Mathematical Modelling Techniques
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Beschreibung:
"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
1.1 The idea of a mathematical model and its relation to other uses of the word
1.2 Relations between models with respect to origins
1.3 Relations between models with respect to purpose and conditions
1.4 How should a model be judged?
2. The Different types of model
2.1 Verbal models and mechanical analogies
2.2 Finite models
2.3 Fuzzy subsets
2.4 Statistical models
2.5 Difference and differential equations
2.6 Stochastic models
3. How to formulate a model
3.1 Laws and conservation principles
3.2 Constitutive relations
3.3 Discrete and continuous models
4. How should a model be manipulated into its most responsive form?
4.1 Introductory suggestions
4.2 Natural languages and notations
4.3 Rendering the variables and parameters dismensionless
4.4 Reducing the number of equations and simplifying them
4.5 Getting partial insights into the form of the solution
4.5.1 The phase plane and competing populations
4.5.2 Coarse numerical methods and their uses
4.5.3 The interaction of easier and more difficult problems
5. How should a model be evaluated?
5.1 Effective presentation of a model
5.2 Extension of models
5.3 Observable quantities
5.4 Comparison of models and prototypes and of models among themselves
Appendices
A. Longitudinal diffusion in a packed bed
B. The coated tube chromatograph and Taylor diffusion
C. The stirred tank reactor
References
Subject index
Name idex
Appendices to the Dover Edition
I. "Re, k and p: A Conversation on Some Aspects of Mathematical Modelling"
II. The Jail of Shape
III. The Mere Notion of a Model
IV. "Ut Simulacrum, Poesis"
V. Manners Makyth Modellers
VI. How to Get the Most Out of an Equation without Really Trying