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2012 | Buch

Homotopy Analysis Method in Nonlinear Differential Equations

verfasst von: Shijun Liao

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

"Homotopy Analysis Method in Nonlinear Differential Equations" presents the latest developments and applications of the analytic approximation method for highly nonlinear problems, namely the homotopy analysis method (HAM). Unlike perturbation methods, the HAM has nothing to do with small/large physical parameters. In addition, it provides great freedom to choose the equation-type of linear sub-problems and the base functions of a solution. Above all, it provides a convenient way to guarantee the convergence of a solution. This book consists of three parts. Part I provides its basic ideas and theoretical development. Part II presents the HAM-based Mathematica package BVPh 1.0 for nonlinear boundary-value problems and its applications. Part III shows the validity of the HAM for nonlinear PDEs, such as the American put option and resonance criterion of nonlinear travelling waves. New solutions to a number of nonlinear problems are presented, illustrating the originality of the HAM. Mathematica codes are freely available online to make it easy for readers to understand and use the HAM.

This book is suitable for researchers and postgraduates in applied mathematics, physics, nonlinear mechanics, finance and engineering.

Dr. Shijun Liao, a distinguished professor of Shanghai Jiao Tong University, is a pioneer of the HAM.

Inhaltsverzeichnis

Frontmatter

Basic Ideas and Theorems

Frontmatter

Chapter 1. Introduction

Abstract

It is well-known that nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) for boundary-value problems are much more difficult to solve than linear ODEs and PDEs, especially by means of analytic methods. Traditionally, perturbation (Van del Pol, 1926; Von Dyke, 1975; Nayfeh, 2000) and asymptotic techniques are widely applied to obtain analytic approximations of nonlinear problems in science, finance and engineering. Unfortunately, perturbation and asymptotic techniques are too strongly dependent upon small/large physical parameters in general, and thus are often valid only for weakly nonlinear problems. For example, the asymptotic/perturbation approximations of the optimal exercise boundary of American put option are valid only for a couple of days or weeks prior to expiry, as shown in Fig. 1.1. Another famous example is the viscous flow past a sphere in fluid mechanics: the perturbation formulas of the drag coefficient are valid only for rather small Reynolds number Re ≪ 1. Thus, it is necessary to develop some analytic approximation methods, which are independent of any small/large physical parameters at all and besides valid for strongly nonlinear problems.

Shijun Liao

Chapter 2. Basic Ideas of the Homotopy Analysis Method

Abstract

The basic ideas and all fundamental concepts of the homotopy analysis method (HAM) are described in details by means of two simple examples, including the concept of the homotopy, the flexibility of constructing equations for continuous variations, the way to guarantee convergence of solution series, the essence of the convergence-control parameter c ₀, the methods to accelerate convergence, and so on. The corresponding Mathematica codes are given in appendixes and free available online. Beginners of the HAM are strongly suggested to read it first.

Shijun Liao

Chapter 3. Optimal Homotopy Analysis Method

Abstract

In this chapter, we describe and compare the different optimal approaches of the homotopy analysis method (HAM). A generalized optimal HAM is proposed, which logically contains the basic optimal HAM with only one convergence-control parameter and also the optimal HAM with an infinite number of parameters. It is found that approximations given by the optimal HAMs converge fast in general. Especially, the basic optimal HAM mostly gives good enough approximations. Thus, the optimal HAMs with a couple of convergence-control parameters are strongly suggested in practice.

Shijun Liao

Chapter 4. Systematic Descriptions and Related Theorems

Abstract

In this chapter, the homotopy analysis method (HAM) is systematically described in details as a whole. Mathematical theorems related to the so-called homotopy-derivative operator and deformation equations are proved, which are helpful to gain high-order approximations. Some theorems of convergence are proved, and the methods to control and accelerate convergence are generally described. A few of open questions are discussed.

Shijun Liao

Chapter 5. Relationship to Euler Transform

Abstract

The so-called generalized Taylor series and homotopy-transform are derived in the frame of the homotopy analysis method (HAM). Some related theorems are proved, which reveal in theory the reason why convergence-control parameter provides us a convenient way to guarantee the convergence of the homotopy-series solution. Especially, it is proved that the homotopy-transform logically contains the famous Euler transform that is often used to accelerate convergence of a series or to make a divergent series convergent. All of these provide us a conner-stone for the concept of convergence-control and the great generality of the HAM.

Shijun Liao

Chapter 6. Some Methods Based on the HAM

Abstract

In this chapter, some analytic and semi-analytic techniques based on the homotopy analysis method (HAM) are briefly described, including the so-called “homotopy perturbation method”, the optimal homotopy asymptotic method, the spectral homotopy analysis method, the generalized boundary element method, and the generalized scaled boundary finite element method. The relationships between these methods with the HAM are also revealed.

Shijun Liao

Mathematica Package BVPh and Its Applications

Frontmatter

Chapter 7. Mathematica Package BVPh

Abstract

The BVPh (version 1.0) is a Mathematica package for highly nonlinear boundary-value/eigenvalue problems with singularity and/or multipoint boundary conditions. It is a combination of the homotopy analysis method (HAM) and the computer algebra system Mathematica, and provides us a convenient analytic tool to solve many nonlinear ordinary differential equations (ODEs) and even some nonlinear partial differential equations (PDEs). In this chapter, we briefly describe its scope, the basic mathematical formulas, and the choice of base functions, initial guess and the auxiliary linear operator, and so on, together with a simple users guide. As open resource, the BVPh 1.0 is given in the appendix of this chapter and free available (Accessed 25 Nov 2011, will be updated in the future) at http://numericaltank.sjtu.edu.cn/BVPh.htm

Shijun Liao

Chapter 8. Nonlinear Boundary-value Problems with Multiple Solutions

Abstract

In this chapter, using three different types of nonlinear boundary-value equations with multiple solutions, we verify the validity of the HAM-based Mathematica package BVPh (version 1.0) for nth-order nonlinear boundary-value equations F in a finite interval 0≤z≤a, subject to the n linear boundary conditions B, (1≤k≤n), where F is a nth-order nonlinear differential operator, F is a linear operator, γ_k is a constant, respectively. Especially, the socalled multiple-solution-control parameter is introduced into initial guess in order to search for multiple solutions. We illustrate that, using the BVPh 1.0 as a tool, multiple solutions of some nonlinear boundary-value equations can be found out by means of such kind of multiple-solution-control parameter.

Shijun Liao

Chapter 9. Nonlinear Eigenvalue Equations with Varying Coefficients

Abstract

Five different types of examples are used to illustrate the validity of the HAM-based Mathematica package BVPh (version 1.0) for nonlinear eigenvalue equations F in a finite interval 0 ≤z ≤a, subject to the n linear boundary conditions B (1 ≤ k ≤ n), where F denotes a nth-order nonlinear ordinary differential operator, ℐ is a linear differential operator, B is a constant, u(z) and α denote eigenfunction and eigenvalue, respectively. These examples verify that, using the BVPh 1.0, multiple solutions of some highly nonlinear eigenvalue equations with singularity and/or multipoint boundary conditions can be found by means of different initial guesses and different types of base functions.

Shijun Liao

Chapter 10. A Boundary-layer Flow with an Infinite Number of Solutions

Abstract

In this chapter, the Mathematica package BVPh (version 1.0) based on the homotopy analysis method (HAM) is used to gain exponentially and algebraically decaying solutions of a nonlinear boundary-value equation in an infinite interval. Especially, an infinite number of algebraically decaying solutions were found for the first time by means of the HAM, which illustrate the originality and validity of the HAM for nonlinear boundary-value problems.

Shijun Liao

Chapter 11. Non-similarity Boundary-layer Flows

Abstract

In this chapter, we illustrate the validity of the HAM-based Mathematica package BVPh (version 1.0) for nonlinear partial differential equations (PDEs) related to non-similarity boundary-layer flows. We show that, using BVPh 1.0, a non-similarity boundary-layer flow can be solved in a rather similar way to that for similarity ones governed by nonlinear ODEs. In other words, in the frame of the HAM, solving non-similarity boundary-layer flows is as easy as similarity ones. This shows the validity of the BVPh 1.0 for some nonlinear PDEs, especially for those related to boundary-layer flows.

Shijun Liao

Chapter 12. Unsteady Boundary-layer Flows

Abstract

In this chapter, we illustrate the validity of the HAM-based Mathematica package BVPh (version 1.0) for nonlinear partial differential equations (PDEs) related to unsteady boundary-layer flows. We show that, using BVPh 1.0, an unsteady boundary-layer flow can be solved in a rather similar way to that for steady-state similarity ones governed by nonlinear ODEs. In other words, in the frame of the HAM, solving unsteady boundary-layer flows is as easy as steady-state ones. This shows the validity of the BVPh 1.0 for some nonlinear PDEs, especially for those related to boundary-layer flows.

Shijun Liao

Applications in Nonlinear Partial Differential Equations

Frontmatter

Chapter 13. Applications in Finance: American Put Options

Abstract

The homotopy analysis method (HAM) is successfully combined with the Laplace transform to solve the famous American put option equation in finance. Unlike asymptotic and/or perturbation formulas that are often valid only a couple of days or weeks prior to expiry, our homotopy approximation of the optimal exercise boundary B(ι) in polynomials of \(\sqrt \tau \) to o(ι^M) may be valid a couple of dozen years, or even a half century, as long as M is large enough. It is found that the homotopyapproximation of B(ι) in polynomial of \(\sqrt \tau \) to o(ι⁴⁸) is often valid in so many years that the well-known theoretical perpetual optimal exercise price is accurate enough thereafter, so that the combination of them can be regarded as an analytic formula valid in the whole time interval 0≤ι<+∞. A practical Mathematica code APOh is provided in the Appendix 13.2 for businessmen to gain accurate enough optimal exercise price of American put option at large expiration-time by a laptop only in a few seconds, which is free available (Accessed 25 Nov 2011, will be updated in the future) at http://numericaltank.sjtu.edu.cn/HAM.htm

Shijun Liao

Chapter 14. Two and Three Dimensional Gelfand Equation

Abstract

Using the two-dimensional (2D) and 3D Gelfand equation as an example, we illustrate that the homotopy analysis method (HAM) can be used to solve a 2nd-order nonlinear partial differential equation (PDE) in a rather easy way by transforming it into an infinite number of the 4th or 6th-order linear PDEs. This is mainly because the HAM provides us extremely large freedom to choose auxiliary linear operator and besides a convenient way to guarantee the convergence of solution series. To the best of our knowledge, such kind of transformation has never been used by other analytic/numerical methods. This illustrates the originality and great flexibility of the HAM for strongly nonlinear problems. It also suggests that we must keep an open mind, since we might have much larger freedom to solve nonlinear problems than we thought traditionally.

Shijun Liao

Chapter 15. Interaction of Nonlinear Water Wave and Nonuniform Currents

Abstract

In this chapter, we illustrate the validity of the homotopy analysis method (HAM) for a complicated nonlinear PDE describing the nonlinear interaction of a periodic traveling wave on a non-uniform current with exponential distribution of vorticity. In the frame of the HAM, the original highly nonlinear PDE with variable coefficient is transferred into an infinite number of much simpler linear PDEs, which are rather easy to solve. Physically, it is found that Stokes’ criterion of wave breaking is still correct for traveling waves on non-uniform currents. It verifies that the HAM can be used to solve some complicated nonlinear PDEs so as to deepen and enrich our physical understanding about some interesting nonlinear phenomena.

Shijun Liao

Chapter 16. Resonance of Arbitrary Number of Periodic Traveling Water Waves

Abstract

In this chapter, we verify the validity of the homotopy analysis method (HAM) for a rather complicated nonlinear PDE describing the nonlinear interaction of arbitrary number of traveling water waves. In the frame of the HAM, the waveresonance criterion for arbitrary number of waves is gained, for the first time, which logically contains the famous Phillips’ criterion for four small amplitude waves. Besides, it is found for the first time that, when the wave-resonance criterion is satisfied and the wave system is fully developed, there exist multiple steady-state resonant waves, whose amplitude might be much smaller than primary waves so that a resonant wave may contain much small percentage of the total wave energy. This example illustrates that the HAM can be used as a tool to deepen and enrich our understandings about some rather complicated nonlinear phenomena.

Shijun Liao

Backmatter

Titel: Homotopy Analysis Method in Nonlinear Differential Equations
verfasst von: Shijun Liao
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-25132-0
Print ISBN: 978-3-642-25131-3
DOI: https://doi.org/10.1007/978-3-642-25132-0