Ragnar Winther - Böcker

“It is impossible to exaggerate the extent to which modern applied mathematics has been shaped and fueled by the g- eral availability of fast computers with large memories. Their impact on mathematics, both applied and pure, is comparable to the role of the telescopes in astronomy and microscopes in biology.” — Peter Lax, Siam Rev. Vol. 31 No. 4 Congratulations! You have chosen to study partial di?erential equations. That decision is a wise one; the laws of nature are written in the language of partial di?erential equations. Therefore, these equations arise as models in virtually all branches of science and technology. Our goal in this book is to help you to understand what this vast subject is about. The book is an introduction to the ?eld. We assume only that you are familiar with - sic calculus and elementary linear algebra. Some experience with ordinary di?erential equations would also be an advantage. Introductory courses in partial di?erential equations are given all over the world in various forms. The traditional approach to the subject is to introduce a number of analytical techniques, enabling the student to - rive exact solutions of some simpli?ed problems. Students who learn about viii Preface computational techniques on other courses subsequently realize the scope of partial di?erential equations beyond paper and pencil.

This book teaches the basic methods of partial differential equations and introduces related important ideas associated with the analysis of numerical methods for those partial differential equations. Standard topics such as separation of variables, Fourier analysis, maximum principles, and energy estimates are included. Numerical methods are introduced in parallel to the classical theory. The numerical experiments are used to illustrate properties of differential equations and theory for finite difference approximations is developed. Numerical methods are included in the book to show the significance of computations in partial differential equations and to illustrate the strong interaction between mathematical theory and the development of numerical methods. Great care has been taken throughout the book to seek a sound balance between the analytical and numerical techniques. The authors present the material at an easy pace with well-organized exercises ranging from the straightforward to the challenging.In addition, special projects are included, containing step by step hints and instructions, to help guide students in the correct way of approaching partial differential equations. The text would be suitable for advanced undergraduate and graduate courses in mathematics and engineering. Necessary prerequisites for this text are basic calculus and linear algebra. Some elementary knowledge of ordinary differential equations is also preferable.

In diesem Buch werden die Grundlagen partieller Differentialgleichungen sowie mit der Untersuchung numerischer Methoden für diese partiellen Differentialgleichungen in Zusammenhang stehende wichtige Überlegungen vorgestellt. Standardthemen wie Trennung der Variablen, Fourier Analysis, Maximumprinzipien und Energieabschätzungen werden behandelt. Numerische Verfahren werden parallel zur klassischen Theorie vorgestellt. Die Eigenschaften von Differentialgleichungen werden anhand numerischer Experimente veranschaulicht, und die Theorie der finiten Differentialapproximationen wird entwickelt. Numerische Verfahren werden eingeführt, um die Bedeutung des Rechnens bei partiellen Differentialgleichungen zu zeigen und die starke Interaktion zwischen mathematischer Theorie und der Entwicklung numerischer Verfahren zu illustrieren. Besonderer Wert wurde im gesamten Buch auf eine möglichst gleichgewichtige Darstellung der analytischen und der numerischen Verfahren gelegt.