Mathematical Modelling Techniques

Rutherford Aris

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486138893
• Topic: Mathematics
• Subtopic: Discrete Mathematics
• Index: Mathematics

"Engaging, elegantly written." — Applied Mathematical Modelling
Mathematical modelling is a highly useful methodology designed to enable mathematicians, physicists and other scientists to formulate equations from a given nonmathematical situation. In this elegantly written volume, a distinguished theoretical chemist and engineer sets down helpful rules not only for setting up models but also for solving the mathematical problems they pose and for evaluating models.
The author begins with a discussion of the term "model, " followed by clearly presented examples of the different types of models (finite, statistical, stochastic, etc.). He then goes on to discuss the formulation of a model and how to manipulate it into its most responsive form. Along the way Dr. Aris develops a delightful list of useful maxims for would-be modellers. In the final chapter he deals not only with the empirical validation of models but also with the comparison of models among themselves, as well as with the extension of a model beyond its original "domain of validity."
Filled with numerous examples, this book includes three appendices offering further examples treated in more detail. These concern longitudinal diffusion in a packed bed, the coated tube chromatograph with Taylor diffusion and the stirred tank reactor. Six journal articles, a useful list of references and subject and name indexes complete this indispensable, well-written guide.
"A most useful, readable-and stimulating-book, to be read both for pleasure and for enlightenment." — Bulletin of the Institute of Mathematics and Its Applications

1

What is a model?

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altr. to Aristophanes of Byzantium
Syrianus in Hermog. (Rabe ij 23)
(0 Menader, o life, which of you has imitated the other?)

1.1 The idea of a mathematical model and its relationship to other uses of of the word.

In these notes the term ‘mathematical model’--usually abbreviated to ‘model’--will be used for any complete and consistent set of mathematical equations which is thought to correspond to some other entity, its prototype. The prototype may be a physical, biological, social, psychological or conceptual entity, perhaps even another mathematical model, though in detailed examples we shall be concerned with a few physico-chemical systems.

Being derived from ‘modus’ (a measure) the word ‘model’ implies a change of scale in its representation and only later in its history did it acquire the meaning of a type of design, as in Cromwell’s New Model Army (1645). still later (1788) came the complacent overtones of the exemplar that Gilbert was to use so effectively for his modern major general, while it is the first years of this century before fashion became so self-conscious as to claim its own models and make possible Kaplan’s double entendre (see quotation at head of Ch. 5). In the sense that we are seeking a different scale of thought or mode of understanding we are using the word in its older meaning. However, the word model (without the adjective ‘mathematical’) has been and is used in a number of senses both by philosophers and scientists as merely glancing through the titles of the bibliography will suggest. Thus Apostel ^[7] distinguishes nine motivations underlying the use of models ranging from the replacement of a theory-less domain of facts by another for which a theory is known (e.g., network theory as a model for neurological phenomena) to the use of a model as a bridge between theory and observation. Suppes in the same volume ^[169] maintains that the logicians concept of a model is the same in the empirical sciences as in mathematics though the use to which they are put is different. The logician’s definition he takes from Tarski ^[172] as: “a possible realization in which all valid sentences of a theory T are satisfied is called a model of T”. This is a non-linguistic entity in which a theory is satisfied and Suppes draws attention to the confusion that can arise when model is used for the set of assumptions underlying a theory, i.e. the linguistic structure which is axiomatized. In our context this suggests that we might usefully distinguish between the prototype (i.e. the physical entity or system being modelled), the precursive assumptions or what the logicians call the theory of the model (i.e. the precise statement of the assumptions of axioms) and the model itself (i.e. the scheme of equations).

The idea of a change of scale which inheres in the notion of a model through its etymology can be variously interpreted. In so far as the prototype is a physical or natural object, the mathematical model represents a change on the scale of abstraction. Certain particularities will have been removed and simplifications made in obtaining the model. For this reason some hard-headed, practical-minded folk seem to regard the model as less “real” than the prototype. However from the logical point of view the prototype is in fact a realization in which the valid sentences of the mathematical model are to some degree satisfied. One could say that the prototype is a model of equations and the two enjoy the happy reciprocality of Menander and life.

The purpose for which a model is constructed should not be taken for granted but, at any rate initially, needs to be made explicit. Apostel (loc. cit.) recognizes this in his formalization of the modelling relationship R(S,P,M,T), which he describes as the subject S taking, in view of a purpose P, the entity M as a model for the prototype T. J. Maynard Smith ^[165] uses the notion of purpose to distinguish mathematical descriptions of ecological systems made for practical purposes from those whose purpose is theoretical. The former he calls ‘simulations’ and points out that their value increases with the amount of particular detail that they incorporate. Thus in trying to predict the population of a pest the peculiarities of its propagation and predilections of its predators would be incorporated in the model with all the specific detail that could be mustered. But ecological theory also seeks to make general statements about the population growth that will discern the broad influence of the several factors that come into play. The mathematical descriptions that serve such theoretical purposes should include as little detail as possible but preserve the broad outline of the problem. These descriptions are called ‘models’ by Smith, who also comments on a remark of Levins ^[114] that the valuable results from such models are the indications, not of what is common to all species or systems, but of the differences between species of systems.

Hesse ^[92] in her excellent little monograph “Models and Analogies in Science” distinguished two basic meanings of the word ‘model’ as it is used in physics and Leatherdale in a very comprehensive discussion of “The Role of Analogy, Model and Metaphor in Science” has at least four. They stem from the methods of “physical analogy” introduced by Kelvin and Maxwell who used the partial resemblance between the laws of two sciences to make one serve as illustrator of the other. In the hands of 19th century English physicists these often took the form of the mechanical analogues that evoked Duhem’s famous passage of Gallic ire and irony. Duhem ^[56] had in mind that a physical theory should be a purely deductive structure from a small number of rather general hypotheses, but Campbell ^[41] claimed that this logical consistency was not enough and that links to or analogies with already established laws must be maintained. Leatherdale’s four types are the formal and informal variants of Hesse’s two. Her “model₁” is a copy, albeit imperfect, with certain features that are positively analogous and certain which are neutral but shorn of all features which are known to be negatively analogous, i.e. definitely dissimilar to the prototype. Her “model₂” is the copy with all its features, good, bad and indifferent. Thus billiard balls in motion, colored and shiny, are a model₂ for kinetic theory, whilst billiard balls in motion obeying perfectly the laws of mechanics but bereft of their colour, shine and all other non-molecular properties constitute a model₁. It is the natural analogies (i.e. the features as yet of unknown relevance) that are regarded by Campbell as the growing points of a theory. In these terms a mathematical model would presumably be a formal model₁.

Brodbeck ^[35], in the context of the social sciences, stresses the aspect of isomorphism and reciprocality when she defines a model by saying that if the laws of one theory have the same form as the laws of another theory, then one may be said to be a model for the other. There remains, of course, the problem of determining whether the two sets of laws are isomorphic. Brodbeck further distinguishes between two empirical theories as models one of the other and the situation when one theory is an “arithmetical structure”. She then goes on to describe three meanings of the term mathematical model according as the modelling theory is (a) any quantified empirical theory, (b) an arithmetic structure or (c) a mere formalization in which descriptive terms are given symbols in the attempt to lay bare the axioms or otherwis...

Table of contents

• Title Page
• Acknowledgments
• Copyright Page
• Preface
• Table of Contents
• 1 - What is a model?
• 2 - The different types of model
• 3 - How to formulate a model
• 4 - How should a model be manipulated into its most responsive form?
• 5 - How should a model be evaluated?
• Appendix A Longitudinal diffusion in a packed bed
• Appendix B The coated tube chromatograph and Taylor diffusion
• Appendix C The stirred tank reactor
• References
• Subject index
• Name index
• Re, k and π : a conversation on some aspects of mathematical modelling
• THE JAIL OF SHAPE - RUTHERFORD ARIS
• THE MERE NOTION OF A MODEL - RUTHERFORD ARIS AND MISCHA PENN
• Ut Simulacrum, Poesis - Rutherford Aris
• FIFTH P. V. DANCKWERTS MEMORIAL LECTURE PRESENTED AT GLAZIERS’ HALL, LONDON, U.K. 16 OCTOBER 1990 - MANNERS MAKYTH MODELLERS
• HOW TO GET THE MOST OUT OF AN EQUATION WITHOUT REALLY TRYING*