Computer algebra techniques in object-oriented mathematical modelling

Abstract

This thesis proposes a rigorous object-oriented methodology, supported by computer algebra software, to generate and relate features in a mathematical model. Evidence shows that there is little heuristic or theoretical research into this problem from any of the three principal modelling methodologies: 'case study’, ‘scenario’ and ‘generic’. This thesis comprises two other major strands: applications of computer algebra software and the efficacy of symbolic computation in teaching and learning. Developing the principal algorithms in computer algebra has sometimes been done at the expense of ease of use. Developers have also not concentrated on integrating an algebra engine into other software. A thorough review of quantitative studies in teaching and learning mathematics highlights a serious difficulty in measuring the effect of using computer algebra. This arises because of the disparate nature of learning with and without a computer. This research tackles relationship formulation by casting the problem domain into object-oriented terms and building an appropriate class hierarchy. This capitalises on the fact that specific problems are instances of generic problems involving prototype physical objects. The computer algebra facilitates calculus operations and algebraic manipulation. In conjunction, I develop an object-oriented design methodology applicable to small-scale mathematical modelling. An object model modifies the generic modelling cycle. This allows relationships between features in the mathematical model to be generated automatically. The software is validated by quantifying the benefits of using the object-oriented techniques, and the results are statistically significant. The principal problem domain considered is Newtonian particle mechanics. Although the Newtonian axioms form a firm basis for a mathematical description of interactions between physical objects, applying them within a particular modelling context can cause problems. The goal is to produce an equation of motion. Applications to other contexts are also demonstrated. This research is significant because it formalises feature and equation-generation in a novel way. A new modelling methodology ensures that this crucial stage in the modelling cycle is given priority and automated.