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This book provides a set of ODE/PDE integration routines in the six most widely used computer languages, enabling scientists and engineers to apply ODE/PDE analysis toward solving complex problems. This text concisely reviews integration algorithms, then analyzes the widely used Runge-Kutta method.

It first presents a complete code before discussing its components in detail, focusing on integration concepts such as error monitoring and control. The format allows scientists and engineers to understand the basics of ODE/PDE integration, then calculate sample numerical solutions within their targeted programming language. The applications discussed can be used as templates for the development of a spectrum of new applications.

Initial value ordinary differential equations (ODEs) and partial differential equations (PDEs) are among the most widely used forms of mathematics in science and engineering. However, insights from ODE/PDE-based models are realized only when solutions to the equations are produced with acceptable accuracy and with reasonable effort.

Most ODE/PDE models are complicated enough (e.g., sets of simultaneous nonlinear equations) to preclude analytical methods of solution; instead, numerical methods must be used, which is the central topic of this book.
The calculation of a numerical solution usually requires that well established numerical integration algorithms are implemented in quality library routines. The library routines in turn can be coded (programmed) in a variety of programming languages. Typically, for a scientist or engineer with
an ODE/PDE- based mathematical model, finding routines written in a familiar language can be a demanding requirement, and perhaps even impossible (if such routines do not exist).
The purpose of this book, therefore, is to provide a set of ODE/PDE integration routines written in six widely accepted and used languages. Our intention is to facilitate ODE/PDE-based analysis by using the library routines to compute reliable numerical solutions to the ODE/PDE system of
interest.

However, the integration of ODE/PDEs is a large subject, and to keep this discussion to reasonable length, we have limited the selection of algorithms and the associated routines. Specifically, we concentrate on explicit (nonstiff) Runge Kutta (RK) embedded pairs. Within this setting, we have provided integrators that are both fixed step and variable step; the latter accept a user-
specified error tolerance and attempt to compute a solution to this required accuracy. The discussion of ODE integration includes truncation error monitoring and control, h and p refinement, stability and stiffness, and explicit and implicit algorithms. Extensions to stiff systems are also discussed and illus-
trated through an ODE application; however, a detailed presentation of stiff (implicit) algorithms and associated software in six languages was judged impractical for a book of reasonable length.

Further, we have illustrated the application of the ODE integration routines to PDEs through the method of lines (MOL). Briefly, the spatial (boundary value) derivatives of the PDEs are approximated algebraically, typically by finite differences (FDs); the resulting system of initial-value ODEs is then solved numerically by one of the ODE routines.