Non-Perturbative Methods for Strongly Nonlinear Problems
1st Edition
Published on 31. May 2006
Book
298 pages
978-3-86624-113-8 (ISBN)
Description
Perturbation method is one of the oldest subjects in mathematics, yet is very much alive and is still evolving. Besides its mathematical importance and its links to other branches of mathematics, it is widely used in all ramifications of modern sciences, but the limitations for weakly nonlinear problems greatly hamper its development and application.
This book is a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximated analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings.
This book serves both as a guide to the perturbation and nonperturtive methods for non-mathematics students, and an aid to the non-specialists in mathematics in discovering the physical meaning of an nonlinear equation arising in a real-life problem.
The methodologies suggested in this book can be easily comprehended with only a basic knowledge of Advanced Calculus, even the reader has no knowledge of homotopy in topology and calculus of variations in pure mathematics. Modern mathematics, e.g., homotopy and calculus of variations, is combined into perturbation methods, leading to a powerful mathematical tool for all students and teachers, scientists and engineers.
This book also contributes a chapter on ancient Chinese methods and their modern applications. Ancient Chinese mathematics, when revisited in the light of new developments, reveals hidden pearls, and can be readily applied to modern problems.
This book is well designed for a course both at the undergraduate and graduate levels, and a valuable reference for scientists and engineers.
This book is a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximated analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings.
This book serves both as a guide to the perturbation and nonperturtive methods for non-mathematics students, and an aid to the non-specialists in mathematics in discovering the physical meaning of an nonlinear equation arising in a real-life problem.
The methodologies suggested in this book can be easily comprehended with only a basic knowledge of Advanced Calculus, even the reader has no knowledge of homotopy in topology and calculus of variations in pure mathematics. Modern mathematics, e.g., homotopy and calculus of variations, is combined into perturbation methods, leading to a powerful mathematical tool for all students and teachers, scientists and engineers.
This book also contributes a chapter on ancient Chinese methods and their modern applications. Ancient Chinese mathematics, when revisited in the light of new developments, reveals hidden pearls, and can be readily applied to modern problems.
This book is well designed for a course both at the undergraduate and graduate levels, and a valuable reference for scientists and engineers.