Shijun Liao - Böcker

This book describes the basic principles of Clean Numerical Simulation (CNS) proposed by the author in 2009, as well as several of its applications. Unlike conventional algorithms, CNS gives in a sufficiently long interval of time a convergent chaotic trajectory whose numerical noise is much lower than the true physical solution, so that one can gain accurately. Thus, CNS provides for the first time an ability to check statistics stability of chaos, leading to a completely new concept of "ultra-chaos," which has both trajectory instability and statistics instability, and thus is of a higher disorder.Notably, it is impossible to repeat experimental results of ultra-chaos even in the statistical sense. However, the reproducibility of physical experiments forms a cornerstone for modern science. Thus, ultra-chaos reveals an incompleteness of the modern science paradigm. In addition, it also reveals statistics stability as a precondition for use of conventional algorithms, including direct numerical simulation (DNS). In Clean Numerical Simulation, several conjectures and open problems are proposed, including a modified fourth Clay millennium problem. Indeed, CNS opens the door for us to enter the "clean" numerical world of chaos and turbulence.

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.