Partial differential equations and solitary waves theory.

*(English)*Zbl 1175.35001
Nonlinear Physical Science. Berlin: Springer; Beijing: Higher Education Press (ISBN 978-3-642-00250-2/hbk; 978-7-04-025480-8; 978-3-642-00251-9/ebook). xix, 741 p. (2009).

This book is devoted to the study of classical and new methods for solving partial differential equations (linear and nonlinear). Only analytical methods are described. They lead to explicit function solutions unlike discretization methods such as finite differences or finite elements methods giving solutions on discrete sets. Adomian decomposition method is the main technique very clearly and rigorously detailed in this book with applications to a lot of concrete partial differential equations. In further chapters, solitons and compactons are used for obtaining solitary waves equations.

Many classical partial differential equations are entirely solved by means of these methods. This work clearly explains the methods and their applications to concrete equations. It can be read by undergraduate and graduate students. But it will be also very useful for researchers in applied mathematics and engineering because it suggests new and powerful techniques avoiding discretization and linearization I warmly recommend this reference book for its simplicity and its rigour.

Many classical partial differential equations are entirely solved by means of these methods. This work clearly explains the methods and their applications to concrete equations. It can be read by undergraduate and graduate students. But it will be also very useful for researchers in applied mathematics and engineering because it suggests new and powerful techniques avoiding discretization and linearization I warmly recommend this reference book for its simplicity and its rigour.

Reviewer: Yves Cherruault (Paris)

##### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35C10 | Series solutions to PDEs |

35G05 | Linear higher-order PDEs |

35G30 | Boundary value problems for nonlinear higher-order PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35Q35 | PDEs in connection with fluid mechanics |