The worlds of statistics and differential equation modeling are separate. Most statisticians ignore the beauty and usefulness of using differential equations in practice and most mathematicians ignore even basic concepts in estimation, testing and uncertainty quantification. This book is a partial solution to the problem and is written at a very low first-year graduate level. The book [the first author and B. W. Silverman, Functional data analysis. 2nd ed. New York, NY: Springer (2005; Zbl 1079.62006)] updates and builds on techniques from the popular functional data analysis.
From a statistical point of view, it starts from nonlinear regression – with which most statisticians will be acquainted with due to classical books such as [D. M. Bates and D. G. Watts, Nonlinear regression analysis and its applications. New York etc.: John Wiley & Sons (1988; Zbl 0728.62062)] among others. Building on these premises, the authors go on illustrating the basics of such “trajectory matching”, as they call traditional nonlinear regression, and “gradient matching”, which roughly means minimizing an integrated squared error instead of a error sum of squares.
From a mathematical point of view, the book reviews several types of linear and nonlinear ordinary differential equations and systems of equations. Some of these are fundamental equations in mathematical physics and several prototypical examples are provided, such as a basic susceptible/infected model in the spread of epidemics and a few compartmental models very common, for example, when studying the pharmacokinetic properties of a drug.
The book does not cover stochastic models, such as applied stochastic differential equations, and does not cover partial differential equations. The authors are very clear in clarifying that from the beginning, another plus to this well-written and much needed contribution.