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Other titles in the Springer Undergraduate Mathematics series:
Analytic Methods for Partial Differential Equations (Springer Undergraduate Mathematics Series)by G. Evans
Synopses & Reviews
The subject of partial differential equations holds an exciting place in mathematics. At one extreme the interest lies in the existence and uniqueness of solutions, and the functional analysis of the proofs of these properties. At the other extreme lies the applied mathematical and engineering quest to find useful solutions, either analytically or numerically, to these important equations which can be used in design and construction. The objective of this book is to actually solve equations rather than discuss the theoretical properties of their solutions. The topics are approached practically, without losing track of the underlying mathematical foundations of the subject. The topics covered include the separation of variables, the characteristic method, D'Alembert's method, integral transforms and Green's functions. Numerous exercises are provided as an integral part of the learning process, with solutions provided in a substantial appendix.
The subject of PDE's holds an exciting place in mathematics. At one extreme, the interest lies in the existence and uniqueness of solutions, and in theoretical treatment of the material. At the other extreme lies the applied mathematical and engineering quest to find useful solutions, which can be used in design and construction. These books aim to actually solve equations rather than discuss the theoretical properties of their solutions, and will serve as a comprehensive, highly accessible introduction to the subject.
Partial differential equations fall into several areas of mathematics: many of the greatest advances in modern science have been based on discovering the underlying PDE for the process in question. In this book, the emphasis is on the practical solution of problems rather than the theoretical background.
This is the practical introduction to the analytical approach taken in Volume 2. Based upon courses in partial differential equations over the last two decades, the text covers the classic canonical equations, with the method of separation of variables introduced at an early stage. The characteristic method for first order equations acts as an introduction to the classification of second order quasi-linear problems by characteristics. Attention then moves to different co-ordinate systems, primarily those with cylindrical or spherical symmetry. Hence a discussion of special functions arises quite naturally, and in each case the major properties are derived. The next section deals with the use of integral transforms and extensive methods for inverting them, and concludes with links to the use of Fourier series.
Includes bibliographical references (p. 293-295) and index.
Table of Contents
Mathematical Preliminaries.- Separation of the Variables.- First Order Equations and Hyperbolic Second Order Equations.- Integral Transform Methods.- Green's Functions.- Appendix.- References.- Index.
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