Numerical analysis using R is a comprehensive and advanced guide for solving ordinary differential equations (ODEs) and partial differential equations (PDEs) in an R framework. The book presents the latest numerical solutions to initial value and boundary value problems often faced in science and engineering. The text consists of 11 chapters and a mathematical aide-mémoire. The latter is a very valuable asset for the book since it can be used as a reference and guide for readers in science and engineering. The programming language used as a framework for all examples and concepts is R. The text assumes some proficiency in R, but does not expect professional-level programming skills.

Each of the chapters deals with a different set of problems. Apart from the first chapter, which is an overview of ODEs and well-known methods from basic calculus like Runge-Kutta, Adams, backward differentiation formulas (BDFs), and others, the next chapters discuss stability, Cauchy problems, well-posed systems, and so on. One of the most useful chapters is 6, which discusses high-resolution schemes for PDEs. The chapter introduces the two major methods for solving monotonic PDEs: the flux limiter method (about 15 different limiter functions are presented), and weighted essentially non-oscillatory (WENO). From the point of view of solving hard problems over non-regular geometries, chapters 7 and 8 are the most interesting. Chapter 7 discusses meshless methods that use radial basis functions. These methods have become very popular and are main methods in solving problems where working on irregular grids is a must, that is, electronic structure calculations. I must stress another value of chapter 7: for many examples, the results are compared to analytic solutions in order to demonstrate the accuracy of the method. Chapter 8 introduces another important concept: conservation laws in the context of evolutionary PDEs. The chapter concludes with a discussion and example calculations of the nonlinear Schrödinger and Boussinesq equations.

In addition to theoretical chapters like the ones discussed above, the book contains several case studies that can serve as illustrations and may also help users see the value in theoretical chapters. As such, chapter 9 presents a case study of a golf ball in flight, chapter 10 describes a problem of explosion, and chapter 11 presents a study of a global carbon cycle.

All chapters include worked examples, and the code is annotated so users can easily download them. The analysis of the data uses Maple and Maxima. In conclusion, this book is an excellent study guide for advanced graduate students, but also can be used by researchers in different fields. Solving differential equations in R [1] is a complementary book.