同伦分析方法进展综述
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摘要
本文简述同伦分析方法基本思想、最新理论进展及其在流体力学、固体力学、一般力学、量子力学、应用数学、金融等科学和工程领域的应用.同伦分析方法不依赖物理小参数,适用范围更广,而且提供了一种简单的途径确保级数解收敛,适用于强非线性问题.同伦分析方法已被成功应用于求解一些具有挑战性的力学问题,并获得一些全新的、从未见报道的解.这些成功的应用,证明了同伦分析方法的普遍有效性和原创性.
In this paper, a brief review of the current advances of the homotopy analysis method(HAM) in theory and applications is given. The HAM is an analytic approximation method for highly nonlinear problems. Traditionally, perturbation methods were widely used. However, perturbation methods are strongly dependent upon the existence of small physical parameters(called perturbation quantity), and besides perturbation approximations often become divergent as perturbation quantity enlarges. However, unlike perturbation methods,the HAM has nothing to do with the existence of small/large physical parameters, since it is based on the homotopy, a basic concept in topology. Especially, the HAM provides a convenient way to guarantee the convergence of solution series. In addition, the HAM provides great freedom to choose the base-functions and the equation-type of high-order equations so that good approximations can be obtained more efficiently. As illustrated in this paper, the HAM has been used to solve some challenging nonlinear problems in nonlinear mechanics,quantum mechanics, applied mathematics, finance and so on.
In this paper, a brief review of the current advances of the homotopy analysis method(HAM) in theory and applications is given. The HAM is an analytic approximation method for highly nonlinear problems. Traditionally, perturbation methods were widely used. However, perturbation methods are strongly dependent upon the existence of small physical parameters(called perturbation quantity), and besides perturbation approximations often become divergent as perturbation quantity enlarges. However, unlike perturbation methods,the HAM has nothing to do with the existence of small/large physical parameters, since it is based on the homotopy, a basic concept in topology. Especially, the HAM provides a convenient way to guarantee the convergence of solution series. In addition, the HAM provides great freedom to choose the base-functions and the equation-type of high-order equations so that good approximations can be obtained more efficiently. As illustrated in this paper, the HAM has been used to solve some challenging nonlinear problems in nonlinear mechanics,quantum mechanics, applied mathematics, finance and so on.
引文
廖世俊.1992.求解非线性问题的同伦分析方法.[博士论文].上海:上海交通大学(Liao S J.1992.The proposed homotopy analysis technique for the solutions of nonlinear problems.[PhD Thesis].Shanghai:Shanghai Jiao Tong University).
刘曾.2015.稳态共振波及非线性波流相互作用研究.[博士论文].上海:上海交通大学(Liu Z.2015.On the study of steady-state resonant waves and wave-current interaction.[PhD Thesis].Shanghai:Shanghai Jiao Tong University).
王记增.2001.正交小波统一理论与方法及其在压电智能结构等力学研究中的应用.[博士论文].兰州:兰州大学.
徐妲莉.2014.同伦分析方法在稳态共振波浪研究中的应用.[博士论文].上海:上海交通大学(Xu D L.2014.Application of homotopy analysis method in steady-state resonant waves.[PhD Thesis].Shanghai:Shanghai Jiao Tong University).
杨兆臣.2017.求解非线性边值问题的小波同伦分析方法及其应用.[硕士论文].上海:上海交通大学(Yang Z C.2017.The wavelet homotopy analysis method for nonlinear boundary value problems and its applicationis.[Master Thesis].Shanghai:Shanghai JiaoTong University).
张丽,王光谦,傅旭东,孙其诚.2009a.低浓度颗粒流Boltzmann方程的同伦分析方法解.科学通报,54:1518-1523(Zhang L,Wang G Q,Fu X D,Sun Q C.2009a.A new solution to Boltzmann equation of dilute granular flow with homotopy analysis method.Chinese Science Bull,54:1518-1523).
张丽,王光谦,傅旭东,孙其诚.2009b.低浓度固液两相流Boltzmann方程的同伦分析方法解.应用基础与工程科学学报,17:811-818(Zhang L,Wang G Q,Fu X D,Sun Q C.2009b.A new solution to Boltzmann equation of dilute solid-liquid two-phase flows with homotopy analysis method.Journal of Basic Science and Engineering,17:811-818).
钟晓旭.2018.应用同伦分析方法求解若干力学和金融学问题.[硕士论文].上海:上海交通大学(Zhong X U.2018.Homotopy analysis method for several problems in mechanics and finance.[Master Thesis].Shanghai:Shanghai Jiao Tong University).
Adomian G.1976.Nonlinear stochastic differential equations.Journal of Mathematical Analysis and Applications,55:441-452.
Adomian G.1994.Solving Frontier Problems of Physics:The Decomposition Method.Kluwer Academic Publishers.
Alobaidi G,Mallier R.2001.On the optimal exercise boundary for an American put option.Journal of Applied Mathematics,1:39-45.
Alomari A K.2012.Modifications of Homotopy Analysis Method for Differential Equations:Modification of Homotopy Analysis Method,Ordinary,Fractional,Delay and Algebraic Differential Equations.LAPLAMBERT Academic Publishing.
Akyildiz F T,Vajravelu,K.2008.Magnetohydrodynamic flow of a viscoelastic fluid.Physics Letters A,372:3380-3384.
Armstrong M A.1983.Basic Topology(Undergraduate Texts in Mathematics).Springer.
Barles G,Burdeau J,Romano M,Samsoen N.1995.Critical stock price near expiration.Mathematical Finance,5:77-95.
Bataineh A S,Noorani M S M,Hashim I.2007.Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method.Physics Letters A,371:72-82.
Bataineh A S,Noorani M S M,Hashim I.2009.Homotopy analysis method for singular IVPs of EmdenFowler type.Communications in Nonlinear Science and Numerical Simulation,14:1121-1131.
Baxter M,Dewasurendra M,Vajravelu K.2017.A method of directly defining the inverse mapping for solutions of coupled systems of nonlinear differential equations.Numerical Algorithm.https://doi.org/10.1007/s11075-017-0359-0
Benney D J.1962.Non-linear gravity wave interactions.Journal of Fluid Mechanics,14:577-584.
Bretherton F P.1964.Resonant interactions between waves.The case of discrete oscillations.Journal of Fluid Mechanics,20:457-479.
Bunch D S,Johnson H.2000.The American put option and its critical stock price.Journal of Finance,5:2333-2356.
Chen S L,Kuang J C.1981.The perturbation parameter in the problem of large deflection of clamped circular plates.Applied Mathematics and Mechanics(English Edition),2:137-154.
Cheng J,Zhu S P,Liao S J.2010.An explicit series approximation to the optimal exercise boundary of American put options.Communications in Nonlinear Science and Numerical Simulation,15:1148-1158.
Chien W Z.1947.Large deflection of a circular clamped plate under uniform pressure.Chinese Journal of Physics,7:102-113.
Cimpean D,Merkin J H,Ingham D B.2006.On a free convection problem over a vertical flat surface in a porous medium.Transport Porous,64:393-411.
Craig W,Nicholls D P.2002 Traveling gravity water waves in two and three dimensions.European Journal of Mechanics B-Fluids,21:615-641.
Evans J D,Kuske R,Keller J B.2002.American options on asserts with dividends near expiry.Mathematical Finance,12:219-237.
Fu Y,Zhao W D,Tao Z.2017.Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs.Discrete&Continuous Dynamical Systems-B.22:3439-3458.
Gauss C F.1830.Principia Generalia Theoriae Figurae Fluidorum in Statu Aequilibrii.
Ghoreishi M,Ismail A I B M,Alomari A K,Bataineh A S.2012.The comparison between Homotopy Analysis Method and Optimal Homotopy Asymptotic Method for nonlinear age-structured population models.Communications in Nonlinear Science and Numerical Simulation,17:1163-1177.
Hedges T S.1995.Regions of validity of analytical wave theories.Ice Proceedings Water Maritime&Energy,112:111-114.
Hunter J K,Vandenbroeck J M.1983.Accurate computations for steep solitary waves.Journal of Fluid Mechanics,136:63-71.
Itik M,Banks S P.2010.Chaos in a three-dimensional cancer model.International Journal of Bifurcation and Chaos,20:71-79.
Karmishin A V,Zhukov A T,Kolosov V G.1990.Methods of dynamics calculation and testing for thinwalled structures(in Russian)//Mashinostroyenie,Moscow.
Laplace P S M D.1805.Trait′e de m′ecanique c′eleste.Suppl′ement au dixieme livre du Trait′e de M′ecanique C′eleste,1-79.
Liang S X,Jeffrey D J.2010.Approximate solutions to a parameterized sixth order boundary value problem.Computers and Mathematics with Applications,59:247-253.
Liao S J.1997.A kind of approximate solution technique which does not depend upon small parameters(II)-An application in fluid mechanics.International Journal of Non-Linear Mechanics,32:815-822.
Liao S J.1999.An explicit,totally analytic approximation of Blasius viscous flow problems.International Journal of Non-Linear Mechanics,34:759-778.
Liao S J.2003a.Beyond Perturbation:Introduction to the Homotopy Analysis Method.Boca Raton:Chapman&Hall/CRC Press.
Liao S J.2003b.A new analytic algorithm of Lane-Emden type equations.Applied Mathematics and Computation,142:1-16.
Liao S J,Pop I.2004.Explicit analytic solution for similarity boundary layer equations.International Journal of Heat and Mass Transfer,47:75-85.
Liao S J.2009.A general approach to get series solution of non-similarity boundary-layer flows.Communications in Nonlinear Science and Numerical Simulation,14:2144-2159.
Liao,S J.2010a.On the relationship between the homotopy analysis method and Euler transform.Communications in Nonlinear Science and Numerical Simulation,15:1421-1431.
Liao,S J.2010b.An optimal homotopy-analysis approach for strongly nonlinear differential equations.Communications in Nonlinear Science and Numerical Simulation,15:2003-2016.
Liao S J.2011.On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves.Communication in Nonlinear Science and Numerical Simulation,16:1274-1303.
Liao S J.2012.Homotopy Analysis Method in Nonlinear Differential Equations.Springer.
Liao S J,Xu D L,Stiassnie Michael.2016.On the steady-state nearly resonant waves.Journal of Fluid Mechanics,794:175-199.
Liao S J,Zhao Y L.2016.On the method of directly defining inverse mapping for nonlinear differential equations.Numerical Algorithm,72:989-1020.
Liao S J.2018.A new non-perturbative approach in quantum mechanics for time-independent Schr¨odinger equations.arXiv:1806.05103.
Liu Y B,Chen Y S.2011.KBM method based on the homotopy analysis.Science China Physics Mechanics&Astronomy,54:1137-1140.
Liu Z,Liao S J.2014.Steady-state resonance of multiple wave interactions in deep water.Journal of Fluid Mechanics,742:664-700.
Liu Z,Xu D L,Li J,Peng T,Alsaedi A,Liao SJ.2015.On the existence of steady-state resonant waves in experiment.Journal of Fluid Mechanics,763:1-23.
Liu Z,Xu D L,Liao S J.2018.Finite amplitude steady-state wave groups with multiple near resonances in deep water.Journal of Fluid Mechanics,835:624-653.
Liu Z,Xu D L,Liao S J.2017.Mass,momentum and energy flux conservation between linear and nonlinear steady-state wave groups.Physics of Fluids,29:127104.
Lyapunov A M.1992.General Problem on Stability of Motion(English translation).Taylor&Francis,London.
Karoui N E,Peng S G,Quenez M C.1997.Backward stochastic differential equations in finance.Mathematical Finance.7:1-71.
Khalid U.2011.Steady Flow in a Williamson Fluid:Basic Concepts of Fluid Mechanics Solution to a Non-linear Ordinary Differential Equation by Homotopy Analysis Method.LAP LAMBERT Academic Publishing.
Keller H B,Reiss E L.1958.Iterative solutions for the non-linear bending of circular plates.Communications on Pure and Applied Mathematics,11:273-292.
Knessl C.2001.A note on a moving boundary problem arising in the American put option.Studies in Applied Mathematics,107:157-183.
Kuske R A,Keller J B.1998.Optional exercise boundary for an American put option.Applied Mathematical Finance,5:107-116.
Madsen P A,Fuhrman D R.2012.Third-order theory for multi-directional irregular waves.Journal of Fluid Mechanics,698:304-334.
Marinca V,Herisanu N.2008.Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer.International Communications in Heat and Mass Transfer,35:710-715.
Massoudi M.2001.Local non-similarity solutions for the flow of a non-Newtonian fluid over a wedge.International Journal of Non-Linear Mechanics,36:961-976.
Mastroberardino A.2011.Homotopy analysis method applied to electrohydrodynamic flow.Communation in Nonlinear Science and Numerical Simulation,16:2730-2736.
Meiron D I,Saffman P G,Yuen H C.1982.Calculation of steady three-dimensional deep-water waves.Journal of Fluid Mechanics,124:109-121.
Nassar C J,Revelli J F,Bowman R J.2011.Application of the homotopy analysis method to the PoissonBoltzmann equation for semiconductor devices.Communation in Nonlinear Science and Numerical Simulation,16:2501-2512.
Nicholls D P,Reitich F.2006.Stable,high-order computation of traveling water waves in three dimensions.European Journal of Mechanics B-Fluids,25:406-424.
Niu Z,Wang C.2010.A one-step optimal homotopy analysis method for nonlinear differential equations.Communications in Nonlinear Science and Numerical Simulation,15:2026-2036.
Niu Z,Liu Z,Cui J F.2016.A Steady-state Trio for Bretherton Equation.Zeitschrift f¨ur Naturforschung A,71:1099-1104.
Peng S G.1991.A nonlinear Feynman-Kac formula and applications//Proceedings of the Symposium on System Sciences and Control Theory.Control Theory,Stochastic Analysis and Applications,173-184.
Phillips O M.1960.On the dynamics of unsteady gravity waves of finite amplitude.Journal of Fluid Mechanics,9:193-217.
Poincar′e H.1892.Les m′ethodes nouvelles de la m′ecanique céleste.Gauthier-Villars.
Sajid M.2009.Some Flow Problems in Differential Type Fluids:Series solutions using homotopy analysis method.VDM Verlag.
Sardany′es J,Rodrigues C,Janu′ario C,Martins N,Gil-G′omez G,Duarte J.2015.Activation of effector immune cells promotes tumor stochastic extinction:A homotopy analysis approach.Applied Mathematics and Computation,252:484-495.
Sen S.1983.Topology and Geometry for Physicists.Academic Press.
Sparrow E M,Quack H,Boerner C J.1970.Local non-similarity boundary-layer solutions.AIAA Journal,8:1936-1942.
Sparrow E M,Yu H S.1971.Local non-similarity thermal boundary-layer solutions.Journal of Heat Transfer,93:328-334.
Sun J L,Cui J F,He Z H,Liu Z.2017.On steady-state multiple resonances for a modified Bretherton equation.Zeitschrift f¨ur Naturforschung A,72:487-491.
Thouless D J.1958.Application of perturbation methods to the theory of nuclear matter.Physical Review,112:906-922.
Tobisch E.2016.New Approaches to Nonlinear Waves.Springer.
Williams J M.1981.Limiting gravity waves in water of finite depth.Philosophical Transactions of the Royal Society A Mathematical Physical&Engineering Sciences,302:139-188.
Vahdati S.2012.Computational Methods for Integral Equations:Linear Legendre Multi-Wavelets and Homotopy Analysis Methods.LAP LAMBERT Academic Publishing.
Vajravelu K,Van Gorder R A.2012.Nonlinear Flow Phenomena and Homotopy Analysis:Fluid Flow and Heat Transfer.Springer.
Van Gorder R A,Vajravelu K.2008.Analytic and numerical solutions to the Lane-Emden equation.Physics Letters A,372:6060-6065.
Van Gorder R A.2012a.Analytical method for the construction of solutions to the F¨oppl-von K′arm′an equations governing deflections of a thin flat plate.International Journal of Non-Linear Mechanics,47:1-6.
Van Gorder R A.2012b.Gaussian waves in the Fitzhugh-Nagumo equation demonstrate one role of the auxiliary function H(x,t)in the homotopy analysis method.Communications in Nonlinear Science and Numerical Simulation,17:1233-1240.
Van Gorder R A.2012c.Control of error in the homotopy analysis of semi-linear elliptic boundary value problems.Numerical Algorithms,61:613-629.
Van Gorder R A.2015.Relation between Lane-Emden solutions and radial solutions to the elliptic heavenly equation on a disk.New Astronomy,37:42-47.
Vincent J J.1931.The bending of a thin circular plate.The London,Edinburgh,and Dublin Philosophical Magazine and Journal of Science,12:185-196.
Wanous K J,Sparrow E M.1965.Heat transfer for flow longitudinal to a cylinder with surface mass transfer.Journal of Heat Transfer,87:317-319.
Xu D L,Lin Z L,Liao S J,Stiassnie M.2012.On the steady-state fully resonant progressive waves in water of finite depth.Journal of Fluid Mechanics,710:379-418.
Xu D L,Lin Z L,Liao S J.2015.Equilibrium states of class-I Bragg resonant wave system.European Journal of Mechanics B/Fluids,50:38-51.
Yabushita K,Yamashita M,Tsuboi K.2007.An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method.Journal of Physics A:Mathematical and Theoretical,40:8403-8416.
Yang Z C,Liao S L.2017a.A HAM-based wavelet approach for nonlinear ordinary differential equations.Communication in Nonlinear Science and Numerical Simulation,48:439-453.
Yang Z C,Liao S L.2017b.A HAM-based wavelet approach for nonlinear partial differential equations:Two dimensional Bratu problem as an application.Communication in Nonlinear Science and Numerical Simulation,53:249-262.
Yang Z J,Liao S L.2017c.On the generalized Wavelet-Galerkin method.Journal of Computational and Applied Mathematics,https://doi.org/10.1016/j.cam.2017.09.042
Yeh K Y,Liu R H,Li S L,Qing Q Y.1965.Nonlinear stabilities of thin circular shallow shells under actions of axisymmetrical uniformly distributed line loads.Journal of Lanzhou University(Natural Science).18:10-33.
Young T.1805.An essay on the cohesion of fluids.Philosophical Transactions of the Royal Society of London,95:65-87.
Zhang J E,Li T C.2006.Pricing and hedging American options analytically:A perturbation method.Working paper,University of Hong Kong.
Zhao Y L,Liao S J.2013.HAM-Based Mathematica Package BVPh 2.0 for Nonlinear Boundary Value Problems.Liao S ed.Advances in the Homotopy Analysis Method,World Scientific Press,Chapter 9:361-416.
Zheng X J,Zhou Y H.1988.On the convergence of the nonlinear equations of circular plate with interpolation iterative method.Chinese Science A,10:1050-1058.
Zheng X J.1990.Large Deflection Theory of Circular Thin Plate and its Application.Jilin Science Technology Press.
Zhong X X,Liao S L.2017a.Analytic solutions of Von K′arm′an plate under arbitrary uniform pressureequations in differential form.Studies in Applied Mathematics,138:371-400.
Zhong X X,Liao S L.2017b.On the homotopy analysis method for backward/forward-backward stochastic differential equations.Numerical Algorithms,76:487-519.
Zhong X X,Liao S L.2018a.Analytic approximations of Von K′arm′an plate under arbitrary uniform pressure-equations in integral form.Science China Physics,Mechanics&Astronomy,61:014611.
Zhong X X,Liao S L.2018b.On the limiting Stokes wave of extreme height in arbitrary water depth.Journal of Fluid Mechanics,843:653-679.
Zhu S P.2006.An exact and explicit solution for the valuation of American put options.Quantitative Finance,6:229-242.
Zou L,Wang Z,Zong Z.2014.Analytical Techniques and Solitary Water Waves.LAP LAMBERT Academic Publishing.
刘曾.2015.稳态共振波及非线性波流相互作用研究.[博士论文].上海:上海交通大学(Liu Z.2015.On the study of steady-state resonant waves and wave-current interaction.[PhD Thesis].Shanghai:Shanghai Jiao Tong University).
王记增.2001.正交小波统一理论与方法及其在压电智能结构等力学研究中的应用.[博士论文].兰州:兰州大学.
徐妲莉.2014.同伦分析方法在稳态共振波浪研究中的应用.[博士论文].上海:上海交通大学(Xu D L.2014.Application of homotopy analysis method in steady-state resonant waves.[PhD Thesis].Shanghai:Shanghai Jiao Tong University).
杨兆臣.2017.求解非线性边值问题的小波同伦分析方法及其应用.[硕士论文].上海:上海交通大学(Yang Z C.2017.The wavelet homotopy analysis method for nonlinear boundary value problems and its applicationis.[Master Thesis].Shanghai:Shanghai JiaoTong University).
张丽,王光谦,傅旭东,孙其诚.2009a.低浓度颗粒流Boltzmann方程的同伦分析方法解.科学通报,54:1518-1523(Zhang L,Wang G Q,Fu X D,Sun Q C.2009a.A new solution to Boltzmann equation of dilute granular flow with homotopy analysis method.Chinese Science Bull,54:1518-1523).
张丽,王光谦,傅旭东,孙其诚.2009b.低浓度固液两相流Boltzmann方程的同伦分析方法解.应用基础与工程科学学报,17:811-818(Zhang L,Wang G Q,Fu X D,Sun Q C.2009b.A new solution to Boltzmann equation of dilute solid-liquid two-phase flows with homotopy analysis method.Journal of Basic Science and Engineering,17:811-818).
钟晓旭.2018.应用同伦分析方法求解若干力学和金融学问题.[硕士论文].上海:上海交通大学(Zhong X U.2018.Homotopy analysis method for several problems in mechanics and finance.[Master Thesis].Shanghai:Shanghai Jiao Tong University).
Adomian G.1976.Nonlinear stochastic differential equations.Journal of Mathematical Analysis and Applications,55:441-452.
Adomian G.1994.Solving Frontier Problems of Physics:The Decomposition Method.Kluwer Academic Publishers.
Alobaidi G,Mallier R.2001.On the optimal exercise boundary for an American put option.Journal of Applied Mathematics,1:39-45.
Alomari A K.2012.Modifications of Homotopy Analysis Method for Differential Equations:Modification of Homotopy Analysis Method,Ordinary,Fractional,Delay and Algebraic Differential Equations.LAPLAMBERT Academic Publishing.
Akyildiz F T,Vajravelu,K.2008.Magnetohydrodynamic flow of a viscoelastic fluid.Physics Letters A,372:3380-3384.
Armstrong M A.1983.Basic Topology(Undergraduate Texts in Mathematics).Springer.
Barles G,Burdeau J,Romano M,Samsoen N.1995.Critical stock price near expiration.Mathematical Finance,5:77-95.
Bataineh A S,Noorani M S M,Hashim I.2007.Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method.Physics Letters A,371:72-82.
Bataineh A S,Noorani M S M,Hashim I.2009.Homotopy analysis method for singular IVPs of EmdenFowler type.Communications in Nonlinear Science and Numerical Simulation,14:1121-1131.
Baxter M,Dewasurendra M,Vajravelu K.2017.A method of directly defining the inverse mapping for solutions of coupled systems of nonlinear differential equations.Numerical Algorithm.https://doi.org/10.1007/s11075-017-0359-0
Benney D J.1962.Non-linear gravity wave interactions.Journal of Fluid Mechanics,14:577-584.
Bretherton F P.1964.Resonant interactions between waves.The case of discrete oscillations.Journal of Fluid Mechanics,20:457-479.
Bunch D S,Johnson H.2000.The American put option and its critical stock price.Journal of Finance,5:2333-2356.
Chen S L,Kuang J C.1981.The perturbation parameter in the problem of large deflection of clamped circular plates.Applied Mathematics and Mechanics(English Edition),2:137-154.
Cheng J,Zhu S P,Liao S J.2010.An explicit series approximation to the optimal exercise boundary of American put options.Communications in Nonlinear Science and Numerical Simulation,15:1148-1158.
Chien W Z.1947.Large deflection of a circular clamped plate under uniform pressure.Chinese Journal of Physics,7:102-113.
Cimpean D,Merkin J H,Ingham D B.2006.On a free convection problem over a vertical flat surface in a porous medium.Transport Porous,64:393-411.
Craig W,Nicholls D P.2002 Traveling gravity water waves in two and three dimensions.European Journal of Mechanics B-Fluids,21:615-641.
Evans J D,Kuske R,Keller J B.2002.American options on asserts with dividends near expiry.Mathematical Finance,12:219-237.
Fu Y,Zhao W D,Tao Z.2017.Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs.Discrete&Continuous Dynamical Systems-B.22:3439-3458.
Gauss C F.1830.Principia Generalia Theoriae Figurae Fluidorum in Statu Aequilibrii.
Ghoreishi M,Ismail A I B M,Alomari A K,Bataineh A S.2012.The comparison between Homotopy Analysis Method and Optimal Homotopy Asymptotic Method for nonlinear age-structured population models.Communications in Nonlinear Science and Numerical Simulation,17:1163-1177.
Hedges T S.1995.Regions of validity of analytical wave theories.Ice Proceedings Water Maritime&Energy,112:111-114.
Hunter J K,Vandenbroeck J M.1983.Accurate computations for steep solitary waves.Journal of Fluid Mechanics,136:63-71.
Itik M,Banks S P.2010.Chaos in a three-dimensional cancer model.International Journal of Bifurcation and Chaos,20:71-79.
Karmishin A V,Zhukov A T,Kolosov V G.1990.Methods of dynamics calculation and testing for thinwalled structures(in Russian)//Mashinostroyenie,Moscow.
Laplace P S M D.1805.Trait′e de m′ecanique c′eleste.Suppl′ement au dixieme livre du Trait′e de M′ecanique C′eleste,1-79.
Liang S X,Jeffrey D J.2010.Approximate solutions to a parameterized sixth order boundary value problem.Computers and Mathematics with Applications,59:247-253.
Liao S J.1997.A kind of approximate solution technique which does not depend upon small parameters(II)-An application in fluid mechanics.International Journal of Non-Linear Mechanics,32:815-822.
Liao S J.1999.An explicit,totally analytic approximation of Blasius viscous flow problems.International Journal of Non-Linear Mechanics,34:759-778.
Liao S J.2003a.Beyond Perturbation:Introduction to the Homotopy Analysis Method.Boca Raton:Chapman&Hall/CRC Press.
Liao S J.2003b.A new analytic algorithm of Lane-Emden type equations.Applied Mathematics and Computation,142:1-16.
Liao S J,Pop I.2004.Explicit analytic solution for similarity boundary layer equations.International Journal of Heat and Mass Transfer,47:75-85.
Liao S J.2009.A general approach to get series solution of non-similarity boundary-layer flows.Communications in Nonlinear Science and Numerical Simulation,14:2144-2159.
Liao,S J.2010a.On the relationship between the homotopy analysis method and Euler transform.Communications in Nonlinear Science and Numerical Simulation,15:1421-1431.
Liao,S J.2010b.An optimal homotopy-analysis approach for strongly nonlinear differential equations.Communications in Nonlinear Science and Numerical Simulation,15:2003-2016.
Liao S J.2011.On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves.Communication in Nonlinear Science and Numerical Simulation,16:1274-1303.
Liao S J.2012.Homotopy Analysis Method in Nonlinear Differential Equations.Springer.
Liao S J,Xu D L,Stiassnie Michael.2016.On the steady-state nearly resonant waves.Journal of Fluid Mechanics,794:175-199.
Liao S J,Zhao Y L.2016.On the method of directly defining inverse mapping for nonlinear differential equations.Numerical Algorithm,72:989-1020.
Liao S J.2018.A new non-perturbative approach in quantum mechanics for time-independent Schr¨odinger equations.arXiv:1806.05103.
Liu Y B,Chen Y S.2011.KBM method based on the homotopy analysis.Science China Physics Mechanics&Astronomy,54:1137-1140.
Liu Z,Liao S J.2014.Steady-state resonance of multiple wave interactions in deep water.Journal of Fluid Mechanics,742:664-700.
Liu Z,Xu D L,Li J,Peng T,Alsaedi A,Liao SJ.2015.On the existence of steady-state resonant waves in experiment.Journal of Fluid Mechanics,763:1-23.
Liu Z,Xu D L,Liao S J.2018.Finite amplitude steady-state wave groups with multiple near resonances in deep water.Journal of Fluid Mechanics,835:624-653.
Liu Z,Xu D L,Liao S J.2017.Mass,momentum and energy flux conservation between linear and nonlinear steady-state wave groups.Physics of Fluids,29:127104.
Lyapunov A M.1992.General Problem on Stability of Motion(English translation).Taylor&Francis,London.
Karoui N E,Peng S G,Quenez M C.1997.Backward stochastic differential equations in finance.Mathematical Finance.7:1-71.
Khalid U.2011.Steady Flow in a Williamson Fluid:Basic Concepts of Fluid Mechanics Solution to a Non-linear Ordinary Differential Equation by Homotopy Analysis Method.LAP LAMBERT Academic Publishing.
Keller H B,Reiss E L.1958.Iterative solutions for the non-linear bending of circular plates.Communications on Pure and Applied Mathematics,11:273-292.
Knessl C.2001.A note on a moving boundary problem arising in the American put option.Studies in Applied Mathematics,107:157-183.
Kuske R A,Keller J B.1998.Optional exercise boundary for an American put option.Applied Mathematical Finance,5:107-116.
Madsen P A,Fuhrman D R.2012.Third-order theory for multi-directional irregular waves.Journal of Fluid Mechanics,698:304-334.
Marinca V,Herisanu N.2008.Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer.International Communications in Heat and Mass Transfer,35:710-715.
Massoudi M.2001.Local non-similarity solutions for the flow of a non-Newtonian fluid over a wedge.International Journal of Non-Linear Mechanics,36:961-976.
Mastroberardino A.2011.Homotopy analysis method applied to electrohydrodynamic flow.Communation in Nonlinear Science and Numerical Simulation,16:2730-2736.
Meiron D I,Saffman P G,Yuen H C.1982.Calculation of steady three-dimensional deep-water waves.Journal of Fluid Mechanics,124:109-121.
Nassar C J,Revelli J F,Bowman R J.2011.Application of the homotopy analysis method to the PoissonBoltzmann equation for semiconductor devices.Communation in Nonlinear Science and Numerical Simulation,16:2501-2512.
Nicholls D P,Reitich F.2006.Stable,high-order computation of traveling water waves in three dimensions.European Journal of Mechanics B-Fluids,25:406-424.
Niu Z,Wang C.2010.A one-step optimal homotopy analysis method for nonlinear differential equations.Communications in Nonlinear Science and Numerical Simulation,15:2026-2036.
Niu Z,Liu Z,Cui J F.2016.A Steady-state Trio for Bretherton Equation.Zeitschrift f¨ur Naturforschung A,71:1099-1104.
Peng S G.1991.A nonlinear Feynman-Kac formula and applications//Proceedings of the Symposium on System Sciences and Control Theory.Control Theory,Stochastic Analysis and Applications,173-184.
Phillips O M.1960.On the dynamics of unsteady gravity waves of finite amplitude.Journal of Fluid Mechanics,9:193-217.
Poincar′e H.1892.Les m′ethodes nouvelles de la m′ecanique céleste.Gauthier-Villars.
Sajid M.2009.Some Flow Problems in Differential Type Fluids:Series solutions using homotopy analysis method.VDM Verlag.
Sardany′es J,Rodrigues C,Janu′ario C,Martins N,Gil-G′omez G,Duarte J.2015.Activation of effector immune cells promotes tumor stochastic extinction:A homotopy analysis approach.Applied Mathematics and Computation,252:484-495.
Sen S.1983.Topology and Geometry for Physicists.Academic Press.
Sparrow E M,Quack H,Boerner C J.1970.Local non-similarity boundary-layer solutions.AIAA Journal,8:1936-1942.
Sparrow E M,Yu H S.1971.Local non-similarity thermal boundary-layer solutions.Journal of Heat Transfer,93:328-334.
Sun J L,Cui J F,He Z H,Liu Z.2017.On steady-state multiple resonances for a modified Bretherton equation.Zeitschrift f¨ur Naturforschung A,72:487-491.
Thouless D J.1958.Application of perturbation methods to the theory of nuclear matter.Physical Review,112:906-922.
Tobisch E.2016.New Approaches to Nonlinear Waves.Springer.
Williams J M.1981.Limiting gravity waves in water of finite depth.Philosophical Transactions of the Royal Society A Mathematical Physical&Engineering Sciences,302:139-188.
Vahdati S.2012.Computational Methods for Integral Equations:Linear Legendre Multi-Wavelets and Homotopy Analysis Methods.LAP LAMBERT Academic Publishing.
Vajravelu K,Van Gorder R A.2012.Nonlinear Flow Phenomena and Homotopy Analysis:Fluid Flow and Heat Transfer.Springer.
Van Gorder R A,Vajravelu K.2008.Analytic and numerical solutions to the Lane-Emden equation.Physics Letters A,372:6060-6065.
Van Gorder R A.2012a.Analytical method for the construction of solutions to the F¨oppl-von K′arm′an equations governing deflections of a thin flat plate.International Journal of Non-Linear Mechanics,47:1-6.
Van Gorder R A.2012b.Gaussian waves in the Fitzhugh-Nagumo equation demonstrate one role of the auxiliary function H(x,t)in the homotopy analysis method.Communications in Nonlinear Science and Numerical Simulation,17:1233-1240.
Van Gorder R A.2012c.Control of error in the homotopy analysis of semi-linear elliptic boundary value problems.Numerical Algorithms,61:613-629.
Van Gorder R A.2015.Relation between Lane-Emden solutions and radial solutions to the elliptic heavenly equation on a disk.New Astronomy,37:42-47.
Vincent J J.1931.The bending of a thin circular plate.The London,Edinburgh,and Dublin Philosophical Magazine and Journal of Science,12:185-196.
Wanous K J,Sparrow E M.1965.Heat transfer for flow longitudinal to a cylinder with surface mass transfer.Journal of Heat Transfer,87:317-319.
Xu D L,Lin Z L,Liao S J,Stiassnie M.2012.On the steady-state fully resonant progressive waves in water of finite depth.Journal of Fluid Mechanics,710:379-418.
Xu D L,Lin Z L,Liao S J.2015.Equilibrium states of class-I Bragg resonant wave system.European Journal of Mechanics B/Fluids,50:38-51.
Yabushita K,Yamashita M,Tsuboi K.2007.An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method.Journal of Physics A:Mathematical and Theoretical,40:8403-8416.
Yang Z C,Liao S L.2017a.A HAM-based wavelet approach for nonlinear ordinary differential equations.Communication in Nonlinear Science and Numerical Simulation,48:439-453.
Yang Z C,Liao S L.2017b.A HAM-based wavelet approach for nonlinear partial differential equations:Two dimensional Bratu problem as an application.Communication in Nonlinear Science and Numerical Simulation,53:249-262.
Yang Z J,Liao S L.2017c.On the generalized Wavelet-Galerkin method.Journal of Computational and Applied Mathematics,https://doi.org/10.1016/j.cam.2017.09.042
Yeh K Y,Liu R H,Li S L,Qing Q Y.1965.Nonlinear stabilities of thin circular shallow shells under actions of axisymmetrical uniformly distributed line loads.Journal of Lanzhou University(Natural Science).18:10-33.
Young T.1805.An essay on the cohesion of fluids.Philosophical Transactions of the Royal Society of London,95:65-87.
Zhang J E,Li T C.2006.Pricing and hedging American options analytically:A perturbation method.Working paper,University of Hong Kong.
Zhao Y L,Liao S J.2013.HAM-Based Mathematica Package BVPh 2.0 for Nonlinear Boundary Value Problems.Liao S ed.Advances in the Homotopy Analysis Method,World Scientific Press,Chapter 9:361-416.
Zheng X J,Zhou Y H.1988.On the convergence of the nonlinear equations of circular plate with interpolation iterative method.Chinese Science A,10:1050-1058.
Zheng X J.1990.Large Deflection Theory of Circular Thin Plate and its Application.Jilin Science Technology Press.
Zhong X X,Liao S L.2017a.Analytic solutions of Von K′arm′an plate under arbitrary uniform pressureequations in differential form.Studies in Applied Mathematics,138:371-400.
Zhong X X,Liao S L.2017b.On the homotopy analysis method for backward/forward-backward stochastic differential equations.Numerical Algorithms,76:487-519.
Zhong X X,Liao S L.2018a.Analytic approximations of Von K′arm′an plate under arbitrary uniform pressure-equations in integral form.Science China Physics,Mechanics&Astronomy,61:014611.
Zhong X X,Liao S L.2018b.On the limiting Stokes wave of extreme height in arbitrary water depth.Journal of Fluid Mechanics,843:653-679.
Zhu S P.2006.An exact and explicit solution for the valuation of American put options.Quantitative Finance,6:229-242.
Zou L,Wang Z,Zong Z.2014.Analytical Techniques and Solitary Water Waves.LAP LAMBERT Academic Publishing.