Introduction to Partial Differential Equations A Computational Appro
“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 t
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“It is impossible to exaggerate the extent to which modern applied mathematics has been shaped and fueled by the general 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 differential equations. That decision is a wise one; the laws of nature are written in the language of partial differential 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 field. We assume only that you are familiar with basic calculus and elementary linear algebra. Some experience with ordinary differential equations would also be an advantage. Introductory courses in partial differential 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 derive exact solutions of some simplified problems. Students who learn about
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Preface
computational techniques on other courses subsequently realize the scope of partial differential equations beyond paper and pencil. Our approach is different. We introduce analytical and computational techniques in the same book and thus in the same course. The main reason for doing this is that the computer, developed to assist scientists in solving partial differential equations, has become commonly available and is currently used in all practical applications of partial differential equations. Therefore, a modern introduction to this topic must focus on methods suitable for computers. But these methods often rely on deep analytical insight into the equations. We must therefore take great care not to throw away basic analytical methods but seek a sound balance between analytical and computational techniques. One advantage of introducing computational techniques is that nonlinear problems can be given more attention than is common in a purely analytical introduction. We have included several examples of nonlinear equations in addition to the standard linear models which are present in any introductory text. In particular we have included a discussion of reaction-diffusion equations. The reason for this is their widespread application as important models in various scientific applications. Our aim is not to discuss the merits of different numerical techniques. There are a huge number of papers in scientific journals comparing different methods to solve various problems. We do not want to include such discussions. Our aim is to demonstrate that computational techniques are simple to use and often give very nice results, not to show that even better results can be obtained if slightly different methods are used. We touch briefly upon some such discussion, but not in any major way, since this really belongs to the field of
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