A method for obtaining difference equations from differential equations which has previously been applied to ordinary differential equations is here applied to the time-dependent heat conduction equation as a representative example of parabolic partial differential equations. Explicit difference equations are derived which are stable for all values of alpha DELTA t/ DELTA x**2 and are also accurate, Implicit algorithms with improved accuracy are also derived, Problems incartesian and cylindrical co-ordinates are treated. (Author abstract(