Shijun Liao Homotopy Analysis Method in Nonlinear Differential Equations
Shijun Liao Homotopy Analysis Method in Nonlinear Differential Equations With 127 figures
Author Shijun Liao Shanghai Jiao Tong University Shanghai 200030, China Email: sjliao@sjtu.edu.cn ISBN 978-7-04-032298-9 Higher Education Press, Beijing ISBN 978-3-642-25131-3 ISBN 978-3-642-25132-0 (ebook) Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011941367 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 (www.springer.com)
To my mother, wife anddaughter
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 1992. 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-
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 http://numericaltank.sjtu.edu.cn/ham.htm http://numericaltank.sjtu.edu.cn/bvph.htm 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
Contents Part I Basic Ideas and Theorems 1 Introduction... 3 1.1 Motivation and purpose..... 3 1.2 Characteristic of homotopy analysis method.... 5 1.3 Outline..... 6 References...... 8 2 Basic Ideas of the Homotopy Analysis Method.... 15 2.1 Concept of homotopy... 15 2.2 Example 2.1: generalized Newtonian iteration formula..... 19 2.3 Example 2.2: nonlinear oscillation...... 26 2.3.1 Analysis ofthe solution characteristic.... 26 2.3.2 Mathematical formulations... 31 2.3.3 Convergence of homotopy-series solution... 38 2.3.4 Essence of the convergence-control parameter c 0... 48 2.3.5 Convergence acceleration by homotopy-padé technique... 56 2.3.6 Convergence acceleration by optimal initial approximation....... 59 2.3.7 Convergence acceleration by iteration.... 63 2.3.8 Flexibility on the choice of auxiliary linear operator..... 69 2.4 Concluding remarks anddiscussions..... 75 Appendix 2.1 Derivation of δ n in (2.57)..... 79 Appendix 2.2 Derivation of (2.55)bythe 2nd approach... 80 Appendix 2.3 Proof of Theorem 2.3... 82 Appendix 2.4 Mathematica code (without iteration) for Example 2.2... 83 Appendix 2.5 Mathematica code (with iteration) for Example 2.2... 87 Problems.... 92 References...... 92
x Contents 3 Optimal Homotopy Analysis Method... 95 3.1 Introduction...... 95 3.2 An illustrative description....101 3.2.1 Basic ideas...101 3.2.2 Different types of optimal methods.... 104 3.3 Systematic description....... 117 3.4 Concluding remarks anddiscussions.... 121 Appendix 3.1 Mathematica code for Blasius flow...122 Problems...126 References.......127 4 Systematic Descriptions and Related Theorems...131 4.1 Brief frame of the homotopy analysis method...131 4.2 Properties of homotopy-derivative...133 4.3 Deformation equations...... 148 4.3.1 A brief history.......148 4.3.2 High-order deformation equations..... 153 4.3.3 Examples..... 165 4.4 Convergence theorems...... 168 4.5 Solution expression...173 4.5.1 Choice of initial approximation...175 4.5.2 Choice of auxiliarylinear operator.....176 4.6 Convergence control and acceleration...179 4.6.1 Optimal convergence-control parameter..... 180 4.6.2 Optimal initial approximation....181 4.6.3 Homotopy-iteration technique.... 181 4.6.4 Homotopy-Padé technique...... 182 4.7 Discussions and open questions..... 183 References.......185 5 Relationship to Euler Transform....189 5.1 Introduction..... 189 5.2 Generalized Taylor series.... 190 5.3 Homotopy transform...210 5.4 Relation between homotopy analysis method and Euler transform.... 215 5.5 Concluding remarks...219 References......220 6 Some Methods Based on the HAM...223 6.1 A brief history of the homotopy analysis method...223 6.2 Homotopy perturbation method.... 225 6.3 Optimal homotopy asymptotic method.... 228 6.4 Spectral homotopy analysis method...230 6.5 Generalized boundary element method.... 230 6.6 Generalized scaled boundary finite element method.... 231
Contents xi 6.7 Predictor homotopy analysis method.... 232 References......233 Part II Mathematica Package BVPh and Its Applications 7 Mathematica Package BVPh...239 7.1 Introduction...239 7.1.1 Scope.... 242 7.1.2 Brief mathematical formulas...... 243 7.1.3 Choice of base function andinitial guess....247 7.1.4 Choice of the auxiliary linear operator......250 7.1.5 Choice of the auxiliary function....252 7.1.6 Choice of the convergence-control parameter c 0...253 7.2 Approximation and iteration of solutions...... 254 7.2.1 Polynomials.... 255 7.2.2 Trigonometric functions...256 7.2.3 Hybrid-base functions..... 257 7.3 A simple users guide of the BVPh 1.0...260 7.3.1 Key modules.... 260 7.3.2 Control parameters.... 261 7.3.3 Input...263 7.3.4 Output.... 264 7.3.5 Global variables..... 264 Appendix 7.1 Mathematica package BVPh (version 1.0)... 265 References.......278 8 Nonlinear Boundary-value Problems with Multiple Solutions...285 8.1 Introduction...285 8.2 Brief mathematical formulas....... 286 8.3 Examples...289 8.3.1 Nonlinear diffusion-reactionmodel..... 289 8.3.2 A three-point nonlinear boundary-value problem...... 296 8.3.3 Channelflows with multiplesolutions...301 8.4 Concluding remarks...306 Appendix 8.1 Input data of BVPh for Example 8.3.1.... 307 Appendix 8.2 Input data of BVPh for Example 8.3.2.... 309 Appendix 8.3 Input data of BVPh for Example 8.3.3.... 310 Problems....312 References......313 9 Nonlinear Eigenvalue Equations with Varying Coefficients...315 9.1 Introduction..... 315 9.2 Brief mathematical formulas...... 317 9.3 Examples...322 9.3.1 Non-uniform beam acted by axial load....... 322 9.3.2 Gelfand equation..... 333
xii Contents 9.3.3 Equation with singularity and varying coefficient....... 337 9.3.4 Multipoint boundary-value problem with multiple solutions.......342 9.3.5 Orr-Sommerfeld stability equation with complex coefficient.....346 9.4 Concluding remarks...350 Appendix 9.1 Input data of BVPh for Example 9.3.1.... 351 Appendix 9.2 Input data of BVPh for Example 9.3.2.... 353 Appendix 9.3 Input data of BVPh for Example 9.3.3.... 354 Appendix 9.4 Input data of BVPh for Example 9.3.4.... 355 Appendix 9.5 Input data of BVPh for Example 9.3.5.... 357 Problems.... 358 References......359 10 A Boundary-layer Flow with an Infinite Number ofsolutions....363 10.1 Introduction..... 363 10.2 Exponentially decaying solutions...365 10.3 Algebraically decaying solutions.... 369 10.4 Concluding remarks...376 Appendix 10.1 Input data of BVPh for exponentially decaying solution.....377 Appendix 10.2 Input data of BVPh for algebraically decaying solution.....378 References......380 11 Non-similarity Boundary-layer Flows.... 383 11.1 Introduction...383 11.2 Brief mathematical formulas...387 11.3 Homotopy-series solution..... 392 11.4 Concluding remarks...396 Appendix 11.1 Input data of BVPh.... 397 References......399 12 Unsteady Boundary-layer Flows...403 12.1 Introduction...403 12.2 Perturbation approximation...406 12.3 Homotopy-series solution.... 408 12.3.1 Brief mathematical formulas...408 12.3.2 Homotopy-approximation....412 12.4 Concluding remarks...417 Appendix 12.1 Input data of BVPh...418 References......420
Contents xiii Part III Applications in Nonlinear Partial Differential Equations 13 Applications in Finance: American Put Options...425 13.1 Mathematical modeling......425 13.2 Brief mathematical formulas...428 13.3 Validity of the explicit homotopy-approximations.... 436 13.4 A practical code for businessmen...443 13.5 Concluding remarks.... 444 Appendix 13.1 Detailedderivation of f n (τ) and g n (τ)...446 Appendix 13.2 Mathematica code for American put option...448 Appendix 13.3 Mathematica code APOh for businessmen...454 References.......457 14 Two and Three Dimensional Gelfand Equation....461 14.1 Introduction...461 14.2 Homotopy-approximations of 2D Gelfand equation...462 14.2.1 Brief mathematical formulas..... 462 14.2.2 Homotopy-approximations...468 14.3 Homotopy-approximations of 3D Gelfand equation.... 474 14.4 Concluding remarks.... 480 Appendix 14.1 Mathematica code of 2D Gelfand equation.... 481 Appendix 14.2 Mathematica code of 3D Gelfand equation.... 485 References.......489 15 Interaction of Nonlinear Water Wave and Nonuniform Currents....493 15.1 Introduction...... 493 15.2 Mathematical modeling...494 15.2.1 Original boundary-value equation...... 494 15.2.2 Dubreil-Jacotin transformation...496 15.3 Brief mathematical formulas...... 497 15.3.1 Solution expression.... 497 15.3.2 Zeroth-order deformation equation..... 498 15.3.3 High-order deformation equation....... 500 15.3.4 Successive solution procedure...502 15.4 Homotopy approximations.... 504 15.5 Concluding remarks.... 515 Appendix 15.1 Mathematica code of wave-current interaction..... 516 References......521 16 Resonance of Arbitrary Number of Periodic Traveling Water Waves...523 16.1 Introduction...523 16.2 Resonance criterion of two small-amplitude primary waves...525 16.2.1 Brief Mathematical formulas.....525 16.2.2 Non-resonant waves...533 16.2.3 Resonant waves..... 538
xiv Contents 16.3 Resonance criterion of arbitrary number of primary waves...... 547 16.3.1 Resonance criterion of small-amplitude waves.... 547 16.3.2 Resonance criterion of large-amplitude waves...... 550 16.4 Concluding remark and discussions..... 553 Appendix 16.1 Detailed derivation of high-order equation.... 555 References......561 Index...563
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