Solving nonlinear problems is inherently difficult, and the stronger the nonlinearity, the more intractable solutions become. Analytic approximations often break down as nonlinearity becomes strong, and even perturbation approximations are valid only for problems with weak nonlinearity.
This book introduces a powerful new analytic method for nonlinear problems-homotopy analysis-that remains valid even with strong nonlinearity. In Part I, the author starts with a very simple example, then presents the basic ideas, detailed procedures, and the advantages (and limitations) of homotopy analysis. Part II illustrates the application of homotopy analysis to many interesting nonlinear problems. These range from simple bifurcations of a nonlinear boundary-value problem to the Thomas-Fermi atom model, Volterra's population model, Von Kármán swirling viscous flow, and nonlinear progressive waves in deep water.
Although the homotopy analysis method has been verified in a number of prestigious journals, it has yet to be fully detailed in book form. Written by a pioneer in its development, Beyond Pertubation: Introduction to the Homotopy Analysis Method is your first opportunity to explore the details of this valuable new approach, add it to your analytic toolbox, and perhaps make contributions to some of the questions that remain open.
Table of Contents
PART I BASIC IDEAS
Relations to Some Previous Analytic Methods
Advantages, Limitations, and Open Questions
PART II APPLICATIONS
Simple Bifurcation of a Nonlinear Problem
Multiple Solutions of a Nonlinear Problem
Nonlinear Eigenvalue Problem
Thomas-Fermi Atom Model
Volterra's Population Model
Free Oscillation Systems with Odd Nonlinearity
Free Oscillation Systems with Quadratic Nonlinearity
Limit Cycle in a Multidimensional System
Blasius' viscous Flow
Boundary-layer Flow with Exponential Property
Boundary-layer Flow with Algebraic Property
Von Kármán Swirling Flow
Nonlinear Progressive Waves in Deep Water
"The author has invested a great amount of effort into determining all the power series and computing the functions depicted in the figures."
-Mathematical Reviews, Issue 2005h
"… an excellent reference to researchers, engineers, and interested individuals in helping them tackle nonlinear problems in an analytical fashion…a good subject index and an outstanding list of bibliography with 136 references cited…very well written and is relatively easy to follow to the mathematically literate person. I highly recommend that it be acquired by interested individuals and libraries throughout."
-Applied Mathematics Review, Vol. 57, No. 5, September 2004
"This monograph offers the opportunity to explore the details of the valuable new approach both in the theory and on many interesting examples. It will be useful to specialists working in applied nonlinear analysis."
-Zentralblatt MATH 1051