Mathematical models in the applied sciences /

In Elementary Mathematical Models, Kalman uses basic growth models (arithmetic, quadratic, geometric, mixed arithmetic-geometric, and logistic) not only to convey the power of mathematics in solving real-world problems but also to motivate the study of the elementary functions usually encountered in college algebra courses. There is a natural evolution from simple hypotheses to difference equations, to their solutions, to the study of the elementary functions associated with the solutions. There is an emphasis on the why of algebra and on manipulation associated with applications rather than for its own sake. Numerical, graphical, and symbolic approaches are used throughout, and the numerous exercises include reading comprehension exercises and group activities as well as more traditional problems. There are solutions to selected exercises. Aimed at students at the college algebra or liberal arts mathematics level, the slow, careful development should be clear even to those with a weak algebraic background. Highly recommended. Lower-division undergraduates. Modeling with Differential and Difference Equations covers a broad spectrum of models from such diverse areas as mechanics, genetics, thermal physics, medicine, economics, and population studies. For each model the relevant background theory is provided along with carefully laid out assumptions. Model development is clear and deliberate--indeed, it is algorithmic, concentrating on the techniques used to set up mathematical models. Although some familiarity with elementary linear algebra and calculus is assumed, the essential theory is provided to analyze and solve the simple differential and difference equations that arise. Introductory examples are well chosen and clearly developed, and exercises reinforce the material well; they vary from the almost trivial to those challenging the reader to develop models that are variants of those presented. Excellent references to classic works. Highly recommended. Undergraduates. Mathematical Models in the Applied Sciences differs markedly from the two books previously discussed and from most other modeling books. The models are more complex and their development is very condensed. A defining characteristic is the emphasis on advanced techniques of analysis. Such techniques as nondimensionalization, scale analysis, and perturbation theory are demonstrated to be unifying threads in the analysis of a wide array of models arising from diverse disciplines. Examples are presented from the physical, biological, physiological, environmental, and industrial sciences. The scope is uniquely broad; many models are unavailable in other modeling books; Fowler culled them from theses and research reports. The notes and references are invaluable guides to the literature, as classic works are cited. This is a demanding text; the exercises are excellent and challenging, and some are at the level of research problems. Applied mathematicians, engineers, and scientists will appreciate this book. Highly recommended. Graduates through professionals. G. J. G. Junevicus Eckerd College