Resolution of Nonlinear, Differential and Partial Differential Equations
Most physical problems can be expressed in the form of mathematical equations (e.g. differential equations, integral equations). Historically, mathematicians had to find analytic solutions to the equations encountered in engineering and related fields (e.g. mechanics, physics, biology). These equations are sometimes highly complex, requiring significant work to be simplified. However, in the mid-20th Century, the introduction of the first computers gave rise to new methods for solving equations:
numerical methods. This new approach allows us to solve the equations that we encounter (when constructing models) as accurately as possible, thereby enabling us to approximate the solutions of the problems that we are studying. These approximate solutions are typically calculated by computers using suitable algorithms.
Practical experience has shown that, compared to standard numerical approaches, a carefully planned and optimized methodology can improve the speed of computation by a factor of 100 or even higher. This can transform a completely unreasonable calculation into a perfectly routine computation, hence our great interest in numerical methods! Clearly, it is important for researchers and engineers to understand the methods that they are using and, in particular, the limitations and advantages associated with each approach. The computations needed by most scientific fields require techniques to represent functions as well as algorithms to
calculate derivatives and integrals, solve differential equations, locate zeros, find the eigenvectors and eigenvalues of a matrix, and much more.
The objective of this book is to present and study the fundamental numerical methods that allow scientific computations to be executed. This involves implementing a suitable methodology for the scientific problem at hand, whether derived from physics (e.g. meteorology, pollution) or engineering (e.g. structural mechanics, fluid mechanics, signal processing).
x Advanced Numerical Methods with Matlab 2
This book is divided into two parts, with two appendices. The first part contains two chapters dedicated to solving nonlinear equations and differential equations. The second part consists of four chapters on the various numerical methods that are used to solve partial differential equations: finite differences, finite elements, finite volumes
and meshless methods. Each chapter starts with a brief overview of relevant theoretical concepts and
definitions, with a range of illustrative numerical examples and graphics. At the end of each chapter, we introduce the reader to the various Matlab commands for implementing the methods that have been discussed. As is often the case, practical applications play an essential role in understanding and mastering these methods.
There is little hope of being able to assimilate them without the opportunity to apply them to a range of concrete examples. Accordingly, we will present various examples and explore them with Matlab. These examples can be used as a starting point for practical exploration.
Matlab is currently widely used in teaching, industry and research. It has become a standard tool in various fields thanks to its integrated toolboxes (e.g. optimization, statistics, control, image processing). Graphical interfaces have been improved considerably in recent versions. One of our appendices is dedicated to introducing readers to Matlab.