Investigating the G-Functions Series Method Adapted to the Numerical Integration of Parabolic PDE

The Method Of Lines (MOL) and Scheifele's G{functions in the design of algorithms adapted for
the numeric integration of parabolic Partial Dierential Equations (PDE) in one space dimension
are applied.
The semi{discrete system of ordinary dierential equations in the time direction, obtained by
applying the MOL to PDE, is solved with the use of a method of adapted series, based on Scheifele's
G{functions. This method integrates exactly unperturbed linear systems of ordinary dierential
equations, with only one G{function.
An implementation of this algorithm is used to approximate the solution of two test problems
proposed by various authors. The results obtained by the Dufort{Frankel, Crank{Nicholson and
the Adapted Series methods versus the analytical solution, showing the results of mistakes made.