This text introduces the reader to concepts and applications for modeling and analyzing natural, social, and technological processes. Connecting mathematics to the applied and natural sciences, the book employs scaling and dimensional analysis, calculus of variations, Green's functions, stability and bifurcation, and more. Final chapters cover wave phenomena (wave propagation, the wave equation, etc.) and mathematical models of continua (kinematics, gas dynamics, and fluid motions in R3).
The book is designed for use as a text. Initially, I thought this would make it suffer when used as a self-directed tool of personal education. Among the things I found daunting were the absence of any exercise solutions or hints and only five pages of index for over 520 pages of text. However, in working through several sections myself I found the examples sufficient in breadth and depth to support the basic exercises. Exercises vary from routine calculations to reinforce foundation techniques to more challenging problems stimulating advanced problem solving. Every section ends with a “Reference and Notes” section that lists additional material making any required additional study for these challenging problems possible, say, in a university library. Even the apparently thin index never failed me to refer back to find specific topics relevant to working through the book.
The organization is such that the first section covers models leading to ordinary differential equations and integral equations, and the second section focuses on partial differential equations and their applications. As such, this is a gateway text for transition into graduate study in mathematical modeling for engineering, and natural sciences. Future mathematicians, scientists and engineers should find the book to be an excellent introductory text for coursework or self-study as well as worth its shelf space for reference.
This new and revised third edition incorporates a new chapter on discrete-time methods, with a section introducing stochastic models to the reader. There is also more linear algebra material (transformations in Rn, eigenvalues, etc.) and updates to content for greater current relevance. Among such updates are extensions into mathematical biology, including matlab programs.
Tom Schulte (http://www.oakland.edu/~tgschult/), a graduate student at Oakland University (http://www.oakland.edu), shares a four bedroom house with three cats while working on the two-body problem since his one wife is in graduate school in the West Indies.
1. Dimensional Analysis, Scaling, & Differential Equations.
2. Perturbation Methods.
3. Calculus of Variations.
4. Eigenvalue Problems, Integral Equations, and Green's Functions.
5. Discrete Models.
6. Partial Differential Equations.
7. Wave Phenomena.
8. Mathematical Models of Continua.