what about the source term in our example with temperature distribution in rodg xtmodels heat generation inside the rod one could think of chemical reactions at microscopic level in some materials as reason to include howeverin most applications with temperature evolutiong is zero and heat generation usually takes place at the boundary (as in our example with td /before continuingwe may consider an example of how the temperature distribution evolves in the rod at time we assume that the temperature is then we suddenly apply device at that keeps the temperature at at this end what happens inside the rodintuitivelyyou think that the heat generation at the end will warm up the material in the vicinity of and as time goes bymore and more of the rod will be heatedbefore the entire rod has temperature of (recall that no heat escapes from the surface of the rodmathematically(with the temperature in kelvinthis example has xd kexcept at the end pointi ks td kand the figure below shows snapshots from four different times in the evolution of the temperature finite difference methods we shall now construct numerical method for the diffusion equation we know how to solve ordinary differential equationsso in way we are able to deal with the time derivative very often in mathematicsa new problem can be solved by reducing it to series of problems we know how to solve in the present caseit means that we must do something with the spatial derivative @ =@ in order