Initial Value Problems for ODEs in Problem Solving Environments

The problem solving environments (PSEs) Maple [8] and Matlab [7] are in very wide use. Although they have much in common, they are clearly distinguished by the emphasis in Maple on algebraic computation and in Matlab on numerical computation. We discuss here a program, IVPsolve, for solving numerically initial value problems (IVPs) for systems of first order ordinary differential equations (ODEs), y′ = f(x, y), in Maple. We draw upon our experience with a number of closely related solvers to illustrate the differences between solving IVPs in general scientific computation (GSC) and in these PSEs. The RKF45 code of Shampine and Watts [10,11] is based on the explicit Runge-Kutta formulas F(4,5) of Fehlberg. It has been widely used in GSC. Translations of this code have been the default solvers in both Maple and Matlab. Neither takes much advantage of the PSE. In developing the Matlab ODE Suite of solvers for IVPs, Shampine and Reichelt [9] exploit fully the PSE, as well as algorithmic advances. IVPsolve is the result of a similar investigation for Maple, though on a much smaller scale. It also uses the F(4,5) formulas for non-stiff problems.