Open Research Online The Open University’s repository of research publications and other research outputs Computer algebra techniques in object-oriented mathematical modelling. Thesis How to cite: Mitic, Peter (1999). Computer algebra techniques in object-oriented mathematical modelling. PhD thesis The Open University. For guidance on citations see FAQs. c 1998 Peter Mitic https://creativecommons.org/licenses/by-nc-nd/4.0/ Version: Version of Record Link(s) to article on publisher’s website: http://dx.doi.org/doi:10.21954/ou.ro.00010236 Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyright owners. For more information on Open Research Online’s data policy on reuse of materials please consult the policies page. oro.open.ac.uk Computer Algebra Techniques in Object-Oriented Mathematical Modelling Peter Mitic Thesis presented for the Degree of Doctor o f Philosophy to the Faculty of Mathematics and Computer Science, * * Discipline of Mathematics ' The Open University May 1999 - j ' j ' O P 2(2) csT8_ OP Gimep : 17 rT\ey ProQuest Number: 27696813 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a com plete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 27696813 Published by ProQuest LLO (2019). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States C ode Microform Edition © ProQuest LLO. ProQuest LLO. 789 East Eisenhower Parkway P.Q. Box 1346 Ann Arbor, Ml 48106- 1346 Acknowledgments I would like to thank several people at the Open University who have made this thesis possible. My supervisors, Peter Thomas and Darrel Ince, guided it over a long eight year period. Karen Vines provided invaluable advice on the statistical aspects and Paula Cole did a great deal of administrative work. Many other people have made their contributions. I am particularly j^telul to Ian Nicol, who did the bulk of the final proof reading. Others have made comments, provided advice, given references, supplied clues, read extracts or have simply given encouragement. These small things are important: they show how small steps fit into an overall scheme. In particular, I am very grateful to everyone who appreciated this work for what it is, and did not encourage deviation. My wife has fed me, done the washing, mowed the lawn and generally organised things. She claims that an average of three children have been present for the duration, but this cannot be mathematically correct since the youngest is now five. Perhaps there were others... Motivations for a long­ term project are sometimes hard to examine, but perhaps my daughter summed it up nicely. She said “If you want to do something, just do it”: not bad for a five-year-old. She might be surprised if she reads this and understands the implications of that statement, but academic pursuit for its own sake seems to be unfeshionable these days. I hope that things will change. 2 8 JUL 1999 d o n k t io n “T C Contents Page Acknowledgments 2 Contents 3 Abstract 5 0 . Introduction and Overview of the Thesis 7 1. Mathematical Modelling Methodologies 13 2. A Review of Computer Algebra Algorithm Development and its impact on Mathematical Modelling 55 3. Teaching and learning Mathematics and Modelling by Computer 77 4. A Summary of Research problems arising 6om Literature Reviews 105 5. The Case for Object-Oriented Modelling 111 6. A new technique and methodology for Mathematical Modelling 141 7. AMK: Object-Oriented software for Particle Mechanics 187 8. Front-end Support for the modelling cycle 217 9. Validation 239 10. Conclusion and Further Research 265 Appendices 281 References 345 Thesis 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-ori^ted 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. Chapter 0 Introduction and Overview of the Thesis 0.0 Introduction The work behind this thesis covers a period during which there has been enormous change in computing. During the 1990s symbolic computation has moved from mainframe computers to micro-computers to hand held calculators. Many people have been reluctant to accept these changes, but they are here to stay. The task is therefore to discover ways of using the available tools to their best advantage, and this thesis must be seen in that context. 0.1 Previous Papers Two papers, (Mitic 95A) and (Mitic 95B), cover aspects of original work for this thesis on object models and user interfaces. 0.2 Problem statement The motivation behind this work lies in teaching a first course in mathematical modelling to undergraduate students, and in teaching mechanics to younger students. It appeared that, despite efforts to ease the modelling process, there were persistent problem-solving strategy difficulties. Of these, it seemed that a significant number could be classified as ‘unable to start’. This was because, having completed preliminary stages, students could not easily identify the salient features in a mathematical model. Even if this stage was done successfully, students were sometimes unable to form relationships between the features to produce a model. The problem was therefore to devise a way of finding features, and of formulating a model by finding relationships between them. Although finding features and formulating relations between them are interlinked, this thesis concentrates more on formulating relations. The aim was to automate this process. Computer algebra software is necessarily involved in this process because algebraic computation is fundamental to mathematics (and hence to modelling) in a way that numeric computation is not. An algebraic model forces functional forms to be explicit, so that functional variation is equally explicit. Routine algebra and calculus operations are needed to do this. 0.3 Literature reviews Chapters 1,2,3 and 5 contain literature reviews covering distinct areas. Chapter 1 contains an analysis of three common mathematical modelling strategies, with a discussion of the strengths and weaknesses of each. The series of ICTMA* conferences is a principal source of information for two of these: ‘scenario’ and ‘generic’ modelling. These, and other sources, indicate that some basic modelling issues (as specified above) are not being addressed. Views on Wiich methodology to use are somewhat polarised, and concentrate on theoretical issues, course content and presentation, presentation of results and other issues related to modelling. The main sources of literature for the third modelling strand, ‘case-study’ modelling, comes fiom standard texts on modelling. From these a surprisingly small number of modelling contexts can be identified, but there is, again, a lack of discussion on relationship formulation. The absence of extensive discussion of relationship formulation and identification of features indicates the gap in knowledge vdiich this thesis fills. The review in Chapter 2 traces the principal stages in algorithm development of computer algebra software from its introduction in the 1960s to the present. Design decisions for computer algebra software created barriers for use in mathematical