Upon encountering the title of this book, one might wonder what type of differential equations R can solve. It turns out that R can be used to address initial value problems (IVPs), using popular and successful methods such as LSODE (Adams method on its core) and RADAU5 (implicit Runge-Kutta), which enable a wide variety of possible implementations. This diversity of implementation in turn permits the use of those methods for solving other types of differential equations: boundary value problems (shooting method), partial differential equations (line method), delay differential equations (for which the trick is how to include the history function), and algebraic differential equations (which may, via some transformations, be solved as IVPs as well).
The book is organized in a “leap-frog” manner. First, a chapter presents useful theory about a type of differential equation, how it is treated in the literature, and the limits of solvability of related discretization methods. This is followed by a chapter on how that type of equation can be implemented in R, along with a well-known example.
This format makes the book an invaluable aid for learning almost from scratch about a quite comprehensive collection of types of differential equations, and their implementation in R, without getting lost in the vastness of available theoretical results. The authors often quote the literature when they want to point the reader to a source for deeper information on a particular issue.
Partial differential equations (PDEs) are solved with the methods of lines (converting the PDE into an IVP) or with finite differences. It is suggested that R may have also implemented the finite volume method, but that is not discussed in this book.
I strongly recommend this work as a companion for most courses on differential equations, as it would enable students to try to implement applications akin to those discussed. Even theoreticians will find a nice review of the history and state of the art of some methods for solving differential equations.