The language of science and engineering is largely mathematical, which, increasingly, requires solving problems that are described by ordinary differential equations (ODEs) and partial differential equations (PDEs). The primary focus of this book is numerical solutions to initial value problems (IVPs) and boundary value problems (BVPs) described by ODEs and PDEs. The solutions are implemented in computer code using the open source R language system.

The intended readership is senior undergraduates and postgraduate students in the subject areas of science, technology, engineering, and mathematics (STEM).The contents should also appeal to engineers and scientists in industry who need practical solutions to real-world problems. The emphasis is on understanding the basic principles of the methods discussed and how they can be implemented in computer code.

The aim of this book is to provide a set of software tools that implement numerical methods that can be applied to a broad spectrum of differential equation problems. Each chapter includes a set of references that provide additional information and insight into the methods and procedures employed. All chapters are more or less complete in themselves, except for a few references to other chapters. Thus each chapter can be studied independently.

It is assumed that the reader has a basic understanding of the Rlanguage, although the computer code is annotated to a level that should make understanding clear. Additional discussion is included in the text for more advanced language constructs. Some basic examples of the use of computer algebra systems are also included that make use of the R interface packages Ryacas and rSymPy.

R is a free, high-level software programming language and software environment that has traditionally been used for statistical computing and graphics. It has been widely used for many years by statisticians for data analysis being particularly effective in handling large data sets. However, recently, packages have been added to R to solve a wider range of numerical problems. In particular, the addition of package deSolve [Soe-10] opened up the language for solving differential equations by adding industrial strength integrators. The package deSolve is used extensively in the R examples provided in this book. For readers wishing to learn about R, The Art of R Programming [Mat-11] is a good introductory text, and The R Book [Cra-11] is a comprehensive description of the R language.