE solver, very few good references http://www.fact-index.com/f/fi/finite_element_method.html FV - http://mathworld.wolfram.com/FiniteVolumeMethod.html The heat conduction equation is: Tt = Txx +Tyy (where Txx implies differentiation with respect to x twice, Tt diferentiate temperature with time.). Very few of the best solution methods are suitable for Tcl - they could be implemented but the sheer volume of data and storage plus solution time. One of the few solution methods that can be implemented easily is a relaxation method for the Heat equation on a 2D rectangular grid. We can approximate (FD!) the above equations using [Euler Methods] forward marching in time with: Tnew(x,y) = Told(x,y) + T(x+1,y)-T(x,y)-T(x,y)+T(x-1,y) +T(x,y+1)-T(x,y)-T(x,y)+T(x,y-1) = Told(x,y) + T(x+1,y)-4 T(x,y)+ T(x-1,y) +T(x,y+1) + T(x,y-1) I am now going to try to develop a simple algorithm to demonstrate how a hot point dissipates. I set one pixel in an image to a very high intensity, then apply the FD equation to observe the spread of the 'heat' = seen as a spreading and cooling area of color. [AM] Could we join forces? I would love a PDE solver in Tcl - even it is not that particularly fast or even remotely useful :) As you have not signed this page with your name nor any of the others on similar subject, I have to take this public route.