Shijun Liao. Homotopy Analysis Method in Nonlinear Differential Equations

Transcription

1 Shijun Liao Homotopy Analysis Method in Nonlinear Differential Equations

2 Shijun Liao Homotopy Analysis Method in Nonlinear Differential Equations With 127 figures

3 Author Shijun Liao Shanghai Jiao Tong University Shanghai , China ISBN Higher Education Press, Beijing ISBN ISBN (ebook) Springer Heidelberg Dordrecht London New York Library of Congress Control Number: c Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012 Thisworkissubject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Lawof September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed on acid-free paper Springer is part of Springer Science+Business Media (

4 To my mother, wife anddaughter

5 Preface It is well-known that perturbation and asymptotic approximations of nonlinear problems often break down as nonlinearity becomes strong. Therefore, they are only valid for weakly nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) in general. The homotopy analysis method(ham)is an analytic approximation method for highly nonlinear problems, proposed by the author in Unlike perturbation techniques, the HAM is independent of any small/large physical parameters at all: one can always transfer a nonlinear problem into an infinite number of linear subproblems by means of the HAM. Secondly, different from all of other analytic techniques, the HAM provides us a convenient way to guarantee the convergence of solution series so that it is valid even if nonlinearitybecomes rather strong. Besides, based on the homotopy in topology, it provides us extremely large freedom to choose equation type of linear sub-problems, base function of solution, initial guess and so on, so that complicated nonlinear ODEs and PDEs can often be solved in a simple way. Finally, the HAM logically contains some traditional methods such as Lyapunov s small artificial method, Adomian decomposition method, the δ-expansion method,and even the Euler transform, so that it has the great generality.therefore, the HAM provides us a useful tool to solve highly nonlinear problems in science, finance and engineering. This book consists of three parts. In Part I, the basic ideas of the HAM, especially its theoretical modifications and developments, are described, including the optimal HAM approaches, the theorems about the so-called homotopy-derivative operator and the different types of deformation equations, the methods to control and accelerate convergence, the relationship to Euler transform, and so on. In Part II, inspirited by so many successful applications of the HAM in different fields and also by the ability of computing with functions instead of numbers of computer algebra system like Mathematica and Maple, a Mathematica package BVPh (version 1.0) is developed by the author in the frame of the HAM for nonlinear boundary-value problems. A dozen of examples are used to illustrate its validity for highly nonlinear ODEs with singularity,multiple solutions and multipoint boundary conditions in either a finite or an infinite interval, and even for some types of non-

6 viii Preface linear PDEs. As an open resource, the BVPh 1.0 is given in this book with a simple users guide and is free available online. In Part III, we illustrate that the HAM can be used to solve some complicated highly nonlinear PDEs so as to enrich and deepen our understandings about these interesting nonlinear problems. For example, By means of the HAM, an explicit analytic approximation of the optimal exercise boundary of American put option was gained, which is often valid for a couple of decades prior to expiry, whereas the asymptotic and perturbation formulas are valid only for a couple of days or weeks in general. A Mathematica code based on such kind of explicit formula is given in this book for businessmen to gain accurate results in a few seconds. In addition, by means of the HAM, the wave-resonance criterion of arbitrary number of traveling gravity waves was found, for the first time, which logically contains the famous Phillips criterion for four waves with small amplitude. All of these show the originality, validity and generality of the HAM for highly nonlinear problems in science, finance and engineering. All Mathematica codes and their input data files are given in the appendixes of this book and available (Accessed 25 Nov 2011, will be updated in the future) either at orat This book is suitable for researchers and postgraduates in applied mathematics, physics, finance and engineering, who are interested in highly nonlinear ODEs and PDEs. I would like to express my gratitude to my collaborators for their valuable discussions and communications, and to my postgraduates for their hard working. Thanks to Natural Science Foundation of China for the financial support. I would like to express my sincere thanks to my parents, wife and daughter for their love, encouragement and support in the past 20 years. Shanghai,China March 2011 Shijun Liao

7 Contents Part I Basic Ideas and Theorems 1 Introduction Motivation and purpose Characteristic of homotopy analysis method Outline References Basic Ideas of the Homotopy Analysis Method Concept of homotopy Example 2.1: generalized Newtonian iteration formula Example 2.2: nonlinear oscillation Analysis ofthe solution characteristic Mathematical formulations Convergence of homotopy-series solution Essence of the convergence-control parameter c Convergence acceleration by homotopy-padé technique Convergence acceleration by optimal initial approximation Convergence acceleration by iteration Flexibility on the choice of auxiliary linear operator Concluding remarks anddiscussions Appendix 2.1 Derivation of δ n in (2.57) Appendix 2.2 Derivation of (2.55)bythe 2nd approach Appendix 2.3 Proof of Theorem Appendix 2.4 Mathematica code (without iteration) for Example Appendix 2.5 Mathematica code (with iteration) for Example Problems References

8 x Contents 3 Optimal Homotopy Analysis Method Introduction An illustrative description Basic ideas Different types of optimal methods Systematic description Concluding remarks anddiscussions Appendix 3.1 Mathematica code for Blasius flow Problems References Systematic Descriptions and Related Theorems Brief frame of the homotopy analysis method Properties of homotopy-derivative Deformation equations A brief history High-order deformation equations Examples Convergence theorems Solution expression Choice of initial approximation Choice of auxiliarylinear operator Convergence control and acceleration Optimal convergence-control parameter Optimal initial approximation Homotopy-iteration technique Homotopy-Padé technique Discussions and open questions References Relationship to Euler Transform Introduction Generalized Taylor series Homotopy transform Relation between homotopy analysis method and Euler transform Concluding remarks References Some Methods Based on the HAM A brief history of the homotopy analysis method Homotopy perturbation method Optimal homotopy asymptotic method Spectral homotopy analysis method Generalized boundary element method Generalized scaled boundary finite element method

9 Contents xi 6.7 Predictor homotopy analysis method References Part II Mathematica Package BVPh and Its Applications 7 Mathematica Package BVPh Introduction Scope Brief mathematical formulas Choice of base function andinitial guess Choice of the auxiliary linear operator Choice of the auxiliary function Choice of the convergence-control parameter c Approximation and iteration of solutions Polynomials Trigonometric functions Hybrid-base functions A simple users guide of the BVPh Key modules Control parameters Input Output Global variables Appendix 7.1 Mathematica package BVPh (version 1.0) References Nonlinear Boundary-value Problems with Multiple Solutions Introduction Brief mathematical formulas Examples Nonlinear diffusion-reactionmodel A three-point nonlinear boundary-value problem Channelflows with multiplesolutions Concluding remarks Appendix 8.1 Input data of BVPh for Example Appendix 8.2 Input data of BVPh for Example Appendix 8.3 Input data of BVPh for Example Problems References Nonlinear Eigenvalue Equations with Varying Coefficients Introduction Brief mathematical formulas Examples Non-uniform beam acted by axial load Gelfand equation

10 xii Contents Equation with singularity and varying coefficient Multipoint boundary-value problem with multiple solutions Orr-Sommerfeld stability equation with complex coefficient Concluding remarks Appendix 9.1 Input data of BVPh for Example Appendix 9.2 Input data of BVPh for Example Appendix 9.3 Input data of BVPh for Example Appendix 9.4 Input data of BVPh for Example Appendix 9.5 Input data of BVPh for Example Problems References A Boundary-layer Flow with an Infinite Number ofsolutions Introduction Exponentially decaying solutions Algebraically decaying solutions Concluding remarks Appendix 10.1 Input data of BVPh for exponentially decaying solution Appendix 10.2 Input data of BVPh for algebraically decaying solution References Non-similarity Boundary-layer Flows Introduction Brief mathematical formulas Homotopy-series solution Concluding remarks Appendix 11.1 Input data of BVPh References Unsteady Boundary-layer Flows Introduction Perturbation approximation Homotopy-series solution Brief mathematical formulas Homotopy-approximation Concluding remarks Appendix 12.1 Input data of BVPh References

11 Contents xiii Part III Applications in Nonlinear Partial Differential Equations 13 Applications in Finance: American Put Options Mathematical modeling Brief mathematical formulas Validity of the explicit homotopy-approximations A practical code for businessmen Concluding remarks Appendix 13.1 Detailedderivation of f n (τ) and g n (τ) Appendix 13.2 Mathematica code for American put option Appendix 13.3 Mathematica code APOh for businessmen References Two and Three Dimensional Gelfand Equation Introduction Homotopy-approximations of 2D Gelfand equation Brief mathematical formulas Homotopy-approximations Homotopy-approximations of 3D Gelfand equation Concluding remarks Appendix 14.1 Mathematica code of 2D Gelfand equation Appendix 14.2 Mathematica code of 3D Gelfand equation References Interaction of Nonlinear Water Wave and Nonuniform Currents Introduction Mathematical modeling Original boundary-value equation Dubreil-Jacotin transformation Brief mathematical formulas Solution expression Zeroth-order deformation equation High-order deformation equation Successive solution procedure Homotopy approximations Concluding remarks Appendix 15.1 Mathematica code of wave-current interaction References Resonance of Arbitrary Number of Periodic Traveling Water Waves Introduction Resonance criterion of two small-amplitude primary waves Brief Mathematical formulas Non-resonant waves Resonant waves

12 xiv Contents 16.3 Resonance criterion of arbitrary number of primary waves Resonance criterion of small-amplitude waves Resonance criterion of large-amplitude waves Concluding remark and discussions Appendix 16.1 Detailed derivation of high-order equation References Index...563

13 Acronyms 2D 3D APO BEM BVP BVPs CPU DNS FDM FEM GBEM HAM IVP IVPs ODE ODEs OHAM PDE PDEs Two Dimensional Three Dimensional American Put Option Boundary Element Method Boundary Value Problem Boundary Value Problems Central Processing Unit Direct Numerical Simulation Finite Difference Method Finite Element Method Generalized Boundary Element Method Homotopy Analysis Method Initial Value Problem Initial Value Problems Ordinary Differential Equation Ordinary Differential Equations Optimal Homotopy Analysis Method Partial Differential Equation Partial Differential Equations