Mathematics shows itself to be truly alive and connected when, with mathematical modeling, real-world systems become subject to analysis, prediction, and, ultimately, understanding.

This text provides an extensive presentation of the general principles and methods of a growing field with a focus on applications in the natural sciences. Each chapter details contains thorough, practical exposition of one layer of this art, supported by interrelated examples. Generally, examples follow a logical progression: from one- to multi-dimensional, as complexity of the system under consideration increases. Often, the chapters are capped with one challenging problem to help the student bring it all together.

The chapters all have supporting exercises, but no solutions are provided. Nevertheless, while obviously suitable as a textbook for an undergraduate semester on mathematical modeling, this work is self-contained and exhaustive enough to be a good guide for self-study in this area. Skipping over much theory, the book exhaustively describes a toolbox containing ODEs, PDEs, the transport theorem for all conservation laws for flowing media, wave equations, basic laws (Newton’s Laws, Maxwell’s Equation, etc.), and more.

How should these tools be used and applied? The explanations involve the applicability of dimensional analysis (Pi or Buckingham’s Theorem), conservation principles (mass, energy, etc.), extracting relations from basic laws, stability and robustness for judging the effects of small perturbations, scaling to reduce the number of parameters, and more basics. Also, identifying discrete versus continuous systems and independent versus dependent variables is covered. This is the thrust of the first four chapters where a variety of natural and physical problems are formulated with spatiotemporal differential equations arising from physical laws and balance and conservation arguments. Chapter five introduces variational modeling, thus laying the groundwork for an approach to optimization and boundary value problems.

With these basic and advanced concepts, the reader is exposed to modeling heat conduction in one dimension, surface and shallow water waves, momentum, mass and heat balance in three dimensions, traffic flow, population models, Brownian motion, and more. The last chapter is devoted to rich case studies in polymer dynamics of molecules in solution, fiber spinning, water waves, and guiding light with waveguide optics.

Tom Schulte is a PhD candidate in Applied Mathematical Sciences at Oakland University in Michigan.