Design and Validation of a Numerical Problem Solving Environment for Ordinary Differential Equations

In the last decade, two distinct directions have emerged in the use of Ordinary Differential Equation (ODE) solver software (Enright ). These are Large Scale Scientific Computation on workstations or large mainframe computers and Problem Solving Environments (PSE’s) on personal computers. Most practicing engineers and scientists, as well as engineering and science students, use numerical software for problem solving, while only a few very specific research applications require large scale computing. Consequently, this work is limited to PSE’s. We have carried out an extensive study of the requirements for a PSE intended for solving ODE’s arising in Chemical and Biochemical engineering applications. In this study we have collected a “Library” of 100 sample problems and solved these problems which contain between one to 50 ordinary differential equations. Many of the problems were taken from the book of Cutlip and Shacham and the rest from our previous publications (Brauner et al., Shacham et al.) as well as from other sources. Mainly initial value problems were considered in this study. Polymath 6.1 (copyrighted by M. Shacham, M. B. Cutlip and M. Elly, http://www.polymath-software.com) and MATLAB (trademark of The Math Works, Inc., http://www.mathworks.com) were used for solving the problems and validating the results. Stiff and non-stiff algorithms were used where needed. All the utilized algorithms included error estimation and step-size control in order to achieve solution of a pre-specified error tolerance. The specific needs for a PSE of ODE’s that were identified in this study included the following: