同伦分析方法进展综述
廖世俊
, 刘曾
A brief review of the homotopy analysis method
LIAO Shijun, LIU Zeng
中图分类号: O34
文献标识码: A
通讯作者:
收稿日期: 2018-03-26
接受日期: 2018-06-27
网络出版日期: 2019-01-15
版权声明: 2019 This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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作者简介:
廖世俊, 上海交通大学“春申”讲席教授, 博士生导师,现任职于上海交通大学船舶海洋与建筑工程学院,上海交通大学物理和天文学院,海洋工程国家重点实验室副主任(2001年—), 教育部长江奖励计划特聘教授(2001年), 国家杰出青年基金获得者(2001年).曾获“上海市第七届自然科学牡丹奖”(2009),“上海市自然科学一等奖”(2009 年, 唯一完成人),“国家自然科学二等奖”(2016年, 唯一完成人), “上海市科技精英”(2017年).廖世俊教授原创性地提出求解强非线性问题的“同伦分析方法” (Homotopyanalysis method, HAM), 撰写2本相关英文专著, 编辑一本英文专著,是“同伦分析方法”的创始人.廖世俊教授提出求解混沌动力系统的高精度数值方法(clean numericalsimulation, CNS),为非线性混沌动力系统提供了一个高精度的、全新的研究工具并与他人合作, 应用CNS和超级计算机,成功获得著名的三体问题两千多个全新的周期解. 迄今共发表150 余篇 SCI论文. 其博士论文、专著和杂志论文共被 SCI检索他引七千余次(H-index为49), 其中18 篇为ESI 高被引用论文,一篇论文入选“2009年中国百篇最具影响国际学术论文”,一篇论文入选“2010年中国百篇最具影响国际学术论文”.连续三年（2014—2016）入选全球高被引用科学家名单(highly-cited researchers).
刘曾, 华中科技大学船舶与海洋工程学院讲师, 硕士生导师.2008—2015年,上海交通大学船舶海洋与建筑工程学院攻读硕士和博士学位,师从廖世俊教授, 其间于2014年访学MIT海洋工程系一年.2015年获上海交通大学工学博士学位.研究方向为非线性海浪动力学、船舶与海洋工程水动力学、同伦分析方法及其在非线性微分方程中的应用.共发表10余篇SCI论文, 其中以第一作者在Journal of Fluid Mechanics发表3篇论文, 在Physics of Fluids上发表1篇论文
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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.
关键词：
Abstract
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.
Keywords：
1 引 言
针对强非线性力学问题,如何构造不依赖物理小参数、具有良好收敛性的解析近似方法长期困扰学术界.廖世俊 (Liao 1992) 将同伦概念引入微分方程解析近似求解,创立了同伦分析方法, 提出了一个求解强非线性力学问题的新途径. 不同于摄动方法, 同伦分析方法不依赖于小(或大)物理参数. 此外,同伦分析方法提供了选取高阶近似级数解表达形式的自由.特别值得强调的是,同伦分析方法还提供了一个简单的途径控制级数解的收敛性.这些优点使得同伦分析方法在解析方法中脱颖而出, 在过去十多年间,研究者们利用同伦分析方法成功求解了许多复杂的强非线性问题,甚至发现了一些全新的解, 这些新解甚至被传统的解析近似方法和数值方法所遗漏. 这些成功的应用,显示了同伦分析方法的原创性和潜力.
本文简要介绍近十年来同伦分析方法理论上的主要进展及其在非线性力学、量子力学、应用数学、金融等科学和工程领域的一些典型应用.
2 同伦分析方法简介
2.1 基本思想简介
同伦分析方法借鉴了拓扑(Armstrong 1983)和微分几何(Sen 1983)中同伦的概念. 考虑非线性微分方程
$$\mathcal{N}[u(x)]=0 (1) $$
其中, $\mathcal{N}$为非线性算子,$u(x)$为未知函数, $x$为自变量.令$q\to[0,1]$为嵌入变量（无物理意义）,选择一个合适的辅助线性算子$\mathcal{L}$, 构造零阶变形方程(Liao 2003a)
$$(1-q)\mathcal{L}[\phi(x;q)-u_0(x)]=c_0q\mathcal{N}[\phi(x;q)] (2)$$
其中$u_0(x)$为初始猜测解,$c_0$为收敛控制参数. 当嵌入变量$q$从$0$增加到$1$时,零阶变形方程(2) 的解$\phi(x;q)$从初始猜测解$u_0(x)$连续变化到原始非线性微分方程之解$u(x)$.这种连续的变化在拓扑理论(Armstrong 1983) 中称为变形(deformation).
将$\phi(x;q)$在$q=0$处进行关于嵌入变量$q$的泰勒级数展开, 即
$$\phi(x;q)=u_0(x)+\sum^{+\infty}_{k=1}u_k(x)q^k (3)$$
若辅助线性算子$\mathcal{L}$,初始猜测解$u_0(x)$和收敛控制参数$c_0$均选取合适, 使得泰勒级数(3)在$q=1$ 处收敛, 即可得到级数解
$$u(x)=\phi(x;1)=u_0(x)+\sum^{+\infty}_{k=1}u_k(x) (4)$$
将泰勒级数(3)代入零阶变形方程(2),合并$q$各次幂的同类项并令相同次幂的系数为零, 可得到高阶变形方程
$$\mathcal{L}[u_k(x)-\chi_ku_{k-1}(x)]=c_0\mathcal{D}_{k-1}{\mathcal{N}[\phi(x;q)]} (5)$$
其中, 当$k\leq 1$时, $\chi_k=0$; 当$k>1$时,$\chi_k=1$,$\mathcal{D}_{k}=\left.\dfrac{1}{k!}\dfrac{\partial^k}{\partial q^k}\right|_{q=0}$ (Liao 2012). 依次求解各阶变形方程,可得到$m$阶级数解
$$U_m(x)=u_0(x)+\sum^{m}_{k=1}u_k(x),\quad m\geq 1 (6)$$
相对于传统的摄动理论和非摄动理论, 如Lyapunov人工小参数法(Lyapunov 1992, Thouless 1958), Adomian分解法(Adomian 1976, 1994)和$\delta $ 展开法(Karmishin et al. 1990)等,上述求解方法的优势在于: (1) 不依赖小参数,无论非线性问题是否含有物理小(或大)参数, 均可获得高阶级数解; (2)确保收敛性, 可通过收敛控制参数确保级数解的收敛; (3)可自由选择辅助线性算子和初始近似,因此可采用更好的基函数来逼近精确解,也方便采用迭代技术加速级数解的收敛.
2.2 最新理论进展
发展至今, 同伦分析方法已拥有一套成熟的理论体系.该方法具有强大生命力和吸引力的一个重要原因在于, 无论其原创者,还是其继承者和使用者都始终站在发展的角度,持续不断地在理论上对其进行改进与完善,并不断地将同伦分析方法应用于不同类型的非线性问题,特别是那些具有强非线性、其他传统的解析近似方法不能很好求解的问题.下面将分别从收敛控制,算子映射、基函数表达和级数收敛性四个方面介绍同伦分析方法的一些最近理论进展.
2.2.1 最优同伦分析方法
在零阶变形方程(2)中引入收敛控制参数$c_0$,在本质上就是在级数解中提供一个额外的自由度,以控制解的收敛区间并调节解的收敛速度. 为进一步引入额外的自由度,廖世俊 (Liao 1999)定义了如下更为广义的零阶变形方程
$$[1-\alpha(q)]\mathcal{L}[\phi(x;q)-u_0(x)]=c_0\beta(q)\mathcal{N}[\phi(x;q)] (7)$$
其中$\alpha(q)$和$\beta(q)$定义为变形函数, 满足
$$\alpha(0)=\beta(0)=0,\quad \alpha(1)=\beta(1)=1 (8)$$
且泰勒级数
$$\alpha(q)=\sum^{+\infty}_{k=1}\alpha_kq^k,\quad \beta(q)=\sum^{+\infty}_{k=1}\beta_kq^k (9)$$
至少在$|q|\leq 1$时收敛.
为确定收敛控制参数$c_0$的取值, 廖世俊 (Liao 1997, 1999,2003a)建议分析待求函数$u(x)$在某些特殊点处的值, 如$u'(0)$和$u"(0)$ 等随$c_0$的变化情况, 并将这类曲线统称为$c_0$ 曲线.大量算例表明, 通过$c_0$曲线可获得收敛的级数解,但却无法确定最佳的收敛参数, 从而得到收敛最快的级数解. Yabushita等(2007) 在利用同伦分析方法求解常微分方程组时,提出可通过控制方程残差平方的极小值来确定最佳收敛控制参数. 随后,Akyildiz 和 Vajravelu(2008) 发现通过控制方程残差平方极小值确定的收敛控制参数,可获得更快速收敛的级数解.理论上更多的收敛控制参数意味着更大的求解自由度和更快的级数解收敛速度.但牛照和汪淳(Niu & Wang 2010)发现更多的收敛控制参数也带来了更多的未知参数,在确定最优控制参数时, 所耗CPU 时间随未知参数个数快速增长.该问题对复杂非线性方程的高阶级数解尤为突出. 为提高计算效率, 廖世俊(Liao 2010b) 提出了包含3个收敛控制参数的最优同伦分析方法.以Blasius方程为例, 廖世俊 (Liao 2010b)定义了基于控制方程残差平方的平均残差余量以高效地确定收敛控制参数,他发现最优同伦分析方法可显著加快级数解的收敛速度,并强烈推荐使用单参数或双参数收敛控制参数(图1). 下面对最优同伦分析方法进行简要介绍.
图1 最优同伦近似解误差$|f"(0)-0.332057|$随CPU计算时间变化. 实线:单参数最优同伦分析方法; 虚线: Marinca 和
在广义零阶变形方程(7)中,定义变形函数$\alpha(q)$和$\beta(q)$依赖于$\kappa$个未知的收敛控制参数
$$\pmb{c}=\{c_1,c_2,\cdots,c_\kappa\} (10)$$
定义平均残差余量
$$E_m=\dfrac{1}{N+1}\sum^{N}_{j=0}\left\{\mathcal{N}\left[\sum^{m}_{k=0}u_k(x_j)\right]\right\}^2 (11)$$
其中$N$为自变量$x$在定义域上的离散点个数.假设$m$阶同伦近似包含$\kappa'+1$个未知的收敛控制参数,其中$\kappa'\leq \kappa$. 那么,$\kappa'+1$个未知收敛控制参数的值可通过如下方式确定:
$$\dfrac{E_m}{c_j}=0,\quad 0\leq j\leq \kappa'\leq \kappa (12)$$
对任意给定的收敛级数
$$\varPi =\sum^{\kappa}_{j=1}c_j\neq 0 (13)$$
可定义
$$\beta(q;\pmb{c})=\dfrac{1}{\varPi }\sum^{\kappa}_{j=1}c_jq^j (14)$$
当$\alpha(q)=q$且
$$\beta(q)=\dfrac{1}{c_0}\sum^{\kappa}_{j=1}c_jq^j,\quad c_0=\varPi=\sum^{\kappa}_{j=1}c_j\neq0 (15)$$
时, 有零阶变形方程
$$[1-q]\mathcal{L}[\phi(x;q)-u_0(x)]=\sum^{\kappa}_{j=1}c_jq^j\mathcal{N}[\phi(x;q)] (16)$$
和高阶变形方程
$$\mathcal{L}[u_m(x)-\chi_mu_{m-1}(x)]=c_0\sum^{\min(m,\kappa)}_{j=1}\beta_j\mathcal{D}_{m-j}\left\{\mathcal{N}\left[\sum^{+\infty}_{n=0}u_{n}(x)q^n\right]\right\} (17)$$
相应的最优同伦近似级数解包含$\min(m,\kappa)$个收敛控制参数.当$\kappa=1$时, 上述推导对应单参数最优同伦分析方法; 当$\kappa=2,3$时, 对应廖世俊(2010b, 2012)提出的双参数或三参数最优同伦分析方法;当$\kappa=+\infty$ 时, 对应Marince和 Herisanu(2008)提出的最优同伦渐进方法于最优同伦分析方法的算例, 可参见 廖世俊(Liao 2010b), Van Gorder(2012a, 2012b, 2012c)和 Ghoreishi等(2012)的工作.
2.2.2 直接定义逆映射法
在科学计算中,计算微分方程$\mathcal{A}\psi=f$的逆算子$\mathcal{A}^{-1}$非常耗时,特别当$\mathcal{A}$ 是一个强非线性算子. 基于同伦分析方法,廖世俊和赵银龙 (Liao & Zhao 2016)提出了直接定义逆映射法(methodof directly defining iverse mapping, MDDiM)以更高效地获得非线性微分方程的解析近似解.直接定义逆映射法通过直接定义逆映射$\mathcal{I}$来求解微分方程$\mathcal{A}\psi=f$,无需计算逆算子$\mathcal{A}^{-1}$. 由于映射比微分算子更为广义,因此直接定义的逆映射$\mathcal{I}$可以不用显性地表达成微分形式.下面通过3个简单算例简要描述直接逆映射法.
考虑非线性特征值问题
$$\mathcal{N}[u,\lambda]=u''(x)+\lambda u(x)+\epsilon u^3(x)=0 (18)$$
及边界条件$u(0)=u(1)=0$和归一化条件$\int^{1}_{0}u^2(x){\rm d}x=1$. 在直接定义逆映射法中, 各阶近似可表达为
$$u_m(x)=\chi_mu_{m-1}(x)+c_0\mathcal{I}[\delta _{m-1}(x)]+a_{m,1}\sin(\pi x) (19)$$
其中$\delta _{m}(x)=\mathcal{D}_m\left\{\mathcal{N}\left[\sum^{+\infty}_{k=0}u_k(x)q^k,\sum^{+\infty}_{k=0}\lambda_kq^k\right]\right\}$,$a_{m,1}$为待求系数. 选取初始解$u_0(x)=\sqrt{2}\sin(\pi x)$,廖世俊和赵银龙 (Liao & Zhao 2016) 发现通过直接定义逆映射
$$\mathcal{I}_1{\sin[(2m-1)\pi x]}=-\dfrac{\sin[(2m-1)\pi x]}{2(m-1)(2m+1+\alpha)\pi ^2} (20)$$
$$\mathcal{I}_2{\sin[m\pi x]}=\dfrac{\sin[m\pi x]}{(1-m)(\sqrt{m}+\beta)(\sqrt{m}+\gamma )\pi ^2} (21)$$
均可以得到收敛的级数解, 其中$\alpha\in(0,8)$, $\beta\in(0,4)$,$\gamma \in(0,4)$. 式(20)中$\alpha=1$时,拟映射$\mathcal{I}_1$对应的辅助算子$\mathcal{L}$可以显性地表达成微分形式.廖世俊和赵银龙 (Liao & Zhao 2016) 通过残差分析发现,当$\alpha=2.2$时, 逆映射$\mathcal{I}_1$给出的级数解收敛最快(图2). 并且对大部分参数组合, 式(20)和式(21)定义的逆映射均不存在对应微分形式的辅助算子.直接定义逆映射法的主要优势在于无需计算辅助算子,在节省计算资源的同时提供了更多获得收敛级数解的可能性.
此外, 不妨考虑经典的Blasius边界层流动控制方程
$$f{"'}(x)+\dfrac{1}{2}f(x)f'(x)=0,\quad f(0)=f'(0)=0,\quad f'(+\infty)=1 (22)$$
令$f(x)=F(z)+x,z=\lambda x$,其中$\lambda>0$为待定常数, 则得到方程
$$\mathcal{N}[F]=F'''+\dfrac{1}{2\lambda^2}(z+\lambda F)F"=0 (23)$$
对应边界条件
$$F(0)=0,\quad F'(0)=-\dfrac{1}{\lambda},\quad F'(+\infty)=0 (24)$$
应用直接定义逆映射法, 各阶近似可表达为
$$F_m(z)=\chi_mF_{m-1}(z)+c_0\mathcal{I}[\delta _{m-1}(z)]+a_{m,0}+\dfrac{a_{m,1}}{(1+z)} (25)$$
其中$\delta _{m}(z)=\mathcal{D}_m\left\{\mathcal{N}\left[\sum^{+\infty}_{k=0}F_k(z)q^k\right]\right\}$,$a_{m,0}$和$a_{m,1}$ 为待定参数, 由边界条件确定.当选取初始解$F_0(z)=\dfrac{1}{\lambda}\left(\dfrac{1}{1+z}-1\right)$时, 廖世俊和赵银龙(Liao & Zhao 2016) 发现通过直接定义逆映射
$$\mathcal{I}\left\{(1+z)^m\right\}=\dfrac{(1+z)^m}{m^3+A_2m^2+A_1m+A_0},\quad m\leq -2 (26)$$
当其中的参数取值为
$ \lambda=\dfrac{1}{3},\quad A_0=\dfrac{1}{3\pi },\quad A_1=\dfrac{\pi }{30},\quad A_2=\dfrac{\pi }{3},\quad c_0=-\dfrac{9}{5} (27)$
$\lambda=\dfrac{1}{3}, \hskip 0.3mm \quad A_0=0,\hskip 2.4mm \quad A_1=0,\hskip 2.5mm \quad A_2=\dfrac{\pi }{3},\quad c_0=-\dfrac{3}{2} (28)$
$ \lambda=\dfrac{1}{3},\quad A_0=\dfrac{1}{10},\quad A_1=\dfrac{\pi }{12},\quad A_2=\dfrac{\pi }{3},\quad c_0=-\dfrac{3}{2} (29)$
时均可得到相同的收敛级数解. 此算例表明在直接定义逆映射法框架下,我们具有较大的自由直接定义不同的逆映射, 但它们都可得到Blasius边界层流动完全相同的级数解.
再者, 考虑二维的Gelfand方程
$$\nabla^2u+\lambda e^u=0,\quad x\in[-1,1],\quad y\in[-1,1] (30)$$
其边界条件为
$$u(x,\pm 1)=f(x,\pm 1),\quad u(\pm 1,y)=f(\pm 1,y) (31)$$
令$u(0,0)=A$且$u=A+w$, 其中$A$为待定常数, 得到方程
$$\mathcal{N}[w,\lambda]=\nabla^2w+\lambda e^Ae^w=0,\quad x\in[-1,1],\quad y\in[-1,1] (32)$$
其对应边界条件为
$$w(x,\pm 1)=-A+f(x,\pm 1),\quad w(\pm 1,y)=-A+f(\pm 1,y),\quad w(0,0)=0 (33)$$
应用直接定义逆映射法, 各阶近似可表达为
$$w_m(x,y)=\chi_mw_{m-1}(x,y)+c_0\mathcal{I}[\delta _{m-1}(x,y)]+w^*_m,\quad m\geq 1 (34)$$
其中$\delta _{m}=\mathcal{D}_m\left\{\mathcal{N}\left[\sum^{+\infty}_{k=0}u_kq^k,\sum^{+\infty}_{k=0}\lambda_kq^k\right]\right\}$,$w^*_m$为由边界条件
$$w_m-\chi_mw_{m-1}=c_1\left\{w_{m-1}+(1-\chi_m)[A-f(x,y)]\right\},\quad \mbox{at} \ x=\pm 1,\quad y=\pm 1 (35)$$
确定的基本解,$c_1$为第二个收敛控制参数. 当选取初始解$w_0(x,y)=0$ 时,廖世俊和赵银龙 (Liao & Zhao 2016) 发现通过直接定义逆映射
$$\mathcal{I}\left[x^my^n\right]=\dfrac{x^{m+2}y^{n+2}}{m^2+B_1m+B_0)(n^2+B_1n+B_0)},\quad m\geq 0,\quad n\geq 0 (36)$$
可以求得
$$f(x,y)=0 (37)$$
$$f(x,y)=\pm\dfrac{(1+x^2)(1+y^2)}{10} (38)$$
$$f(x,y)=\pm\dfrac{x^2-x^2y^2+y^2}{2} (39)$$
$$f(x,y)=\cos(x)+\cos(y) (40)$$
$$f(x,y)=\cos[\sin(x)]-\exp(y^2) (41)$$
时收敛的级数解,其中$B_0$, $B_1$, $c_0$, $c_1$的取值随$f(x,y)$相应变化(图3).值得注意的是, 当$B_0=\pi /2$ 且$B_1=\pi $ 时,辅助算子$\mathcal{L}=\mathcal{I}^{-1}$不存在微分表达形式.该算例表明, 直接定义逆映射法由于不需考虑微分算子的存在性,因而提供了更为广义且高效的途径来求解非线性微分方程. 值得强调的是,映射是比微分算子更为广义的概念,因此有理由认为基于映射概念的直接定义逆映射法包含基于微分算子的传统同伦分析方法.关于直接定义逆映射法的更多算例, 可参考Baxter等(2017)的文献.
图3 直接逆映射法(36)求得Gelfand方程(30)特征值$\lambda$随$A$ 变化. 实线: $f(x,y)=\cos x+\cos y$; 虚线:$f(x,y)=\cos[\sin(x)]-\exp(y^2)$. $B_0=\pi /2$, $B_1=\pi $,$c_0=1$和$c_1=-1$
2.2.3 小波--同伦分析方法
在传统的同伦分析方法中, 高阶变形方程右端项的计算效率较低,且解表达因受辅助线性算子影响,相应基函数被限制为常见的初等函数及其组合. 为突破上述限制,杨兆臣和廖世俊 (Yang & Liao 2017a, 2017b, 2017c)基于同伦分析方法和现代小波理论, 采用小波作为同伦分析方法的基函数,提出了一种求解非线性边值问题的新方法, 即小波--同伦分析方法.作为例子, 不妨考虑一维 Bratu 方程
$$\dfrac{{\rm d}^2u(x)}{{\rm d}x^2}+\lambda\exp[u(x)]=0,\quad x\in(0,1) (42)$$
边界条件$u(0)=u(1)=0$,分析$\lambda<3.513830719$时方程的下分支解.在引入变换$V(x)=\exp\left[-\dfrac{1}{2}u(x)\right]$ 后,原方程可表达为
$$\mathcal{N}[V(x)]=V(x)\dfrac{{\rm d}^2V(x)}{{\rm d}x^2}-\left[\dfrac{{\rm d}V(x)}{{\rm d}x}\right]^2-\dfrac{\lambda}{2}=0,\quad x\in (0,1) (43)$$
其对应边界条件$V(0)=V(1)=1$. 关于$m$阶近似$V_m(x)$的高阶变形方程为
$$\mathcal{L}[V_m(x)-\chi_mV_{m-1}(x)]=c_0\mathcal{D}_{m-1}\left\{\mathcal{N}\left[\sum_{k=1}^{+\infty}V_k(x)q^k\right]\right\},\quad m\geq 1 (44)$$
与传统同伦分析方法中的解表达原则、解存在原则和系数遍历原则(Liao 2003a)对应, 杨兆臣和廖世俊 (Yang & Liao 2017a)在小波--同伦分析方法中提出了边界适应原则、解存在原则和阶数匹配原则,以指导初始猜测解和辅助线性算子的选取. 根据上述原则,选择最简单的初始猜测解$V_0(x)=1$, 和辅助线性算子
$$\mathcal{L}[\phi(x;q)]=\dfrac{\partial^2\phi(x;q)}{\partial x^2}+\kappa_1 \dfrac{\partial\phi(x;q)}{\partial x}+\kappa_0 \phi(x;q) (45)$$
其中, $\kappa_0$ 和$\kappa_1$ 为可选参数.与传统同伦分析方法不同, 在小波同伦分析方法中,解表达在多分辨分析框架下由小波基给出,解表达形式不受辅助线性算子影响. 采用多分辨分析与小波逼近对$m$阶变形方程两端进行展开, 得
$$\sum^{2^j-1}_{k=1}\left[V_m\left(\dfrac{k}{2^j}\right)-\chi_mV_{m-1}\left(\dfrac{k}{2^j}\right)\right]\mathcal{L}[\psi_{j,k}(x)]=c_0\sum^{2^j}_{k=0}R_m\left(\dfrac{k}{2^j}\right)\psi_{j,k}(x) (46)$$
其中$\psi_{j,k}(x)$为广义Coiflet小波(王记增 2001)尺度函数
$$R_{m}\left(\dfrac{k}{2^j}\right)=\sum^{m-1}_{s=0}\left[V_s\left(\dfrac{k}{2^j}\right)V"_{m-1-s}\left(\dfrac{k}{2^j}\right)-V'_s\left(\dfrac{k}{2^j}\right)V'_{m-1-s}\left(\dfrac{k}{2^j}\right)\right]-\dfrac{\lambda}{2}(1-\chi_m) (47)$$
在式(47)两边分别同时乘以$\varphi_{j,l}$ $\left(l=1,2,\cdots,2^j-1\right)$并在$ [0,1]$ 区间上进行积分, 得$m$阶小波伽辽金形变方程
$$\pmb A ^{\rm T} \left(\hat{\pmb V} _m-\chi_m \hat{\pmb V} _{m-1}\right)=c_0 \; \pmb B ^{\rm T} \; \hat{\pmb R} _m, (48)$$
其中$(\cdot)^{\rm T}$为转置算子
$$\hat{\pmb V} _k=\left\{V_k\left(\dfrac{1}{2^j}\right), V_k\left(\dfrac{2}{2^j}\right), \cdots, V_k\left(\dfrac{2^j-1}{2^j}\right)\right\}^{\rm T} (49)$$
为 $(2^j-1)$维列向量
$$\hat{\pmb R} _m=\left\{R_m\left(\dfrac{0}{2^j}\right), R_m\left(\dfrac{1}{2^j}\right),R_m\left(\dfrac{2}{2^j}\right), \cdots, R_m\left(\dfrac{2^j}{2^j}\right)\right\}^{\rm T} (50)$$
为$(2^j+1)$维列向量
$$\pmb A =\left\{a_{k,l}=\int _0^1\mathcal{L}\left[\varphi_{j,k}(x)\right] \varphi_{j,l}(x){\rm d} x \right\}_{k,\ l=1}^{k,\ l=2^j-1} (51)$$
和
$$\pmb B = \left\{b_{k,l}=\int _0^1\varphi_{j,k}(x) \varphi_{j,l}(x){\rm d} x \right\}_{k=0,\ l=1}^{k=2^j,\ l=2^j-1} (52)$$
分别为 $(2^j-1) \times (2^j-1)$和$(2^j+1) \times (2^j-1)$矩阵.注意到矩阵$\pmb A$和$\pmb B$不随阶数$m$ 的变化而变化,因此小波--同伦分析方法可高效地求解$m$阶小波伽辽金变形方程(48).
当参数$\lambda=1$时,选取辅助线性算子$\mathcal{L}[\phi(x;q)]=\dfrac{\partial^2\phi(x;q)}{\partial x^2}$ , 图4 给出了不同分辨率水平下, 均方误差Err$SQ_m$随阶数$m$ 的变化情况. 在不同分辨率水平下, 均方误差Err$SQ_m$随着阶数的增加而快速减小, 且当分辨率增大时均方误差进一步减小.这表明小波同伦分析方法在非线性微分方程 求解中的有效性. 此外,通过调节分辨率水平,可以方便地平衡通过小波同伦分析方法获得高阶级数解时的效率与精度问题.
图4 Frank-Kamenetskii参数$\lambda=1$, 控制收敛参数$c_0 =-1/2$, 辅助线性算子 $\mathcal{L}[\phi(x;q)]=\dfrac{\partial^2\phi(x;q)}{\partial x^2}$ 时, 不同分辨率水平$j$ 下一维Bratu 方程小波同伦逼近的均方误差随阶数$m$的变化
传统的同伦分析方法具有自由选择辅助线性算子的优势,但基函数与解表达的选取限制了辅助线性算子的选取.小波同伦分析方法采用小波基函数进行小波逼近,消除了辅助线性算子对解表达的限制,提高了解表达对不同辅助线性算子的适应性. 当辅助线性算子(45)中$\kappa_1=0$, $\kappa_0=[-5,5]$或$\kappa_0=5$,$\kappa_1=[0,5]$时, 相应的均方误差均随着阶数$m$ 的增加而快速减小.与传统的同伦分析方法相比,小波同伦分析方法对不同的辅助线性算子具有更强的适应性,主要体现为小波形式的解表达对不同辅助线性算子的普遍适应性、不同辅助线性算子下小波同伦解的快速收敛性以及不同辅助线性算子下运算的高效性三大方面,这一方面使辅助线性算子的选取变得更加简单,另一方面也使小波同伦分析方法在强非线性问题的求解上具备了更大的潜力.
此外, 以二维Bratu方程为例, 杨兆臣和廖世俊 (Yang & Liao 2017b)针对偏微分方程高阶变形方程右端项中偏导数的计算问题,提出了基于截面曲线的快速算法.该算法采用截面曲线替代空间曲面进行小波逼近,在不影响级数解收敛性的前提下显著地减少了高阶变形方程右端项的项数、提高了高阶变形方程的求解速度(图5). 针对非齐次边界条件, 杨兆臣 (Yang 2017) 在王记增等的基础上,充分利用广义Coiflet 小波之特性,在小波逼近框架下提出了处理非齐次边界条件的一般方法,进一步完善了小波同伦分析方法的理论体系.
图5 Frank-Kamenetskii参数$\lambda=1$, 分辨率水平$j=5$, 控制收敛参数$c_0 = -0.4$时, 不同阶数($m$) 小波同伦解对应的截面曲线$(u(x,1/2))$
针对变系数非线性微分方程, 杨兆臣和廖世俊 (Yang & Liao 2017c)提出了广义小波伽辽金方法.该方法对做等价变换后的变系数函数和待求函数整体进行小波逼近,以避免复杂连接系数的计算,极大地拓展了小波同伦分析方法和传统小波伽辽金方法的适用范围. 例如,考虑区间$[0,1]$上变系数常微分方程
$$\sum_{n=0}^N \dfrac{{\rm d} ^{n}\left(b_{n}(x)u(x)\right)}{{\rm d} x^n}=r(x), \quad 0<x<1 (53)$$
其中, $r(x)$ 和$b_{n}(x)$ 为已知函数,方程满足若干边界条件以使得解的存在唯一性得到保证.与传统小波伽辽金方法不同,广义小波伽辽金方法并不直接对未知函数$u(x)$ 作小波展开,而是将函数$b_n(x) u(x)$ 作为整体进行展开.小波基函数会根据不同边界条件而进行修正. 对变系数偏微分方程
$$\sqrt{x^2+y^2+1}\left(\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial x^2}\right) +\dfrac{1}{ \sqrt{x^2+y^2+1}}\left(x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y}-2u\right)=0 (54)$$
及边界条件
$$\left. \begin{array}{l} u(0,y)=\sqrt{1+y^2},\quad u(1,y)=\sqrt{2+y^2} \\ u(x,0)=\sqrt{1+x^2},\quad u(x,1)=\sqrt{2+x^2} \end{array} \right\} (55)$$
杨兆臣和廖世俊 (Yang & Liao 2017c)通过广义小波伽辽金方法成功求得基于小波逼近的级数解(图6). 更多算例可参见杨兆臣和廖世俊 (Yang & Liao 2017c)和杨兆臣(Yang 2017)的文献.
2.2.4 同伦变换
对级数$\sum^{+\infty}_{k=0}u_k$,定义同伦变换$\mathcal{T}\left(c_0,\pmb{\alpha},\pmb{\beta}\right)$,得到级数$\{\mu_m\}$
$$\mu_m=u_0+\sum^{m}_{k=1}u_kT_{m,k}(c_0,\pmb{\alpha},\pmb{\beta}) (56)$$
其中
$$T_{m,k}(c_0,\pmb{\alpha},\pmb{\beta})=(-c_0)^k\sum^{m-k}_{n=0}\sum^{n}_{r=0}{k+r-1 \choose r}(1+c_0)^r\sum^{n-r}_{s=0}\alpha_{k,k+s}\beta_{r,n-s} (57)$$
$$\alpha_{m,k}=\sum^{k-1}_{i=m-1}\alpha_{m-1,i}\alpha_{1,k-i}\quad (m\geq 2,k\geq m) (58)$$
$$\beta_{0,0}=1,\quad \beta_{0,k}=0\;(k\geq 1),\quad \;\beta_{m,k}=\sum^{k-1}_{i=m-1}\beta_{m-1,i}\beta_{1,k-i}\;\;(m\geq2,k\geq m) (59)$$
$A(q)$和$B(q)$为满足$A(0)=B(0)=0$,$A(1)=B(1)=1$的变形函数,其麦克劳林级数$\sum^{+\infty}_{k=1}\alpha_{1,k}p^k$和$\sum^{+\infty}_{k=1}\beta_{1,k}q^k$在区间$|q|\leq 1$内绝对收敛. 廖世俊 (Liao 2010a)发现当$c_0=-q,\alpha_{1,1}=\beta_{1,1}=1$且$\alpha_{1,k}=\beta_{1,k}=0$ $(k>1 )$ 时,对级数$\sum^{+\infty}_{k=0}u_k$进行同伦变换$\mathcal{T}\left(c_0,\pmb{\alpha},\pmb{\beta}\right)$等价于对有限序列$s_m=\sum^{m}_{k=0}u_k$进行欧拉变换$\mathcal{E}(q)$. 欧拉变换可被视为同伦变换的一个特列.进一步, 廖世俊(Liao 2010a) 发现: 由式(56)定义的同伦变换可通过同伦分析方法在某些特定初始解和辅助线性算子组合下获得.因此, 在特定参数下, 欧拉变化等价于同伦分析方法. 众所周知,欧拉变换可加速级数收敛或使发散的级数收敛,这一方面解释了为什么在同伦分析方法框架内级数解的收敛性可得到保证,另一方面也表明相比欧拉变化,同伦分析方法提供了一种更为广义的途径来加速级数收敛或使发散的级数收敛.
3 应用举例
下面将通过一些具体算例说明同伦分析方法在过去十多年的应用中所体现出来的普适性、优越性和独特性.
3.1 同伦分析方法的普适性
3.1.1 大挠度圆薄板方程
求解均布外载荷作用下的大挠度圆薄板方程是固体力学中的经典问题.首先考虑微分形式大挠度圆薄板方程
$${\cal N}_1[\varphi(y),S(y)] = y^{2}\dfrac{{\rm d}^{2}\varphi(y)}{{\rm d}y^{2}}-\varphi(y)S(y)-Qy^{2}=0 (60)$$
$${\cal N}_2[\varphi(y),S(y)] = y^{2}\dfrac{{\rm d}^{2}S(y)}{{\rm d}y^{2}}+\dfrac{1}{2}\varphi^{2}(y)=0 (61)$$
对应边界条件
$$\varphi(0)=S(0)=0 (62)$$
$$\varphi(1)=\dfrac{\lambda}{\lambda-1}\cdot\dfrac{{\rm d}\varphi(y)}{{\rm d}y}\bigg{|}_{y=1},\quad \;\;\;S(1)= \dfrac{\mu}{\mu-1}\cdot\dfrac{{\rm d}S(y)}{{\rm d}y}\bigg{|}_{y=1} (63)$$
和无量纲的中心挠度
$$W(y)=-\int _{y}^{1}\dfrac{1}{\varepsilon}\varphi(\varepsilon){\rm d}\varepsilon (64)$$
对固定夹紧边界条件$\lambda=0$, $\mu=20/7$, 钟晓旭和廖世俊 (Zhong & Liao 2017a)发现 $25$ 阶同伦级数解可在$a=w(0)/h = 3.03$ 处收敛,而摄动方法的有效域仅为$a < 2.44$ (Chen & Kuang 1981). 进一步,在同伦分析方法框架内采用迭代技术以加快收敛, 钟晓旭和廖世俊 (Zhong & Liao 2017a) 得到了在更大变形区域$a > 20$ 的收敛级数解(图7), 这表明同伦分析方法具有求解强非线性问题的能力.对其他三类边界条件(可移夹紧, 简单支承, 简单铰链支承),上述求解策略依然有效. 此外, 钟晓旭和廖世俊 (Zhong & Liao 2017a)还证明: 对微分形式大挠度圆薄板方程(60)和(61), 摄动方法(Vincent 1931, Chien 1947) 和修正迭代法(Yeh et al.1965)均为同伦分析方法的特例.此工作表明了同伦分析方法求解微分形式大挠度圆薄板方程的有效性和优越性.
此外, 考虑积分形式大挠度圆薄板方程
$${\cal N}_1[\varphi(y),S(y)] =\varphi(y)+\int_{0}^{1}\dfrac{1}{\varepsilon^{2}}K(y,\varepsilon)S(\varepsilon)\varphi(\varepsilon){\rm d}\varepsilon+\int_{0}^{1}K(y,\varepsilon) Q{\rm d}\varepsilon=0 (65)$$
$${\cal N}_2[\varphi(y),S(y)] =S(y) -\dfrac{1}{2}\int_{0}^{1}\dfrac{1}{\varepsilon^{2}}G(y,\varepsilon)\varphi^2(\varepsilon){\rm d}\varepsilon =0 (66)$$
钟晓旭和廖世俊(Zhong & Liao 2018a)发现当外载荷$Q$或中心挠度$\int_{0}^{1}\dfrac{1}{\varepsilon}\tilde{\varPhi}(\varepsilon,q) {\rm d}\varepsilon=-a$ 给定时,通过基于同伦分析方法的迭代可获得比内插迭代法(Keller & Reiss 1958,Zheng 1990) 收敛更快的级数解. 此外, 对给定外载荷$Q$,钟晓旭和廖世俊 (Zhong & Liao 2018a)证明了内插迭代法(Keller &Reiss 1958) 是一阶同伦迭代法的特例.内插迭代法已被证明对任意大均布外载荷均能给出收敛的结果(Zheng &Zhou 1988), 这从理论上间接证明了该同伦 分析方法的收敛性. 因此,同伦分析方法不仅可以获得任意大变形的圆形板收敛的级数解,而且求解大挠度圆薄板方程的三类传统方法(摄动方法(Vincent 1931,Chien 1947), 修正迭代法(Yeh et al. 1965)和内插迭代法(Keller &Reiss 1958))均可被统一到同伦分析方法框架中.
3.1.2 倒向型/正倒向型随机微分方程
倒向型随机微分方程广泛存在于科学与工程领域, 如随机控制,股票市场和化学反应等. 在期货投资领域,正向随机微分方程的解将今天的确定状态（初始条件）变为未来的不确定的状态以研究其统计规律,而倒向型随机微分方程的解则是将未来的（也可是不确定的）目标变为当今的明确的解从而来帮助制定当前的决策.对于高维问题, 数值方法通常会遇到所谓的"维度灾难".如何提高计算效率以快速求解高维正倒向型随机微分方程一直是研究者们十分关心的一个问题.
对非耦合正倒向型随机微分方程, 考虑经Feynman-Kac公式(Peng 1991,Karoui 1997)变换后的偏微分方程
$${\mathcal N}[u]=\dfrac{\partial u}{\partial t}+\dfrac{1}{2{\rm d}^2}\sum_{i=1}^{{\rm d}}\left({\rm e}^{-2x_i^2}\cdot\dfrac{\partial^2 u}{\partial x_i^2}\right)+\dfrac{u}{{\rm d}^2}-\dfrac{1}{{\rm d}}\sum_{i=1}^{{\rm d}}\left[x_i^2\sum_{\mbox{\tiny$\begin{array}{c}j=1\\j\neq i\end{array}$}}^{{\rm d}}\prod_{\mbox{\tiny$\begin{array}{c}k=1\\k\neq i\\k \neq j\end{array}$}}^{{\rm d}}(x_k+t)\right]-$$ $$\dfrac{1}{{\rm d}^3}\sum_{i=1}^{{\rm d}}\left[(x_i^2+{\rm e}^{-2x_i^2})\prod_{\mbox{\tiny$\begin{array}{c}k=1\\k\neq i\end{array}$}}^{{\rm d}}(x_k+t)\right]=0 (67)$$
和边界条件
$$u(t,\pmb x )\bigg{|}_{t=1}=\dfrac{1}{{\rm d}}\sum_{j=1}^{{\rm d}}\left[x_{j}^2 \prod_{\mbox{\tiny$\begin{array}{c}k=1\\k\neq j\end{array}$}}^{{\rm d}}(x_{k}+1)\right] (68)$$
其中,向量$\pmb x = \big{\{} x_1, x_2, \cdots, x_d \big{\}} $.
钟晓旭和廖世俊 (Zhong & Liao 2017b)发现对任意收敛控制参数$c_0\in[-1.1,-0.6]$, 同伦级数解均收敛.经测试, 在$d=6$ 时同伦分析方法耗时5s即可获得误差为$10^{-7}$量级的级数解. 在相同计算配置下达到同等精度,基于稀疏网格的谱方法则需耗时18481s (Fu et al. 2017),其计算效率比同伦分析方法低3000余倍(图8). 对于不同维度$d$,利用同伦分析方法仅需五阶近似即可获得高精度级数解,特别对高维情况$(d=12)$,同伦分析方法可快速得到精确结果(耗时3084s),这揭示了同伦分析方法求解高维正倒向型随机微分方程的巨大潜力. 此外,钟晓旭和廖世俊(Zhong & Liao 2017b) 利用相同的求解策略,在同伦分析框架下成功地求解了三种倒向型随机微分方程和正倒向型随机微分方程.研究发现, 在同伦分析方法框架内,倒向型随机微分方程与正倒向型随机微分方程的求解并没有本质区别.
图8 高维正倒向型随机微分方程,同伦分析方法和基于稀疏网格的谱方法(
3.1.3 非相似边界层流动
当相似解存在时,边界层流动控制方程可通过相似变换简化为相对易于求解的常微分方程.对非相似边界层流动,不存在恒等变换将非线性偏微分方程转换为非线性常微分方程, 因而,其求解更困难.数值方法在求解非相似边界层问题时需将无穷域问题替换为有界域问题,这种替换给数值结果常给数值结果带来不确定性.解析方法可求解无穷域上的偏微分方程, 但如摄动法(Cimpean et al.2006), 局部相似法(Sparrow & Yu 1971, Massoudi 2001)和局部非相似法(Wanous & Sparrow 1965, Sparrow et al. 1970,Sparrow & Yu 1971) 等, 均无法给出对所有物理变量均有效的近似解.
对拉升平板上的非相似边界层流动, 考虑控制方程
$$\mathcal{N}[f]=\dfrac{f}{\eta}+\sigma_1(\xi)f\dfrac{f}{\eta}+\sigma_2(\xi)\left(\dfrac{f}{\xi}\dfrac{f}{\eta}-\dfrac{f}{\eta}\dfrac{f}{\xi}{\eta}\right)=0 (69)$$
和边界条件
$$f(\xi,0)=0,\quad f_\eta(\xi,0)=U_w(\xi),\quad f_\eta(\xi,+\infty)=0 (70)$$
其中$\sigma_1(\xi)=\dfrac{1}{2}[\sigma^2(x)]'$,$\sigma_2(\xi)=\varGamma'(s)\sigma^2(x)$.
当$U_w=x/(1+x)$和$x/(1+x)-0.5x^2/(1+x)^2$时, 廖世俊（Liao 2009)发现运用同伦分析方法, 选取$\sigma(x)=\sqrt{1+x}$和$\xi=\varGamma(x)=x/(1+x)$ 可得到收敛的同伦级数解.和传统的解析方法不同,同伦分析方法给出的级数解在整个流动区间内均收敛.对应的非相似边界层流动在$x$ 趋于$0$ 和$+\infty$时与相似边界层流动重合(图9). 注意到, 在边界层流动中,沿边界层厚度方向速度梯度远大于沿边界层方向速度梯度, 因此,对非相似边界层流动, 廖世俊(Liao 2009) 选取了和相似边界层流动(Liao & Pop 2004))相同的辅助线性算子. 这意味着, 在同伦分析方法框架下,可以采取与相似边界层问题类似的求解策略来求解非相似边界层流动问题.基于自由选取辅助线性算子的优势, 同伦分析方法提供了足够的自由度,以将一个非线性偏微分方程转化为无穷多个线性的常微分方程.
图9 当$x/(1+x)-0.5x^2/(1+x)^2$时非相似边界层厚度$\delta /\sqrt{\nu}$.实线: 30 阶同伦近似解; 圆: 20阶同伦近似解; 虚线:$\delta (x)=2.28537\sqrt{\nu x}$; 点划线: $\delta (x)=\sqrt{\nu}$
3.1.4 边值问题软件包BVPh
鉴于同伦分析方法的广泛应用得益于符号推导软件,如Mathemtaica和Maple等, 廖世俊教授研究团队(Liao 2012, Zhao & Liao 2013)基于同伦分析方法开发了针对强非线性边值问题和特征值问题的Mathematica软件包BVPh. 该软件包旨在提供一个解析解工具,以方便地近似求解尽可能多的非线性边值问题.软件包求解功能包括（但不限于）搜寻强非线性边值问题多解,求解无穷域边值问题、包含奇性的边值问题和多点边值问题等.
软件包第一版BVPh1.0由廖世俊于2012年推出(Liao 2012),主要求解常微分方程. 软件包第二版BVPh2.0 由赵银龙和廖世俊于2013年联合推出(Zhao & Liao 2013),该版本可求解有限域或半无穷域上耦合的常微分方程组,更新了一些算法以提高软件包的计算效率并增加了用户友好度以方便使用.BVPh2.0的用户手册和软件包源代码均可在网站(http://numericaltank.sjtu.edu.cn/BVPh.htm)免费下载, 更多关于BVPh2.0的介绍, 可参考赵银龙和廖世俊(Zhao & Liao 2013). BVPh自推出后被国内外同行广泛访问和下载,对同伦分析方法在各学科领域的应用起到了推动作用.
至今, 同伦分析方法已被中、美、英等十几国的其他研究小组广泛应用,累计发表两千余篇英文论文(数据来自web of science)和 7本英文专著(Sajid 2009, Khalid 2011, Alomari 2012, Liao 2012,Vahdati 2012, Vajravelu & Van Gorder 2012, Zou et al. 2014),国内各研究领域课题组相继完成了几十 篇相关学位论文,这些研究工作不仅进一步丰富和发展了同伦分析方法,而且也体现了同伦分析方法在非线性问题求解上的普适性.
3.2 同伦分析方法的优越性
本节将通过同伦分析方法在金融,半导体, 生物和物理等领域的应用体现其在方程求解过程中的优越性.
3.2.1 美式期权问题 考虑无量纲的Black-Scholes方程
$$-\dfrac{\partial V}{\partial \tau}+S^2\dfrac{\partial^2V}{\partial S^2}+\gamma S\dfrac{\partial V}{\partial S}-\gamma V=0 (71)$$
初/边值条件为
$$V(S,0)=0,\quad \quad V(S,\tau)\to 0,\;\; \text{当}\;S\to+\infty (72)$$
$$\dfrac{\partial V(B(\tau),\tau)}{\partial S}=-1,\quad \quad V(B(\tau),\tau)=1-B(\tau) (73)$$
其中, $V(S,\tau)$为美式期权价值, $S$为标的物价格,$B$为最佳执行边界. 特别是, 边界条件(73)需在未知的边界$B(\tau)$上得到满足, 这给求解带来一定困难.
廖世俊 (Liao 2012)对待求函数$V(S,\tau)$和$B(\tau)$分别定义变形函数$\phi(S,\tau;q)$和$\varLambda(\tau;q)$,将未知函数$\phi(S,\tau;q)$在未知边界$S=\varLambda(\tau;q)$上关于$q$进行级数展开并引入Laplace变换,最终推导出最佳执行边界$B(\tau)$ 的$N$ 阶近似显性表达式
$$B(\tau)\approx\sum^{N}_{k=0}B_k(\tau)=\sum^{2M}_{k=0}b_k(\sqrt{\tau})^k (74)$$
通过算例对比分析, 廖世俊(Liao 2012)发现:基于摄动方法或其他渐进方法的传统解析方法, 如Barle等(1995), Kuske和 Keller(1998), Alobaidi 和 Mallier(2001), Evans等(2002),Zhang 和 Li(2006)和Knessl(2001)等, 仅能给出最佳执行边界$B(\tau)$有效期为数天或数周的级数解,而同伦近似解的有效期可达到数十年甚至半个世纪,其收敛区间为传统级数解的1000倍(图10). 廖世俊(Liao 2012)发现最佳执行边界$B(\tau)$的最大有效区间和多项式(74)中$\tau$的幂次$M$有关: 当$M$越大时, 多项式$B(\tau)$ 的有效区间也越大.例如, 取关于$B(\tau)$的10阶同伦近似解, 此时$M=48$, 廖世俊(Liao 2012) 发现此幂级数表达式在整个时间轴上有效.
图10 最佳执行边界$B(\tau)$. 虚线A:
同伦分析方法在美式期权问题上的应用历程如下: Zhu(2006)首次运用同伦分析方法研究美式期权问题(71)$\sim$(73),得到了包含双重积分的高阶级数解, 其有效期为数周或数月. 随后,成钧等(Cheng et al. 2010) 运用同伦分析方法得到了$B(\tau)$关于$\tau$ 的一个6 次多项式, 其与数值结果吻合良好, 有效期为数年.廖世俊(Liao 2012)进一步优化了成钧等(Cheng et al. 2010)的解法,获得了有效期为数十年甚至半个世纪的级数解.该研究显示了同伦分析方法在求解以美式期权为代表的金融问题上的优势.
3.2.2 Poisson-Boltzmann方程
描述半导体器件静电势分布的一维Poisson-Boltzmann方程为
$$\dfrac{{\rm d}^2\psi}{{\rm d}x^2}=\dfrac{-q_e}{\epsilon_s}\left[N_A\exp\left(\dfrac{-\psi}{kT}\right)-N_A-\dfrac{N^2_i}{N_A}\exp\left(\dfrac{\psi}{kT}\right)+\dfrac{N^2_i}{N_A}\right] (75)$$
对应边界条件
$$\left.\dfrac{{\rm d}\psi}{{\rm d}x}\right|_{x=0}=0,\quad \left.\psi\right|_{x=0}=\psi_0 (76)$$
通过引入变换$\psi=\psi_0+kT\ln v$, Christopher等(2011)得到如下方程
$$vv"(z+1)^2+vv'(z+1)-(v')^2(z+1)^2+\alpha v-\beta v^2-\gamma v^3=0 (77)$$
和边界条件
$$v'|_{z=0}=0,\quad v|_{z=0}=1 (78)$$
遵循同伦分析方法3个基本原则: 解表达原则、完备性原则和解存在原则,Christopher等(2011)选取混合型解表达$v(z)=\sum^{+\infty}_{m=0}\sum^{+\infty}_{n=0}c_{mn}z^m\exp(-nz)$和辅助函数$H(z)=\exp(-z)$,并运用最优同伦分析方法得到了收敛的解析级数解. Christopher等(2011)发现, 不同于其他传统解析方法,在同伦分析方法框架内求解策略可方便扩展至更一般化的非线性静电势分布问题.此外, 相比数值方法, 如Runge-Kutta 算法等,同伦级数解可直接提供任意点处的导数值, 给结果分析带来便利.
3.2.3 电磁流动问题
圆柱管道内电磁流动控制方程为
$$\dfrac{{\rm d}^2w}{{\rm d}r^2}+\dfrac{1}{r}\dfrac{{\rm d}w}{{\rm d}r}+H^2\left(1-\dfrac{w}{1-\alpha w}\right)=0 (79)$$
对应边界条件
$$w'(0)=0,\quad w(1)=0 (80)$$
式(79)中非线性部分包含分数项,对求解级数近似解构成了一个很大的挑战. Mastroberardino(2011)通过同伦分析方法成功获得方程(79)$\sim$(80)对所有物理参数均收敛的解析近似解.基于同伦分析方法的级数解与数值结果吻合良好(图11),而基于同伦摄动方法得到的级数解则发散,仅同伦分析方法能够获得对所有物理参数都收敛的解. Mastroberardino(2011)发现同伦分析方法可通过使用收敛控制参数调节和控制解的收敛,因而是获得强非线性问题解析解的最有效方法.这进一步说明了同伦分析方法相对于其他解析方法在求解强非线性微分方程时的优势.
3.2.4 癌细胞增长模型
Sardanyés等(2015)运用基于同伦分析方法的步进同伦分析方法成功求解了癌细胞增长非线性模型(Itik & Banks 2010)
$$\dfrac{x_1}{t}=x_1(1-x_1)-\alpha_{12}x_1x_2-\alpha_{13}x_1x_3 (81)$$
$$\dfrac{x_2}{t}=r_2x_2(1-x_2)-\alpha_{21}x_1x_2 (82)$$
$$\dfrac{x_3}{t}=\dfrac{r_3x_1x_3}{x_1+k_3}-\alpha_{31}x_1x_3-d_3x_3 (83)$$
其中$x_1$,$x_2$和$x_3$分别描述肿瘤细胞、健康细胞与免疫细胞各自存活量的百分比.在分析癌细胞增长模型的双周期解和混沌解后, Sardanyés等(2015)总结到收敛控制参数$c_0$提供了一个方便的途径控制级数的收敛,这是同伦分析方法与其他方法的本质区别.
3.2.5 量子力学模型
考虑量子力学中无量纲的关于一维非线性振动的薛定谔方程
$${\rm H}\psi_n(\xi)=\left[-\dfrac{1}{2}\dfrac{{\rm d}^2}{{\rm d}\xi^2}+\dfrac{1}{2}\xi^2+\beta\xi^4\right]\psi_n(\xi)=E_n\psi_n(\xi) (84)$$
其中, ${\rm H}$为哈密尔顿算子,$\psi_n(\xi)$和$E_n$分别表示未知的特征函数和特征值. 廖世俊(Liao2018)运用同伦分析方法求解方程(84),发现传统摄动方法求得的结果仅对非常小的扰动$(\beta <0.02)$才或许有效(图12),而同伦分析方法即使在大扰动时$(\beta=5)$依然能得到收敛的解(图13). 此算例表明, 相比量子力学中广泛采用的微扰方法,同伦分析方法可以获得量子力学中大扰动薛定谔方程的解,从而可以将收敛的理论解直接与实验结果比较, 有益于更严格地验证理论,或提高实验测量精度. 因此,同伦分析方法可望为量子力学薛定谔方程的求解提供一种可确保获得收敛级数解的全新途径,该方法即使对很大的扰动也有效.
3.3 新解之发现
任意一种全新的方法, 都应该给出一些从未见报道的、全新的解.本节通过平衡态精确共振水波、平衡态近似共振水波和极限水波3个算例展现同伦分析方法在一些挑战性问题上所获得的全新的解.
3.3.1 平衡态精确共振水波之发现
波浪共振与波能谱演化密切相关,共振波研究对海洋平台、波能发电等海洋工程都具有重要理论价值.Phillips(1960) 应用摄动方法提出波浪共振条件,发现当共振条件满足时共振波波幅将线性增加. Benney(1962)随后导出波能分布的周期解, 发现共振波能谱可周期性变化.但平衡态共振波系,即共振条件满足时波能谱不随时间变化的波系是否存在,半个世纪以来依然是一个未知问题. Madsen 和 Fuhrman(2012)发现摄动方法无法处理共振所导致的奇性问题,因此通过摄动方法无法求解平衡态共振波.传统解析方法在求解此问题时失效,阻碍了对波浪能谱演化机制的全面理解!
廖世俊 (Liao 2011)应用同伦分析方法成功求解了精确的非线性水波方程,首次获得深水中处于平衡态共振波系的理论解! 廖世俊(Liao 2011)发现平衡态共振波系有多解存在并分析了波能分布(图14).徐妲莉等(Xu et al. 2012)随后应用同伦分析方法将廖世俊(Liao 2011)的工作推广至有限水深, 并应用Zakharov 方程用数值方法验证,证实平衡态共振波系在有限水深中依然存在. 刘曾和廖世俊(Liu & Liao2014)进一步考虑包含多个波分量相互作用组成的四波共振波系和六波共振波系,应用同伦分析方法, 发现在深水中依然存在多波平衡态共振波系;研究发现, 当波浪场从三维转向二维或波系非线性增加时,近似共振波分量将包含越来越多的波能. 对周期底部边界, 徐妲莉等(Xu et al. 2015) 应用同伦分析方法研究第一类Bragg 共振,发现平衡态共振波存在,且当传播方向、水深、底部边界高度和波系非线性等参数变化时,波系能量分布存在分叉现象. 上述这些对平衡态精确共振水波的成功求解,有赖于同伦分析方法选取初始猜测解的自由.同伦分析方法不依赖物理小参数, 对初始解中各项的量级不作限制,因此可以在初始解中增加幅值与基波不同量级的共振项,正是此共振项系数的确定使得在同伦分析方法框架内可消除方程求解过程中由共振导致的奇性.
图14 共振波分量所占能量百分比$a_{2,-1}^2/\varPi$随无量纲频率$\sigma_1/\sqrt{gk_1}$变化.$k_2/k_1=0.8925$且$\sigma_1/\sqrt{gk_1}=\sigma_2/\sqrt{gk_2}$
此外, 刘曾等(Liu et al.2015)首次在海洋工程国家重点实验室的深水池通过实验验证了深水平衡态共振波系的存在性.对由多组短峰波所组成的平衡态共振波系, 刘曾等(Liu et al.2015)发现实测数据和理论预测吻合良好(图15),有力地支撑了平衡态共振波系的理论工作.更多关于平衡态精确共振水波的介绍, 可参考徐妲莉(2014)和刘曾(2015)的博士论文及廖世俊研究小组收录在Tobisch(2016)编辑的专著.平衡态精确共振水波之发现, 历经了同伦分析方法的原创性提出,到平衡态共振波系的理论发现,再到平衡态共振波系的物理实验验证全过程,它很好地体现了同伦分析方法的原创性,因为一个全新的方法终究能带来全新的结果.
3.3.2 平衡态近似共振水波之发现
当线性共振条件近似满足时,近似共振会导致非线性水波求解过程中出现小分母问题,而小分母问题是力学中著名的难题, 许多科学家, 如 Poincaré(1892), Meiron等(1982), Craig 和 Nicholls(2002), Nicholls 和Reitich(2006) 均发现很难从数学上处理小分母问题.
廖世俊等(Liao et al. 2016)应用同伦分析方法,首次成功获得了平衡态近似共振波系. 对近似共振
$$m_*\pmb{k}_1+n_*\pmb{k}_2=\pmb{k}_0,\quad m_*\omega_1+n_*\omega_2=\omega_0+{\rm d}\omega (85)$$
廖世俊等(Liao et al.2016)选取了一个不同于控制方程线性项的更为广义的辅助线性算子
$$\mathcal{L}_b[\varphi]=\omega_1^2\dfrac{\partial^2\varphi}{\partial\xi^2_1}+\mu_1\omega_1\omega_2\dfrac{\partial^2\varphi}{\partial\xi_1 \partial\xi_2}+\omega_2^2\dfrac{\partial^2\varphi}{\partial\xi^2_2}+g\dfrac{\partial\varphi}{\partial z} (86)$$
其中${\rm d}\omega$为频率偏移量
$$\mu_1=\dfrac{g|m_*\pmb{k}_1+n_*\pmb{k}_2|-(m_*^2\omega_1^2+n_*^2\omega_2^2)}{m_*n_*\omega_1\omega_2} (87)$$
辅助线性算子(86)可将平衡态近似共振波的小分母问题转化为平衡态精确共振波的零分母问题,进而通过在初始解中增加共振项来消除零分母.在辅助线性算子(86)中令$\mu_1=2$, 即得到廖世俊(Liao 2011)、徐妲莉等(Xu et al. 2012)以及刘曾和廖世俊 (Liu & Liao 2014)针对平衡态精确共振波系(式(85)中${\rm d}\omega=0)$所选取的辅助线性算子. 该策略本质上讲,就是利用同伦分析方法提供的选取辅助线性算子之自由,将平衡态近似共振问题转化为已解决的平衡态精确共振问题!
此外, 当非线性增加时, 刘曾和廖世俊 (Liu & Liao 2014)发现波能将分布在越来越多的波分量中, 对有限振幅平衡态共振波系,需考虑多波耦合的共振作用. 对多波近似共振
$$m_{*,\iota}\pmb{k}_1+n_{*,\iota}\pmb{k}_2=\pmb{k}_{0,\iota}, \quad m_{*,\iota}\omega_1+n_{*,\iota}\omega_2=\omega_{0,\iota}+{\rm d}\omega_{\iota}, \quad \iota=1,2,\cdots,l (88)$$
在同伦分析方法框架下, 刘曾等(Liu et al. 2018)通过选取辅助线性算子
$$\mathcal{L}_c[\varphi]=\omega_1^2\dfrac{\partial^2\varphi}{\partial\xi^2_1}+\mu_2(m,n)\omega_1\omega_2\dfrac{\partial^2\varphi}{\partial\xi_1\partial\xi_2}+\omega_2^2\dfrac{\partial^2\varphi}{\partial\xi^2_2}+g\dfrac{\partial\varphi}{\partial z} (89)$$
将所有由近似共振导致的小分母问题转化成精确共振对应的零分母问题,其中
$$\mu_2(m,n)=\left\{ \begin{array}{ll} \dfrac{g|m\pmb{k}_1+n\pmb{k}_2|-(m^2\omega_1^2+n^2\omega_2^2)}{mn\omega_1\omega_2}, & m=m_{*,\iota},\quad n=n_{*,\iota}\\ 2, & \text{else} \end{array}\right. (90)$$
基于辅助线性算子(89), 刘曾(Liu等 2018)在初始猜测解中同时考虑基波项, 精确共振项和近似共振项.对弱非线性波系, 发现波能可显著地在不同近似共振波分量间转移.随着非线性增加, 越来越多的波分量参与共振,平衡态共振波系解的个数增加,有效频段变宽且频谱图的非对称性越来越明显(图16).这表明平衡态共振波系存在的可能性随着非线性增加而增加.
基于刘曾等(Liu et al.2018)在同伦分析方法框架内提出的针对多波平衡态共振的求解策略,刘曾等(Liu et al. 2017)对比分析了质量,动量和能量通量守恒时同向传播的线性和非线性平衡态波系.发现非线性波系中的近似共振波分量显著地影响了波浪场分布:非线性自由液面波峰更陡、波谷更深且波节点更平坦,相应波峰和波谷处速度随非线性的增加而增加得更快(图17).该研究工作再一次强调了近似共振的重要性.
上述对平衡态近似共振波系的成功求解,有赖于同伦分析方法选取辅助线性算子的自由.针对近似共振条件选取有针对性的辅助线性算子,可有效消除（多个）近似共振导致的小分母问题.
在水波共振研究工作的早期, Bretherton(1964)针对振动问题提出了Bretherton方程以阐述共振发生时的数学本质.为回答平衡态共振波系在水波以外色散介质中的存在性问题, 牛照等(Niu et al. 2016)应用同伦分析方法考虑了Bretherton 方程的三波共振解.牛照等(Niu et al. 2016)发现平衡态共振波解存在,且当基波波数差或波系非线性增加时, 所有波分量的幅值随之同时增加.随后, 孙江龙等(Sun et al. 2017)应用同伦分析方法研究了改进的Bretherton 方程,发现平衡态共振波系依然存在且存在耦合的多波共振. 牛照等(Niu et al. 2016) 和孙江龙等(Sun et al.2017)的工作表明平衡态共振波系可存在于除水波以外的其他色散介质中.上述平衡态共振解的求解策略对其他色散介质依然适用.
3.3.3 极限斯托克斯波
Schwartz(1974)和Cokelet(19771)等学者发现, 在极浅水中,即使通过Padé近似和Shank非线性变换等外推技术,依然无法通过摄动理论获得极限波高水波的级数解. 特别是,很难通过传统解析方法或数值方法求得带有$120^\circ$波峰的自由液面.因此, 必须在不同水深下采取不同的波浪模型.求解任意水深的斯托克斯波主要有如下两个难点:(1)当水深很浅或波高很高时(甚至达到极限波高),波浪方程的非线性非常强, 高阶傅里叶系数很难收敛;(2)随着傅里叶系数项数的增加, 解析方法的计算效率会急剧下降.
钟晓旭和廖世俊 (Zhong & Liao 2018c)采用同伦分析方法成功克服了这两个难点.通过调节收敛控制参数$c_0$, 钟晓旭和廖世俊(Zhong & Liao 2018c)在无需借助任何近似技巧（如帕德近似和其他外推技巧）的情况下,成功获得任意水深下带有$120^\circ$波峰的极限斯托克斯波的收敛级数解. 钟晓旭和廖世俊 (Zhong & Liao 2018c)的工作表明, 在同伦分析方法框架内,斯托克斯波理论适用于任意水深下的波浪,包括深水和有限水深下的周期波,浅水下的椭圆余弦波和极浅水下的孤立波(图18).通过引入迭代技巧并选取较优的初始猜测解,该级数解在任意水深(包括孤立波存在的极浅水深)通过数百次迭代依然保持收敛.同伦分析方法提供了一个有效且高效的方式来求解任意水深下的极限斯托克斯波.这个例子表明, 同伦分析方法可以用来攻克一些经典的力学难题.
图18 极限斯托克斯波$H/d\sim \lambda/d$.-·-斯托克斯波理论和椭圆余弦波理论分界线, 即Ursell 数 $H\lambda^2/d^3=40$ (
4 总结与展望
本文简述了近年来同伦分析方法的理论进展和一些典型应用.首先从方法论角度对同伦分析方法进行了简要介绍, 从收敛控制,算子映射、基函数表达和级数收敛性四个方面介绍了同伦分析方法的最新理论进展.接着从方法的普适性,优越性和原创性3个层面介绍了同伦分析方法的一些应用,表明同伦分析方法不仅使用范围广, 通常优于传统解析方法,而且针对一些挑战性问题, 甚至可以获得从未见报道的新解.
同伦分析方法在理论和应用上的发展历程表明其具有如下3个显著优点:首先, 不同于摄动方法, 同伦分析方法不依赖物理小(或大)参数,适用范围广; 其次,同伦分析方法提供了选取初始猜测解和辅助线性算子的自由,并提供了一个简单的途径调节和控制级数解的收敛性,因而适用于强非线性问题. 此外,同伦分析方法在理论上包含其他传统非摄动方法、任意阶牛顿迭代和欧拉变换,体现了同伦分析方法的兼容性.
经过二十多年来理论上的不断发展和完善以及其广泛的成功应用,同伦分析方法的普适性和优越性已逐步得到越来越多学者的认可.同伦分析方法不仅被成功应用于大变形 Von Kármán板、任意水深中的极限波浪、美式期权、倒向型随机微分方程、量子力学中具有大扰动的薛定谔方程等具有挑战性的力学问题,而且成功获得一些全新的解（如首次理论上预测了定常共振波浪,且该理论预测已得到实验证实）.这些成功的应用显示了同伦分析方法的原创性和潜力. 我们坚信,同伦分析方法将被愈来愈多地应用于求解非线性力学、量子力学、应用数学和金融等科学和工程领域中一些具有挑战性的难题.
致谢
The authors have declared that no competing interests exist.
参考文献
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稳态共振波及非线性波流相互作用研究 . [博士论文].URL 摘要
非线性波波共振作用和非线性波流相互作用是导致波浪场能量谱发生变化的重要因素,相关问题广泛存在于工程实际问题中,已有的大量研究工作主要从初边值问题角度研究这两种因素对波浪场能量谱随时间演化的影响。本论文尝试从边值问题角度出发寻求对应的稳态解现象,以理论分析为主,辅以试验验证的方式分析了稳态共振波的存在性并揭示了其能量分布特性,此外提出了非线性波流相互作用模型并考虑了涡量对频率(相速度)的影响。本文主要研究了如下六个问题,分别描述如下:1.从已有的特殊四波共振开始,研究了一般的四波共振,多个波分量耦合的四波共振及由高阶波波作用形成的六波共振。在以上三种情况中,我们发现稳态共振均存在且有多解,此外当参加共振的波分量个数增加时稳态共振波多解的个数有增加的趋势。2.以特殊的四波共振和六波共振为例,研究了当参数变化时稳态共振波的能量分布情况。确认了能量分布在参数空间内的连续变化,当波形从二维趋向一维或当波系中各波分量幅值增加(总能量增加)时,近似共振分量将包含越来越多的能量。3.通过线性稳定性分析发现当扰动不与任何未扰动波分量发生共振作用时,稳态共振波是稳定的。考虑到小振幅的初始扰动需要足够长的时间发展才有可能影响到波系能量分布,因此至少理论上讲,稳态共振波可能是亚稳定的,这为实验观测提供了可能性。4.为消除了稳态共振波必须满足Phillips线性共振条件的限制,在同伦分析方法框架中提出了分段的线性算子。通过此分段线性算子,得到了仅满足非线性共振条件的稳态共振波,扩展了稳态共振波的存在空间,且发现当波系中波分量幅值增加(总能量增加)时稳态共振波的存在区域随之扩大。5.在物理实验误差允许范围内,通过对照组实验的定量对比讨论了稳态共振波的存在性。观察到了由主波分量间的相位差而产生的对称组和非对称组波形。此外,定性对比上实测数据和理论结果吻合良好。6.考虑了无来流时双色波中基波相速度受非线性波波作用的影响,给出了考虑波波作用时相速度的高阶近似。分析了有旋来流对基波相速度的影响,发现来流(平均涡量)对非线性波波作用的影响由来流特征斜率决定,而来流表面速度大小则增强或减弱此效应。
On the study of steady-state resonant waves and wave-current interaction . [PhD Thesis].URL 摘要
非线性波波共振作用和非线性波流相互作用是导致波浪场能量谱发生变化的重要因素,相关问题广泛存在于工程实际问题中,已有的大量研究工作主要从初边值问题角度研究这两种因素对波浪场能量谱随时间演化的影响。本论文尝试从边值问题角度出发寻求对应的稳态解现象,以理论分析为主,辅以试验验证的方式分析了稳态共振波的存在性并揭示了其能量分布特性,此外提出了非线性波流相互作用模型并考虑了涡量对频率(相速度)的影响。本文主要研究了如下六个问题,分别描述如下:1.从已有的特殊四波共振开始,研究了一般的四波共振,多个波分量耦合的四波共振及由高阶波波作用形成的六波共振。在以上三种情况中,我们发现稳态共振均存在且有多解,此外当参加共振的波分量个数增加时稳态共振波多解的个数有增加的趋势。2.以特殊的四波共振和六波共振为例,研究了当参数变化时稳态共振波的能量分布情况。确认了能量分布在参数空间内的连续变化,当波形从二维趋向一维或当波系中各波分量幅值增加(总能量增加)时,近似共振分量将包含越来越多的能量。3.通过线性稳定性分析发现当扰动不与任何未扰动波分量发生共振作用时,稳态共振波是稳定的。考虑到小振幅的初始扰动需要足够长的时间发展才有可能影响到波系能量分布,因此至少理论上讲,稳态共振波可能是亚稳定的,这为实验观测提供了可能性。4.为消除了稳态共振波必须满足Phillips线性共振条件的限制,在同伦分析方法框架中提出了分段的线性算子。通过此分段线性算子,得到了仅满足非线性共振条件的稳态共振波,扩展了稳态共振波的存在空间,且发现当波系中波分量幅值增加(总能量增加)时稳态共振波的存在区域随之扩大。5.在物理实验误差允许范围内,通过对照组实验的定量对比讨论了稳态共振波的存在性。观察到了由主波分量间的相位差而产生的对称组和非对称组波形。此外,定性对比上实测数据和理论结果吻合良好。6.考虑了无来流时双色波中基波相速度受非线性波波作用的影响,给出了考虑波波作用时相速度的高阶近似。分析了有旋来流对基波相速度的影响,发现来流(平均涡量)对非线性波波作用的影响由来流特征斜率决定,而来流表面速度大小则增强或减弱此效应。
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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].
小波理论在数值分析方面具有很大的应用潜力,但现有方法还中足以毫无困难地应用到固体力学领域一些并非问题的求解中.针对这种现状,该文给出一些基于小波理论的数值方法,并应用这些方法求解了方板结构的弯曲与振动;建立了压电智能方板结构动力控制的小波模型及数值仿真程序等.
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同伦分析方法在稳态共振波浪研究中的应用 . [博士论文].
波的共振现象存在于海洋工程和光学等领域,因此研究波共振的相关问题有着重要的理论和实际意义。在海洋工程领域,已有大量工作研究了线性和非线性波浪共振的问题,而这些工作主要集中在讨论共振波波幅的演化趋势以及共振波浪系统内部分量之间的能量周期交换。本文基于同伦分析方法,重点描述了一种稳态共振波浪系统,其中当共振条件满足时各个波分量能够达到一种平衡,互相之间没有能量交换,同时分析了这种稳态共振波浪系统与之前人们发现的波浪系统之间的联系。本文主要研究了如下问题:(1)研究了平坦底部之上水深有限情况下的稳态共振波浪系统。利用同伦分析方法求解了控制这种非线性波浪系统的非线性方程,发现了多个稳态的共振波浪系统,其中波分量的波幅是常数,各个波分量之间没有能量交换,即这种稳态共振波浪系统的能量谱与时间无关。此外,还发现共振波分量在有些稳态共振波浪系统中只含有小部分能量,而在另外一些稳态共振波浪系统中又占有了绝大部分能量。通过Zakharov方程的研究,证明了同伦分析方法所获得解的正确性,同时明确了稳态解与非稳态解之间的关系。(2)研究了具有单个基波的表面波浪系统与周期变化的非平坦底部之间所发生的共振,该底部在很大范围内分布着呈余弦函数变化的波纹,所考虑的波浪与底部之间的共振属于第一类布拉格共振。同样着眼于各波分量间没有能量交换的稳态共振波浪系统,发现了两种完全不同的稳态共振波浪系统,其中基波和共振波分量加起来都占据了系统的绝大部分能量。在第一种稳态共振波浪系统中,基波和共振波分量具有相同的波幅,而在另一种稳态共振波浪系统中,两者的波幅却截然不同。首次发现了这两种稳态共振波浪系统关于基波的传播角度、平均水深、波纹底部波陡以及非线性强弱这四者之间存在分叉现象。这些关于稳态第一类布拉格共振波的结果,以前从未见报道,证实稳态共振波及其多解存在的普遍性。(3)通过求解Korteweg-de Vries(KdV)方程研究了浅水中当有无穷多个共振发生时的稳态波浪系统。在同伦分析方法的框架内,通过适当地选取辅助线性算子建立合理的线性高阶形变方程,简单高效地克服了由于共振产生的所有奇点,成功地获得了当有无穷多个共振发生时的稳态波浪系统。
Application of homotopy analysis method in steady-state resonant waves . [PhD Thesis].
波的共振现象存在于海洋工程和光学等领域,因此研究波共振的相关问题有着重要的理论和实际意义。在海洋工程领域,已有大量工作研究了线性和非线性波浪共振的问题,而这些工作主要集中在讨论共振波波幅的演化趋势以及共振波浪系统内部分量之间的能量周期交换。本文基于同伦分析方法,重点描述了一种稳态共振波浪系统,其中当共振条件满足时各个波分量能够达到一种平衡,互相之间没有能量交换,同时分析了这种稳态共振波浪系统与之前人们发现的波浪系统之间的联系。本文主要研究了如下问题:(1)研究了平坦底部之上水深有限情况下的稳态共振波浪系统。利用同伦分析方法求解了控制这种非线性波浪系统的非线性方程,发现了多个稳态的共振波浪系统,其中波分量的波幅是常数,各个波分量之间没有能量交换,即这种稳态共振波浪系统的能量谱与时间无关。此外,还发现共振波分量在有些稳态共振波浪系统中只含有小部分能量,而在另外一些稳态共振波浪系统中又占有了绝大部分能量。通过Zakharov方程的研究,证明了同伦分析方法所获得解的正确性,同时明确了稳态解与非稳态解之间的关系。(2)研究了具有单个基波的表面波浪系统与周期变化的非平坦底部之间所发生的共振,该底部在很大范围内分布着呈余弦函数变化的波纹,所考虑的波浪与底部之间的共振属于第一类布拉格共振。同样着眼于各波分量间没有能量交换的稳态共振波浪系统,发现了两种完全不同的稳态共振波浪系统,其中基波和共振波分量加起来都占据了系统的绝大部分能量。在第一种稳态共振波浪系统中,基波和共振波分量具有相同的波幅,而在另一种稳态共振波浪系统中,两者的波幅却截然不同。首次发现了这两种稳态共振波浪系统关于基波的传播角度、平均水深、波纹底部波陡以及非线性强弱这四者之间存在分叉现象。这些关于稳态第一类布拉格共振波的结果,以前从未见报道,证实稳态共振波及其多解存在的普遍性。(3)通过求解Korteweg-de Vries(KdV)方程研究了浅水中当有无穷多个共振发生时的稳态波浪系统。在同伦分析方法的框架内,通过适当地选取辅助线性算子建立合理的线性高阶形变方程,简单高效地克服了由于共振产生的所有奇点,成功地获得了当有无穷多个共振发生时的稳态波浪系统。
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| [5] |
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].
正科学和工程中普遍存在由非线性常、偏微分方程控制的边值问题,这些非线性问题的有效求解对深入认识相关问题的本质具有非常重要的意义。然而,由于非线性边值问题大多具有非常复杂的性质乃至存在多解情况,故求解起来远远比线性问题要复杂得多。同伦分析方法(HAM)自提出以来,由于具有不依赖于
The wavelet homotopy analysis method for nonlinear boundary value problems and its applicationis . [Master Thesis].
正科学和工程中普遍存在由非线性常、偏微分方程控制的边值问题,这些非线性问题的有效求解对深入认识相关问题的本质具有非常重要的意义。然而,由于非线性边值问题大多具有非常复杂的性质乃至存在多解情况,故求解起来远远比线性问题要复杂得多。同伦分析方法(HAM)自提出以来,由于具有不依赖于
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| [6] |
低浓度颗粒流Boltzmann方程的同伦分析方法解 .
同伦分析方法(homotopy analysis method, HAM)是求解强非线性问题的有力手段. 针对颗粒流的动理学理论中的非线性微分积分方程——?Boltzmann方程, 采用 HAM方法选取局域Maxwell速度分布函数作为初始猜测解, 得到了低浓度颗粒流的Boltzmann方程的一阶近似解, 与传统的Chapman-Enskog方法得到的一阶近似解表达式的结构一致, 初步显示了HAM方法求解Boltzmann方程的有效性, 为一般Boltzmann方程的HAM方法求解奠定了基础.
A new solution to Boltzmann equation of dilute granular flow with homotopy analysis method .
同伦分析方法(homotopy analysis method, HAM)是求解强非线性问题的有力手段. 针对颗粒流的动理学理论中的非线性微分积分方程——?Boltzmann方程, 采用 HAM方法选取局域Maxwell速度分布函数作为初始猜测解, 得到了低浓度颗粒流的Boltzmann方程的一阶近似解, 与传统的Chapman-Enskog方法得到的一阶近似解表达式的结构一致, 初步显示了HAM方法求解Boltzmann方程的有效性, 为一般Boltzmann方程的HAM方法求解奠定了基础.
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| [7] |
低浓度固液两相流Boltzmann方程的同伦分析方法解 .
同伦分析方法(Homotopy analysis method, HAM)是求解强非线性问题的有力手段.针对两相流动理学理论中的非线性微分积分方程--Boltzmann方程,本文采用HAM方法选取Maxwell 速度分布函数作为初始猜测解,求解得到了低浓度固液两相流的BGK模型Boltzmann方程的一阶近似解,与传统的Chapman-Enskog方法得 到的一阶近似解表达式的结构一致,显示了HAM方法求解Boltzmann方程的有效性,为一般Boltzmann方程的HAM方法求解奠定了基础.
A new solution to Boltzmann equation of dilute solid-liquid two-phase flows with homotopy analysis method .
同伦分析方法(Homotopy analysis method, HAM)是求解强非线性问题的有力手段.针对两相流动理学理论中的非线性微分积分方程--Boltzmann方程,本文采用HAM方法选取Maxwell 速度分布函数作为初始猜测解,求解得到了低浓度固液两相流的BGK模型Boltzmann方程的一阶近似解,与传统的Chapman-Enskog方法得 到的一阶近似解表达式的结构一致,显示了HAM方法求解Boltzmann方程的有效性,为一般Boltzmann方程的HAM方法求解奠定了基础.
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| [8] |
应用同伦分析方法求解若干力学和金融学问题 . [硕士论文].
Homotopy analysis method for several problems in mechanics and finance . [Master Thesis].
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| [9] |
Nonlinear stochastic differential equations . |
| [10] |
Solving Frontier Problems of Physics: The Decomposition Method .
The Adomian decomposition method enables the accurate and efficient analytic solution of nonlinear ordinary or partial differential equations without the need to resort to linearization of perturbation approaches. It unifies the treatment of linear and nonlinear, ordinary or partial differential equations, or systems of such equations, into a single basic method, which is applicable to both initial and boundary-value problems. The author applies the method to many problems of physics. Annotation copyright by Book News, Inc., Portland, OR
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| [11] |
On the optimal exercise boundary for an American put option .
An American put option is a derivative financial instrument that gives its holder the right but not the obligation to sell an underlying security at a pre-determined price. American options may be exercised at any time prior to expiry at the discretion of the holder, and the decision as to whether or not to exercise leads to a free boundary problem. In this paper, we examine the behavior of the free boundary close to expiry. Working directly with the underlying PDE, by using asymptotic expansions, we are able to deduce this behavior of the boundary in this limit.
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| [12] |
Modifications of Homotopy Analysis Method for Differential Equations: Modification of Homotopy Analysis Method, Ordinary, Fractional, Delay and Algebraic Differential Equations . |
| [13] |
Magnetohydrodynamic flow of a viscoelastic fluid .
The non-linear differential equation for the magnetohydrodynamic Poiseuille flow of Phan-Thein–Tanner (PTT) conducting fluid is derived. Using the homotopy analysis method (HAM), the series solution is developed and its convergence is discussed. Also, the results are presented graphically and the effects of non-dimensional parameters on the flow field are analyzed. The results obtained reveal many interesting behaviors that warrant further study on the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.
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| [14] |
Basic Topology(Undergraduate Texts in Mathematics) . |
| [15] |
Critical stock price near expiration .
We study the critical price of an American put option near expiration in the Black-Scholes model. Our main result is an estimate for the difference ( t )- K between the critical price at time t and the exercise price as t approaches the maturity of the option.
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| [16] |
Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method .
In this Letter, we apply the homotopy-perturbation method (HPM) to obtain approximate analytical solutions of the time-dependent Emden–Fowler type equations. We also present a reliable new algorithm based on HPM to overcome the difficulty of the singular point at x = 0. The analysis is accompanied by some linear and nonlinear time-dependent singular initial value problems. The results prove that HPM is very effective and simple.
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| [17] |
Homotopy analysis method for singular IVPs of Emden-Fowler type .
In this paper, approximate and/or exact analytical solutions of singular initial value problems (IVPs) of the Emden–Fowler type in the second-order ordinary differential equations (ODEs) are obtained by the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. It is shown that the solutions obtained by the Adomian decomposition method (ADM) and the homotopy-perturbation method (HPM) are only special cases of the HAM solutions.
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| [18] |
A method of directly defining the inverse mapping for solutions of coupled systems of nonlinear differential equations . |
| [19] |
Non-linear gravity wave interactions .
ABSTRACT In earlier papers Phillips (1960) and Longuet-Higgins (1962) have investigated phase velocity effects and possible resonances associated with the interactions of gravity waves. In this note the problem is discussed from a different viewpoint which demonstrates more clearly the energy-sharing mechanism involved. Equations governing the time dependence of the resonant modes are obtained, rather than the initial growth rate as has been found previously.
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| [20] |
Resonant interactions between waves. The case of discrete oscillations .
ABSTRACT The mathematical basis for resonance is investigated using a model equation describing one-dimensional dispersive waves interacting weakly through a quadratic term. If suitable time-invariant boundary conditions are imposed, possible oscillations of infinitesimal amplitude are restricted to a discrete set of wave-numbers. An asymptotic expansion valid for small amplitude shows that oscillations of different wave-number interact primarily in independent resonant trios. Energy is redistributed between members of a trio over a characteristic time inversely proportional to the amplitude of the oscillations in a periodic manner. The period depends on the initial conditions but is in general finite. Cubic interactions through resonant quartets are also discussed. The methods used are valid for a fairly wide class of equations describing weakly non-linear dispersive waves, but the expansion procedure used here fails for a continuous spectrum.
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| [21] |
The American put option and its critical stock price .
We derive an expression for the critical stock price for the American put. We start by expressing the put price as an integral involving first-passage probabilities. This approach yields intuition for Merton's result for the perpetual put. We then consider the finite-lived case. Using (1) the fact that the put value ceases to depend on time when the critical stock price is reached and (2) the result that an American put equals a European put plus an early-exercise premium, we derive the critical stock price. We approximate the critical-stock-price function to compute accurate put prices.
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| [22] |
The perturbation parameter in the problem of large deflection of clamped circular plates .
In the problems of large deflection of clamped circular plates under uniformly distributed loads, various perturbation parameters relating to load, deflection, slope of deflection, membrane force, etc. are studied. For a general perturbation parameter, the variational principle is used for the solution of such a problem. The applicable range of these perturbation parameters are studied in detail. In the case of uniformly loaded plate, perturbation parameter relating to central deflection seems to be the best among all others. The method of determination of perturbation solution by means of variational principle can be used to treat a variety of problems, including the large deflection problems under combine loads.
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| [23] |
An explicit series approximation to the optimal exercise boundary of American put options .
This paper derives an explicit series approximation solution for the optimal exercise boundary of an American put option by means of a new analytical method for strongly nonlinear problems, namely the homotopy analysis method (HAM). The Black holes equation subject to the moving boundary conditions for an American put option is transferred into an infinite number of linear sub-problems in a fixed domain through the deformation equations. Different from perturbation/asymptotic approximations, the HAM approximation can be applicable for options with much longer expiry. Accuracy tests are made in comparison with numerical solutions. It is found that the current approximation is as accurate as many numerical methods. Considering its explicit form of expression, it can bring great convenience to the market practitioners.
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| [24] |
Large deflection of a circular clamped plate under uniform pressure .
The problem of large deflection of a clamped circular plate under uniform pressure is studied by the method of successive approximation in terms of the parameter representing the ratio of the center deflection to the thickness. The tedious numerical computations, involved in Way's power series solution are thus avoided. The yielding condition at the edge checks very well with the experimental results given by McPherson, Ramberg and Levy. The method may be easily extended to any other boundary conditions and loading details.
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| [25] |
On a free convection problem over a vertical flat surface in a porous medium .
The problem of the free convection boundary-layer flow over a semi-infinite vertical flat surface in a porous medium is considered, in which the surface temperature has a constant value T1at the...
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| [26] |
Traveling gravity water waves in two and three dimensions. European Journal of Mechanics B |
| [27] |
American options on asserts with dividends near expiry . |
| [28] |
Efficient spectral sparse grid approximations for solving multi-dimensional forward backward SDEs .
Abstract: This is the second part in a series of papers on multi-step schemes for solving coupled forward backward stochastic differential equations (FBSDEs). We extend the basic idea in our former paper [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve high-dimensional FBSDEs, by using the spectral sparse grid approximations. The main issue for solving high dimensional FBSDEs is to build an efficient spatial discretization, and deal with the related high dimensional conditional expectations and interpolations. In this work, we propose the sparse grid spatial discretization. We use the sparse grid Gaussian-Hermite quadrature rule to approximate the conditional expectations. And for the associated high dimensional interpolations, we adopt an spectral expansion of functions in polynomial spaces with respect to the spatial variables, and use the sparse grid approximations to recover the expansion coefficients. The FFT algorithm is used to speed up the recovery procedure, and the entire algorithm admits efficient and high accurate approximations in high-dimensions, provided that the solutions are sufficiently smooth. Several numerical examples are presented to demonstrate the efficiency of the proposed methods.
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| [29] |
Principia Generalia Theoriae Figurae Fluidorum in Statu Aequilibrii .
Vires ascensionem vel depressionem fluidorum in tubis capillaribus gubernantes primus acute et accurate enumeravit sagax Clairaut , sed quum legem virium omnino intactam liquerit, nihil fructus ad explicationem mathematicam phaenomenorum ex illa enumeratione nasci potuit. Attractio vulgaris quadrato distantiae reciproce proportionalis, quae omnes motus coelestes tam felici successu explicat, nullius usus est nec in phaenomenis capillaribus, nec in phaenomenis adhaesionis et cohaesionis explicandis; calculus enim recte institutus facile docet, ad normam illius legis attractionem cuiusvis corporis, quocum experimenta instituere licet, i. e. cuius moles respectu totius terrae pro nihilo haben potest, in punctum ubicunque vel adeo in contactu positum, evanescere respectu gravitatis*). Recte hinc concluditur, illam attractionis legem in distantiis minimis naturae baud amplius consentaneam esse, sed modificationem quandam postulare, sive quod eodem redit, corporum particulas praeter illam vim attractivam exer-cere aliam in distantiis minimis tantum conspicuam. Phaenomena omnia con-spirant ad arguendum, hancce alteram vis attractivae partem ( attractionem molecularem ) in distantiis vel minimis quas mensurare licet insensibilem esse, dum in distantiis insensibilibus partem priorem (quadrato distantiae reciproce proportionalem) longe superare possit.
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| [30] |
The comparison between Homotopy Analysis Method and Optimal Homotopy Asymptotic Method for nonlinear age-structured population models .
This paper presents comparison between Homotopy Analysis Method (HAM) and Optimal Homotopy Asymptotic Method (OHAM) for the solution of nonlinear age-structured population models. Three examples have been presented to illustrate and compare these methods. In OHAM the convergence region can be easily adjusted and controlled. Comparison between our solution and the exact solution shows that the both methods are effective and accurate in solving nonlinear age-structured population models with HAM being the more accurate for the same number of terms. It was also found that OHAM require more CPU time.
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| [31] |
Regions of validity of analytical wave theories . |
| [32] |
Accurate computations for steep solitary waves .
ABSTRACT Finite-amplitude solitary waves in water of arbitrary uniform depth are considered. A numerical scheme based on series truncation is presented to calculate the highest solitary wave. It is found that the ratio of the amplitude of the wave versus the depth is 0.83322. This value is compared with the values obtained by previous investigators. In addition, another numerical scheme based on an integral-equation formulation is derived to compute solitary waves of arbitrary amplitude. These calculations confirm and extend the calculations of Byatt-Smith & Longuet-Higgins (1976) for very steep waves.
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| [33] |
Chaos in a three-dimensional cancer model .
In this study, we develop a new dynamical model of cancer growth, which includes the interactions between tumour cells, healthy tissue cells, and activated immune system cells, clearly leading to chaotic behavior. We explain the biological relevance of our model and the ways in which it differs from the existing ones. We perform equilibria analysis, indicate the conditions where chaotic dynamics can be observed, and show rigorously the existence of chaos by calculating the Lyapunov exponents and the Lyapunov dimension of the system. Moreover, we demonstrate that Shilnikov's theorem is valid in the parameter range of interest.
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| [34] |
Methods of dynamics calculation and testing for thin-walled structures (in Russian)// |
| [35] |
Traité de mécanique céleste .
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| [36] |
Approximate solutions to a parameterized sixth order boundary value problem .
In this paper, the homotopy analysis method (HAM) is applied to solve a parameterized sixth order boundary value problem which, for large parameter values, cannot be solved by other analytical methods for finding approximate series solutions. Convergent series solutions are obtained, no matter how large the value of the parameter is. [All rights reserved Elsevier].
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| [37] |
A kind of approximate solution technique which does not depend upon small parameters(II)-An application in fluid mechanics .
In this paper, the non-linear approximate technique called Homotopy Analysis Method proposed by Liao is further improved by introducing a non-zero parameter into the traditional way of constructing a homotopy. The 2D viscous laminar flow over an infinite flat-plain governed by the non-linear differential equation f′''(η) + f(η)f″(η)2 = 0 with boundary conditions f (0) = f ′(0) = 0, f ′(+ ∞) = 1 is used as an example to describe its basic ideas. As a result, a family of approximations is obtained for the above-mentioned problem, which is much more general than the power series given by Blasius [ Z. Math. Phys . 36 , 1(1908)] and can converge even in the whole region η 03 [0, + ∞). Moreover, the Blasius' solution is only a special case of ours. We also obtain the second-derivative of f ( η ) at η = 0, i.e. f ″(0) = 0.33206, which is exactly the same as the numerical result given by Howarth [ Proc. Roy. Soc. London A164 , 547 (1938)].
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| [38] |
An explicit, totally analytic approximation of Blasius viscous flow problems .
By means of using an operator A to denote non-linear differential equations in general, we first give a systematic description of a new kind of analytic technique for non-linear problems, namely the homotopy analysis method (HAM), Secondly, we generally discuss the convergence of the related approximate solution sequences and show that, as long as the approximate solution sequence given by the HAM is convergent, it must converge to one solution of the non-linear problem under consideration. Besides, we illustrate that even though a non-linear problem has one and only one solution, the sole solution might have an infinite number of expressions. Finally, to show the validity of the HAM, we apply it to give an explicit, purely analytic solution of the 2D laminar viscous flow over a semi-infinite flat plate. This explicit analytic solution is valid in the whole region 畏 = [0, + 瀅 and can give, the first time in history (to our knowledge), an analytic value f(0) = 0.33206, which agrees very well with Howarth's numerical result. This verifies the validity and great potential of the proposed homotopy analysis method as a new kind of powerful analytic tool
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| [39] |
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| [40] |
A new analytic algorithm of Lane-Emden type equations .
An reliable, ease-to-use analytic algorithm is provided for Lane–Emden type equation which models many phenomena in mathematical physics and astrophysics. This algorithm logically contains the well-known Adomian decomposition method. Different from all other analytic techniques, this algorithm itself provides us with a convenient way to adjust convergence regions even without Páde technique. Some applications are given to show its validity.
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| [41] |
Explicit analytic solution for similarity boundary layer equations .
In this paper the homotopy analysis method for strongly non-linear problems is employed to give two kinds of explicit analytic solutions of similarity boundary-layer equations. The analytic solutions are explicitly expressed by recurrence formulas for constant coefficients and can give accurate results in the whole regions of physical parameters.
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| [42] |
A general approach to get series solution of non-similarity boundary-layer flows .
An analytic method for strongly non-linear problems, namely the homotopy analysis method (HAM), is applied to give convergent series solution of non-similarity boundary-layer flows. As an example, the non-similarity boundary-layer flows over a stretching flat sheet are used to show the validity of this general analytic approach. Without any assumptions of small/large quantities, the corresponding non-linear partial differential equation with variable coefficients is transferred into an infinite number of linear ordinary differential equations with constant coefficients. More importantly, an auxiliary artificial parameter is used to ensure the convergence of the series solution. Different from previous analytic results, our series solutions are convergent and valid for all physical variables in the whole domain of flows. This work illustrates that, by means of the homotopy analysis method, the non-similarity boundary-layer flows can be solved in a similar way like similarity boundary-layer flows. Mathematically, this analytic approach is rather general in principle and can be applied to solve different types of non-linear partial differential equations with variable coefficients in science and engineering. [All rights reserved Elsevier].
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| [43] |
On the relationship between the homotopy analysis method and Euler transform .
A new transform, namely the homotopy transform, is defined for the first time. Then, it is proved that the famous Euler transform is only a special case of the so-called homotopy transform which depends upon one non-zero auxiliary parameter 67 and two convergent series ∑ k = 1 + ∞ α 1 , k = 1 and ∑ k = 1 + ∞ β 1 , k = 1. In the frame of the homotopy analysis method, a general analytic approach for highly nonlinear differential equations, the so-called homotopy transform is obtained by means of a simple example. This fact indicates that the famous Euler transform is equivalent to the homotopy analysis method in some special cases. On one side, this explains why the convergence of the series solution given by the homotopy analysis method can be guaranteed. On the other side, it also shows that the homotopy analysis method is more general and thus more powerful than the Euler transform.
|
| [44] |
An optimal homotopy-analysis approach for strongly nonlinear differential equations .
In this paper, an optimal homotopy-analysis approach is described by means of the nonlinear Blasius equation as an example. This optimal approach contains at most three convergence-control parameters and is computationally rather efficient. A new kind of averaged residual error is defined, which can be used to find the optimal convergence-control parameters much more efficiently. It is found that all optimal homotopy-analysis approaches greatly accelerate the convergence of series solution. And the optimal approaches with one or two unknown convergence-control parameters are strongly suggested. This optimal approach has general meanings and can be used to get fast convergent series solutions of different types of equations with strong nonlinearity.
|
| [45] |
On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves .
The basic ideas of a homotopy-based multiple-variable method is proposed and applied to investigate the nonlinear interactions of periodic traveling waves. Mathematically, this method does not depend upon any small physical parameters at all and thus is more general than the traditional multiple-scale perturbation techniques. Physically, it is found that, for a fully developed wave system, the amplitudes of all wave components are finite even if the wave resonance condition given by Phillips (1960) is exactly satisfied. Besides, it is revealed that there exist multiple resonant waves, and that the amplitudes of resonant wave may be much smaller than those of primary waves so that the resonant waves sometimes contain rather small part of wave energy. Furthermore, a wave resonance condition for arbitrary numbers of traveling waves with large wave amplitudes is given, which logically contains Phillips four-wave resonance condition but opens a way to investigate the strongly nonlinear interaction of more than four traveling waves with large amplitudes. This work also illustrates that the homotopy multiple-variable method is helpful to gain solutions with important physical meanings of nonlinear problems, if the multiple-variables are properly defined with clear physical meanings.
|
| [46] |
Homotopy Analysis Method in Nonlinear Differential Equations .
全书分三个部分。第一部分描述同伦分析方法的基本思想和相关理论。第二部分给出基于同伦分析方法和计算机代数软件 Mathematica 开发的软件包 BVPh 1.0 及其应用举例。第三部分给出同伦分析方法求解非线性偏微分方程的一些典型例子，如美式期权问题、任意多个波浪的共振条件等。
|
| [47] |
On the steady-state nearly resonant waves .
The steady-state nearly resonant water waves with time-independent spectrum in deep water are obtained from the full wave equations for inviscid, incompressible gravity waves in the absence of surface tension by means of a analytic approximation approach based on the homotopy analysis method (HAM). Our strategy is to mathematically transfer the steady-state nearly resonant wave problem into the steady-state exactly resonant ones. By means of choosing a generalized auxiliary linear operator that is a little different from the linear part of the original wave equations, the small divisor, which is unavoidable for nearly resonant waves in the frame of perturbation methods, is avoided, or moved far away from low wave frequency to rather high wave frequency with physically negligible wave energy. It is found that the steady-state nearly resonant waves have nothing fundamentally different from the steady-state exactly resonant ones, from physical and numerical viewpoints. In addition, the validity of this HAM-based analytic approximation approach for the full wave equations in deep water is numerically verified by means of the Zakharov090005s equation. A thought experiment is discussed, which suggests that the essence of the so-called 090004wave resonance090005 should be reconsidered carefully from both of physical and mathematical viewpoints.
|
| [48] |
On the method of directly defining inverse mapping for nonlinear differential equations .
In scientific computing, it is time-consuming to calculate an inverse operator \(\mathcal {A}^{-1}\) of a differential equation \(\mathcal {A}\phi = f\) , especially when \(\mathcal {A}\) is a highly nonlinear operator. In this paper, based on the homotopy analysis method (HAM), a new approach, namely the method of directly defining inverse mapping (MDDiM), is proposed to gain analytic approximations of nonlinear differential equations. In other words, one can solve a nonlinear differential equation \(\mathcal {A}\phi = f\) by means of directly defining an inverse mapping \(\mathcal J\) , i.e. without calculating any inverse operators. Here, the inverse mapping \(\mathcal {J}\) is even unnecessary to be explicitly expressed in a differential form, since “mapping” is a more general concept than “differential operator”. To guide how to directly define an inverse mapping \(\mathcal {J}\) , some rules are provided. Besides, a convergence theorem is proved, which guarantees that a convergent series solution given by the MDDiM must be a solution of problems under consideration. In addition, three nonlinear differential equations are used to illustrate the validity and potential of the MDDiM, and especially the great freedom and large flexibility of directly defining inverse mappings for various types of nonlinear problems. The method of directly defining inverse mapping (MDDiM) might open a completely new, more general way to solve nonlinear problems in science and engineering, which is fundamentally different from traditional methods.
|
| [49] |
A new non-perturbative approach in quantum mechanics for time-independent Schrödinger equations .URL 摘要
Abstract: A new non-perturbative approach is proposed to solve time-independent Schr枚dinger equations in quantum mechanics and chromodynamics (QCD). It is based on the homotopy analysis method (HAM), which was developed by the author for highly nonlinear equations since 1992 and has been widely applied in many fields. Unlike perturbative methods, this HAM-based approach has nothing to do with small/large physical parameters. Besides, convergent series solution can be obtained even if the disturbance is far from the known status. A nonlinear harmonic oscillator is used as an example to illustrate the validity of this approach for disturbances that might be more than hundreds larger than the possible superior limit of the perturbative approach. This HAM-based approach could provide us rigorous theoretical results in quantum mechanics and chromodynamics (QCD), which can be directly compared with experimental data. Obviously, this is of great benefit not only for improving the accuracy of experimental measurements but also for validating physical theories.
|
| [50] |
KBM method based on the homotopy analysis .
http://link.springer.com/article/10.1007%2Fs11433-011-4250-z
|
| [51] |
Steady-state resonance of multiple wave interactions in deep water .
The steady-state resonance of multiple surface gravity waves in deep water was investigated in detail to extend the existing results due to Liao (Commun. Nonlinear Sci. Numer. Simul., vol.0216, 2011, pp.021274–1303) and Xuet al.(J. Fluid Mech., vol.02710, 2012, pp.02379–418) on steady-state resonance from a quartet to more general and coupled resonant quartets, together with higher-order resonant interactions. The exact nonlinear wave equations are solved without assumptions on the existence of small physical parameters. Multiple steady-state resonant waves are obtained for all the considered cases, and it is found that the number of multiple solutions tends to increase when more wave components are involved in the resonance sets. The topology of wave energy distribution in the parameter space is analysed, and it is found that the steady-state resonant waves indeed form a continuum in the parameter space. The significant roles of the near-resonance and nonlinearity were also revealed. It is found that all of the near-resonant components as a whole contain more and more wave energy, as the wave patterns tend from two dimensions to one dimension, or as the nonlinearity of the steady-state resonant wave system increases. In addition, the linear stability of the steady-state resonant waves is analysed. It is found that the steady-state resonant waves are stable, as long as the disturbance does not resonate with any components of the basic wave. All of these findings are helpful to enrich and deepen our understanding about resonant gravity waves.
|
| [52] |
On the existence of steady-state resonant waves in experiment . |
| [53] |
Finite amplitude steady-state wave groups with multiple near resonances in deep water . |
| [54] |
Mass, momentum and energy flux conservation between linear and nonlinear steady-state wave groups . |
| [55] |
General Problem on Stability of Motion(English translation). Taylor & Francis , |
| [56] |
Backward stochastic differential equations in finance .
We are concerned with different properties of backward stochastic differential equations and their applications to nance. These equations, rst introduced by Pardoux and Peng (1990), are useful for the theory of contingent claim valuation, especially cases with constraints and for the theory of recursive utilities, introduced by Dufe and Epstein (1992a, 1992b).
|
| [57] |
Steady Flow in a Williamson Fluid: Basic Concepts of Fluid Mechanics Solution to a Non-linear Ordinary Differential Equation by Homotopy Analysis Method . |
| [58] |
Iterative solutions for the non-linear bending of circular plates . |
| [59] |
A note on a moving boundary problem arising in the American put option .
We consider an American put option, under the Black–Scholes model. This corresponds to a moving boundary problem for a PDE. We convert the problem to a nonlinear integral equation for the moving boundary, which corresponds to the optimal exercise of the option. We use singular perturbation methods to compute the moving boundary, as well as the full solution to the PDE, in various asymptotic limits. We consider times close to the expiration date, as well as systems where the interest rate is large or small, relative to the volatility of the asset for which the option is sold.
|
| [60] |
Optional exercise boundary for an American put option .
The optimal exercise boundary near the expiration time is determined for an American put option. It is obtained by using Green's theorem to convert the boundary value problem for the price of the option into an integral equation for the optimal exercise boundary. This integral equation is solved asymptotically for small values of the time to expiration. The leading term in the asymptotic solution is the result of Barles et al. An asymptotic solution for the option price is obtained also.
|
| [61] |
Third-order theory for multi-directional irregular waves .
A new third-order solution for multi-directional irregular water waves in finite water depth is presented. The solution includes explicit expressions for the surface elevation, the amplitude dispersion and the vertical variation of the velocity potential. Expressions for the velocity potential at the free surface are also provided, and the formulation incorporates the effect of an ambient current with the option of specifying zero net volume flux. Harmonic resonance may occur at third order for certain combinations of frequencies and wavenumber vectors, and in this situation the perturbation theory breaks down due to singularities in the transfer functions. We analyse harmonic resonance for the case of a monochromatic short-crested wave interacting with a plane wave having a different frequency, and make long-term simulations with a high-order Boussinesq formulation in order to study the evolution of wave trains exposed to harmonic resonance.
|
| [62] |
Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer .
We consider one of the newest analytical methods, the Optimal Homotopy Asymptotic Method (OHAM), to solve nonlinear equations arising in heat transfer. Two specific applications are considered: cooling of a lumped system with variable specific heat and the temperature distribution equation in a thick rectangular fin radiation to free space. Results obtained by OHAM, which does not need small parameters are compared with numerical results and a very good agreement was found. This method provides us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary. The results reveal that the proposed method is explicit, effective and easy to use.
|
| [63] |
Local non-similarity solutions for the flow of a non-Newtonian fluid over a wedge .
The boundary layer and heat transfer equations for a non-Newtonian fluid, represented by a power-law model, over a porous wedge is studied. The free stream velocity, the surface temperature variations, and the injection velocity at the surface are assumed variables. Similar and non-similar solutions are presented and the restrictions for these cases are studied. The results are presented for velocity and temperature profiles for various values of the dimensionless numbers. The effects of the different parameters on the skin friction co-efficient and the local heat transfer co-efficient are also studied.
|
| [64] |
Homotopy analysis method applied to electrohydrodynamic flow .
http://linkinghub.elsevier.com/retrieve/pii/S1007570410005344
|
| [65] |
Calculation of steady three-dimensional deep-water waves .
Steady three-dimensional symmetric wave patterns for finite-amplitude gravity waves on deep water are calculated from the full unapproximated water-wave equations as well as from an approximate equation due to Zakharov. These solutions are obtained as bifurcations from plane Stokes waves. The results are in good agreement with the experimental observations of Su.
|
| [66] |
Application of the homotopy analysis method to the Poisson-Boltzmann equation for semiconductor devices .
This paper describes the application of a recently developed analytic approach known as the homotopy analysis method to derive an approximate solution to the nonlinear Poisson–Boltzmann equation for semiconductor devices. Specifically, this paper presents an analytic solution to potential distribution in a DG-MOSFET (Double Gate-Metal Oxide Semiconductor Field Effect Transistor). The DG-MOSFET represents one of the most advanced device structures in semiconductor technology and is a primary focus of modeling efforts in the semiconductor industry.
|
| [67] |
Stable, high-order computation of traveling water waves in three dimensions .
The Euler equations of free-surface ocean dynamics constitute a model of central importance in fluid mechanics due to the wide range of physical phenomena they are intended to represent, from shoaling and breaking of waves in nearshore regions to energy and momentum transport in the open ocean. From a mathematical perspective, these equations present rather unique challenges for analysis and simulation as they couple the subtleties of nonlinear wave equations (balancing nonlinearity with dispersion in the absence of dissipation) to the difficulties of free-boundary problems. In this paper a new, stable high-order boundary perturbation algorithm for the numerical simulation of traveling water waves is described. Its performance is compared to that of classical surface deformation algorithms and it is shown that the new scheme displays significantly enhanced conditioning properties and a lower computational cost, which enable very accurate predictions of physical observables such as velocity, energy, height/steepness, and shape.
|
| [68] |
A one-step optimal homotopy analysis method for nonlinear differential equations .
In this paper, a one-step optimal approach is proposed to improve the computational efficiency of the homotopy analysis method (HAM) for nonlinear problems. A generalized homotopy equation is first expressed by means of a unknown embedding function in Taylor series, whose coefficient is then determined one by one by minimizing the square residual error of the governing equation. Since at each order of approximation, only one algebraic equation with one unknown variable is solved, the computational efficiency is significantly improved, especially for high-order approximations. Some examples are used to illustrate the validity of this one-step optimal approach, which indicate that convergent series solution can be obtained by the optimal homotopy analysis method with much less CPU time. Using this one-step optimal approach, the homotopy analysis method might be applied to solve rather complicated differential equations with strong nonlinearity.
|
| [69] |
A Steady-state Trio for Bretherton Equation . |
| [70] |
A nonlinear Feynman-Kac formula and applications//Proceedings of the Symposium on System Sciences and Control Theory. Control Theory, |
| [71] |
On the dynamics of unsteady gravity waves of finite amplitude .
On the dynamics of unsteady gravity waves of finite amplitude Part 1. The elementary interactions - Volume 9 Issue 2 - O. M. Phillips
|
| [72] |
Les méthodes nouvelles de la mécanique céleste .
|
| [73] |
Some Flow Problems in Differential Type Fluids: Series solutions using homotopy analysis method . |
| [74] |
Activation of effector immune cells promotes tumor stochastic extinction: A homotopy analysis approach . |
| [75] |
|
| [76] |
Local non-similarity boundary-layer solutions . |
| [77] |
Local non-similarity thermal boundary-layer solutions . |
| [78] |
On steady-state multiple resonances for a modified Bretherton equation .
In this article, a modified Bretherton equation is considered to further check if steady-state multiple resonances exist not only for water waves but also for other dispersive medium. The linear resonance condition analysis shows that different components may interact with each other so multiple resonances may happen. Convergent steady-state solutions are obtained by solution procedure based on the homotopy analysis method (HAM) and the collocation method. Amplitude spectrum analysis confirms that more components indeed join the resonance as the nonlinearity increases. This article suggests that steady-state multiple resonance may exist in other dispersive system.
|
| [79] |
Application of perturbation methods to the theory of nuclear matter .
A generalized perturbation theory is developed in such a way that it can be applied to a many-body problem with strong forces between the particles. The Brueckner expression for the energy is shown to be the first-order term in a particular case of this expansion. Some of the higher-order terms in the expansion are studied, and the importance of self-consistency in the energy denominator of Brueckner's equation and of the use of the exclusion principle in intermediate states is assessed. A possible simplification of the methods used is suggested, which involves solving the Brueckner equation for the hard core, and using normal perturbation theory for the attractive part of the potential. The methods developed are used to analyze some details of previously published calculations. The lack of equality between the Fermi energy and the binding energy in the nuclear matter calculations shows that there must be a rearrangement energy. A simple formula for the rearrangement energy is derived, and its importance for single-particle excited states, such as occur in the optical model, is shown. The relation between the rearrangement energy and the departure of the system from a degenerate Fermi-gas state is shown. The effect of the rearrangement energy on the ground-state energy is indirect, but it is as important as the self-consistency condition. The rearrangement energy seems to come mainly from the hard core, and simple numerical estimates of the rearrangement energy from a hard core potential show that it is somewhat less than 16 Mev at the Fermi surface. The ground-state energy is reduced by perhaps 1 Mev. There seems to be a discrepancy between the calculated and observed energy dependence of the real part of the optical model potential.
|
| [80] |
New Approaches to Nonlinear Waves .
ABSTRACT The book details a few of the novel methods developed in the last few years for studying various aspects of nonlinear wave systems. The introductory chapter provides a general overview, thematically linking the objects described in the book. Two chapters are devoted to wave systems possessing resonances with linear frequencies (Chapter 2) and with nonlinear frequencies (Chapter 3). In the next two chapters modulation instability in the KdV-type of equations is studied using rigorous mathematical methods (Chapter 4) and its possible connection to freak waves is investigated (Chapter 5). The book goes on to demonstrate how the choice of the Hamiltonian (Chapter 6) or the Lagrangian (Chapter 7) framework allows us to gain a deeper insight into the properties of a specific wave system. The final chapter discusses problems encountered when attempting to verify the theoretical predictions using numerical or laboratory experiments. All the chapters are illustrated by ample constructive examples demonstrating the applicability of these novel methods and approaches to a wide class of evolutionary dispersive PDEs, e.g. equations from Benjamin-Oro, Boussinesq, Hasegawa-Mima, KdV-type, Klein-Gordon, NLS-type, Serre, Shamel , Whitham and Zakharov. This makes the book interesting for professionals in the fields of nonlinear physics, applied mathematics and fluid mechanics as well as students who are studying these subjects. The book can also be used as a basis for a one-semester lecture course in applied mathematics or mathematical physics.
|
| [81] |
Limiting gravity waves in water of finite depth .
Progressive, irrotational gravity waves of constant form exist as a two-parameter family. The first parameter, the ratio of mean depth to wavelength, varies from zero (the solitary wave) to infinity (the deep-water wave). The second parameter, the wave height or amplitude, varies from zero (the infinitesimal wave) to a limiting value dependent on the first parameter. For limiting waves the wave crest ceases to be rounded and becomes angled, with an included angle of 120 degrees. Most methods of calculating finite-amplitude waves use either a form of series expansion or the solution of an integral equation. For waves nearing the limiting amplitude many terms (or nodal points) are needed to describe the wave form accurately. Consequently the accuracy even of recent solutions on modern computers can be improved upon, except at the deep-water end of the range. The present work extends an integral equation technique used previously in which the angled crest of the limiting wave is included as a specific term, derived from the well known Stokes corner flow. This term is now supplemented by a second term, proposed by Grant in a study of the flow near the crest. Solutions comprising 80 terms at the shallow-water end of the range, reducing to 20 at the deep-water end, have defined many field and integral properties of the flow to within 1 to 2 parts in 10. It is shown that without the new crest term this level of accuracy would have demanded some hundreds of terms while without either crest term many thousands of terms would have been needed. The practical limits of the computing range are shown to correspond, to working accuracy, with the theoretical extremes of the solitary wave and the deep-water wave. In each case the results agree well with several previous accurate solutions and it is considered that the accuracy has been improved. For example, the height:depth ratio of the solitary wave is now estimated to be 0.833 197 and the height:wavelength ratio of the deep-water wave to be 0.141 063. The results are presented in detail to facilitate further theoretical study and early practical application. The coefficients defining the wave motion are given for 22 cases, five of which, including the two extremes, are fully documented with tables of displacement, velocity, acceleration, pressure and time. Examples of particle orbits and drift profiles are presented graphically and are shown for the extreme waves to agree very closely with simplified calculations by Longuet-Higgins. Finally, the opportunity has been taken to calculate to greater accuracy the long-term Lagrangian-mean angular momentum of the maximum deep-water wave, according to the recent method proposed by Longuet-Higgins, with the conclusion that the level of action is slightly above the crest.
|
| [82] |
Computational Methods for Integral Equations: Linear Legendre Multi-Wavelets and Homotopy Analysis Methods . |
| [83] |
Nonlinear Flow Phenomena and Homotopy Analysis: Fluid Flow and Heat Transfer .
|
| [84] |
Analytic and numerical solutions to the Lane-Emden equation .
In this Letter, we present analytical solutions to the Lane–Emden equation y ″ ( x ) + 2 x y ′ ( x ) + f ( y ) = 0 describing the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules. Solutions are obtained by using the traditional power series approach and by using the Homotopy Analysis Method (HAM). We show that the series solutions obtained by the Homotopy Analysis Method converge in a larger interval than in the case of the corresponding traditional series solutions. Furthermore, we obtained numerical solutions (using Runge–Kutta–Fehlberg 4-5 technique) which are used to validate the analytical solutions.
|
| [85] |
Analytical method for the construction of solutions to the Föppl-von Kármán equations governing deflections of a thin flat plate . |
| [86] |
Gaussian waves in the Fitzhugh-Nagumo equation demonstrate one role of the auxiliary function $H(x, t)$ in the homotopy analysis method . |
| [87] |
c. Control of error in the homotopy analysis of semi-linear elliptic boundary value problems .
Abstract(). After outlining the methods in general, we consider three applications. First, we apply the method of minimized residual error in order to determine optimal values of the convergence control parameter to obtain solutions exhibiting central symmetry for the Yamabe equation in three or more spatial dimensions. Secondly, we apply the method of minimizing error functionals in order to obtain optimal values of the convergnce control parameter for the homotopy analysis solutions to the Brinkman–Forchheimer equation. Finally, we carefully selected the auxiliary function () in order to obtain an optimal homotopy solution for Liouville’s equation.
|
| [88] |
Relation between Lane-Emden solutions and radial solutions to the elliptic heavenly equation on a disk .
We provide a transformation between a type of solution to a Lane–Emden equation of second kind and a solution of the elliptic Heavenly equation on a disk. By doing so, we show that any solution of this Lane–Emden equation of second kind corresponds to an infinite family of solutions to the Heavenly equation. This Lane–Emden equation is naturally formulated as a boundary value problem, which makes it somewhat distinct from the initial value problem versions in the literature. We obtain simple analytical solutions of this Lane–Emden equation and associated boundary value problem, and then we use these analytical solutions to construct a family of solutions for the elliptic Heavenly equation. The obtained solutions are radial solutions to the Heavenly equation; that is, they exhibit radial symmetry. In effect, we obtain a relation between radially-symmetric self-dual gravitational instantons and the Lane–Emden approximation to the structure of a neutron star. In other words, the radially symmetric neutron star under the Lane–Emden model can be seen as a special type of gravitational instanton.
|
| [89] |
The bending of a thin circular plate . |
| [90] |
Heat transfer for flow longitudinal to a cylinder with surface mass transfer . |
| [91] |
On the steady-state fully resonant progressive waves in water of finite depth . |
| [92] |
Equilibrium states of class-I Bragg resonant wave system .
61The class-I Bragg resonant waves are solved analytically.61Multiple equilibrium-state resonant wave systems with time-independent wave spectrum are found.61Bifurcations with respect to wave propagation angle, water depth, bottom slope and nonlinearity are found.
|
| [93] |
An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method .
We consider the problem of two-dimensional projectile motion in which the resistance acting on an object moving in air is proportional to the square of the velocity of the object (quadratic resistance law). It is well known that the quadratic resistance law is valid in the range of the Reynolds number: 1 x 10 2 x 10(for instance, a sphere) for practical situations, such as throwing a ball. It has been considered that the equations of motion of this case are unsolvable for a general projectile angle, although some solutions have been obtained for a small projectile angle using perturbation techniques. To obtain a general analytic solution, we apply Liao's homotopy analysis method to this problem. The homotopy analysis method, which is different from a perturbation technique, can be applied to a problem which does not include small parameters. We apply the homotopy analysis method for not only governing differential equations, but also an algebraic equation of a velocity vector to extend the radius of convergence. Ultimately, we obtain the analytic solution to this problem and investigate the validation of the solution
|
| [94] |
A HAM-based wavelet approach for nonlinear ordinary differential equations .
Based on the homotopy analysis method (HAM) and the generalized Coiflet-type orthogonal wavelet, a new analytic approximation approach for solving nonlinear boundary value problems (governed by nonlinear ordinary differential equations), namely the wavelet homotopy analysis method (wHAM), is proposed. The basic ideas of the wHAM are described using the one-dimensional Bratu equation as an example. This method not only keeps the main advantages of the normal HAM, but also possesses some new properties and advantages. First of all, the wHAM possesses high computational efficiency. Besides, based on multi-resolution analysis, it provides us a convenient way to balance the accuracy and efficiency by simply adjusting the resolution level. Furthermore, different from the normal HAM, the wHAM provides us much larger freedom to choose the auxiliary linear operator. In addition, just like the normal HAM, iteration can greatly accelerate the computational efficiency of the wHAM without loss of accuracy.
|
| [95] |
A HAM-based wavelet approach for nonlinear partial differential equations: Two dimensional Bratu problem as an application . |
| [96] |
On the generalized Wavelet-Galerkin method . |
| [97] |
Nonlinear stabilities of thin circular shallow shells under actions of axisymmetrical uniformly distributed line loads .
In 1939, the importance of the nonlinear features in the shell buckling problem was first pointed out in a most spectacular manner by Von Karman and Chien, but the mathematical difficulty is so great that progress has been slow after the first attempts. According to our experience, we should face the difficult barrier of solving nonlinear partial differential equations and an effective, simple, accurate method is required. With this in mind, we suggest the modified iteration method. four cases are considered to verify it. Here is one of the cases in which nonlinear stabilities of thin circular shallow shells under actions of axisymmetrical uniformly distributed line loads are considered. As special cases, we have also investigated large deflections of plates under the same loads. All these results are presented in such a form that direct application in design is possible. Two stability curves coincide with the experimental results given by D. G.Ashwell and Chien Wei-chang. The results have also been compared with several other writers'.
|
| [98] |
An essay on the cohesion of fluids . |
| [99] |
Pricing and hedging American options analytically: A perturbation method .
This paper studies the critical stock price of American options with continuous dividend yield. We solve the integral equation and derive a new analytical formula in a series form for the critical stock price. American options can be priced and hedged analytically with the help of our critical-stock-price formula. Numerical tests show that our formula gives very accurate prices. With the error well controlled, our formula is now ready for traders to use in pricing and hedging the S&P 100 index options and for the Chicago Board Options Exchange to use in computing the VXO volatility index.
|
| [100] |
HAM-Based Mathematica Package BVPh 2.0 for Nonlinear Boundary Value Problems . Liao S ed.
Abstract We issue the new version BVPh 2.0 of the Mathematica package BVPh, a free software based on the homotopy analysis method (HAM) for nonlinear boundary-value and eigenvalue problems. The aim of the package BVPh is to develop a kind of analytic tool for as many nonlinear boundary value problems (BVPs) as possible such that multiple solutions of highly nonlinear BVPs can be conveniently found out, and that the infinite interval and singularities of governing equations and/or boundary conditions at multi-points can be easily resolved. Unlike its previous versions, BVPh 2.0 works for systems of coupled nonlinear ordinary differential equations. It is user-friendly and free available online (http://numericaltank.sjtu.edu.cn/BVPh.htm). Different from numerical packages (such as BVP4c), it is based on the idea omputing numerically with functions instead of numbers . Especially, unlike other packages, the convergence of results given by the BVPh 2.0 is guaranteed by means of the so-called convergence-control parameter in the frame of the homotopy analysis method. In this chapter, we briefly describe how to install and use the BVPh 2.0 with a simple user's guide. Five typical examples (governed by up to four coupled ODEs) are used to illustrate the validity of the BVPh 2.0, and the corresponding input data of these examples for the BVPh 2.0 are free available online (http://numericaltank.sjtu.edu.cn/BVPh.htm). The BVPh 2.0 indeed provides us with an easy-to-use tool to efficiently solve various types of coupled linear/nonlinear ordinary differential equations. 2014 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
|
| [101] |
On the convergence of the nonlinear equations of circular plate with interpolation iterative method . |
| [102] |
|
| [103] |
Analytic solutions of Von Kármán plate under arbitrary uniform pressure-equations in differential form .
The large deflection of a circular thin plate under uniform external pressure is a classic problem in solid mechanics, dated back to Von K{\'a}rm{\'a}n \cite{Karman}. {This problem is reconsidered in this paper using an analytic approximation method, namely the homotopy analysis method (HAM).} Convergent series solutions are obtained for four types of boundary conditions with rather high nonlinearity, even in the case of $w(0)/h>20$, where $w(0)/h$ denotes the ratio of central deflection to plate thickness. Especially, we prove that the previous perturbation methods for an arbitrary perturbation quantity (including the Vincent's [2] and Chien's [3] methods) and the modified iteration method [4] are only the special cases of the HAM. However, the HAM works well even when the perturbation methods become invalid. All of these demonstrate the validity and potential of the HAM for the Von K{\'a}rm{\'a}n's plate equations, and show the superiority of the HAM over perturbation methods for highly nonlinear problems
|
| [104] |
On the homotopy analysis method for backward/forward-backward stochastic differential equations .
In this paper, an analytic approximation method for highly nonlinear equations, namely the homotopy analysis method (HAM), is employed to solve some backward stochastic differential equations (BSDEs) and forward-backward stochastic differential equations (FBSDEs), including one with high dimensionality (up to 12 dimensions). By means of the HAM, convergent series solutions can be quickly obtained with high accuracy for a FBSDE in a 6-dimensional case, within less than 1 % CPU time used by a currently reported numerical method for the same case [34]. Especially, as dimensionality enlarges, the increase of computational complexity for the HAM is not as dramatic as this numerical method. All of these demonstrate the validity and high efficiency of the HAM for the backward/forward-backward stochastic differential equations in science, engineering, and finance.
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| [105] |
Analytic approximations of Von Kármán plate under arbitrary uniform pressure-equations in integral form .
Analytic approximations of the Von Kármán’s plate equations in integral form for a circular plate under external uniform pressure to arbitrary magnitude are successfully obtained by means of the homotopy analysis method (HAM), an analytic approximation technique for highly nonlinear problems. Two HAM-based approaches are proposed for either a given external uniform pressure Q or a given central deflection, respectively. Both of them are valid for uniform pressure to arbitrary magnitude by choosing proper values of the so-called convergence-control parameters c 1 and c 2 in the frame of the HAM. Besides, it is found that the HAM-based iteration approaches generally converge much faster than the interpolation iterative method. Furthermore, we prove that the interpolation iterative method is a special case of the first-order HAM iteration approach for a given external uniform pressure Q when c 1 = 61 θ and c 2 = 611, where θ denotes the interpolation iterative parameter. Therefore, according to the convergence theorem of Zheng and Zhou about the interpolation iterative method, the HAM-based approaches are valid for uniform pressure to arbitrary magnitude at least in the special case c 1 = 61 θ and c 2 = 611. In addition, we prove that the HAM approach for the Von Kármán’s plate equations in differential form is just a special case of the HAM for the Von Kármán’s plate equations in integral form mentioned in this paper. All of these illustrate the validity and great potential of the HAM for highly nonlinear problems, and its superiority over perturbation techniques.
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| [106] |
HAM approach for post-buckling problems of a large deformed elastic beam . |
| [107] |
On the limiting Stokes wave of extreme height in arbitrary water depth .
<div class="abstract" data-abstract-type="normal">Both Schwartz (<span class='italic'>J. Fluid Mech., vol.0262 (3), 1974, pp.02553–578) and Cokelet (<span class='italic'>Phil. Trans. R. Soc. Lond., vol.02286 (1335), 1977, pp.02183–230) failed to gain convergent results for limiting Stokes waves in extremely shallow water by means of perturbation methods, even with the aid of extrapolation techniques such as the Padé approximant. In particular, it is extremely difficult for traditional analytic/numerical approaches to present the wave profile of limiting waves with a sharp crest of <span class='inlineFormula'> <span class='alternatives'> <span class='mathjax-tex-wrapper' data-mathjax-type='texmath'> <span class='tex-math mathjax-tex-math mathjax-on'> $120^{\circ }$ included angle first mentioned by Stokes in the 1880s. Thus, traditionally, different wave models are used for waves in different water depths. In this paper, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear equations, we successfully gain convergent results (and especially the wave profiles) of the limiting Stokes waves with this kind of sharp crest in arbitrary water depth, even including solitary waves of extreme form in extremely shallow water, without using any extrapolation techniques. Therefore, in the frame of the HAM, the Stokes wave can be used as a unified theory for all kinds of waves, including periodic waves in deep and intermediate depths, cnoidal waves in shallow water and solitary waves in extremely shallow water.
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| [108] |
An exact and explicit solution for the valuation of American put options .
In this paper, an exact and explicit solution of the well-known Black choles equation for the valuation of American put options is presented for the first time. To the best of the author's knowledge, a closed-form analytical formula has never been found for the valuation of American options of finite maturity, although there have been quite a few approximate solutions and numerical approaches proposed. The closed-form exact solution presented here is written in the form of a Taylor's series expansion, which contains infinitely many terms. However, only about 30 terms are actually needed to generate a convergent numerical solution if the solution of the corresponding European option is taken as the initial guess of the solution series. The optimal exercise boundary, which is the main difficulty of the problem, is found as an explicit function of the risk-free interest rate, the volatility and the time to expiration. A key feature of our solution procedure, which is based on the homotopy-analysis method, is the optimal exercise boundary being elegantly and temporarily removed in the solution process of each order, and, consequently, the solution of a linear problem can be analytically worked out at each order, resulting in a completely analytical and exact series-expansion solution for the optimal exercise boundary and the option price of American put options.
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| [109] |
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