Beyond Perturbations - An Introduction to Homotopy Analysis Methods

Author: Liao Shijun
Publisher:
Publish Date: 2006-04-01
Features: This book systematically describes an analytical approximate method for solving nonlinear ordinary and partial differential equations—the homotopy analysis method. This method not only overcomes the limitation of perturbation methods that rely on small parameters but also logically includes other non-perturbation methods, such as Lyapunov artificial small parameter method, Adomian decomposition method, and δ-expansion method. Therefore, it has a wide range of applications and is more general. Unlike other analytical approximate methods, this approach provides a simple way to regulate and control the convergence region and convergence rate of the obtained series solutions, making it suitable for strongly nonlinear problems. Part I (Chapters 1–5) of this book systematically describes the theoretical framework of the "homotopy analysis method"; Part II (Chapters 6–18) presents its applications in various nonlinear problems, including nonlinear bifurcation, multiple solutions, nonlinear eigenvalues, the Thomas-Fermi atomic model, the Volterra ecological model, nonlinear vibration, multidimensional dynamical system limit cycles, boundary layers, viscous flow, and nonlinear waves. This book can serve as a reference for scientists, engineers, and researchers whose work involves nonlinear problems, as well as for teachers and graduate students in the fields of mechanics and applied mathematics.