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同伦分析方法:一种新的求解非线性问题的近似解析方法 总被引:9,自引:0,他引:9
本文描述了一种称为“同伦分析方法”(HAM)的新的求解非线性问题的近似解析方法之基本思想·不同于摄动展开方法,“同伦分析方法”的有效性不依赖于所研究的非线性方程中是否含有小参数·因此,该方法提供了一个强有力的分析非线性问题的新工具·作为示例,我们应用一个典型的非线性问题来说明该方法的有效性及其巨大潜力· 相似文献
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Jundi Ding Hongyou Yin 《高等学校计算数学学报(英文版)》2007,16(2):155-163
In this paper, we present a new homotopy method for the nonlinear complementarity problems. Without the regularity or non-singulary assumptions for▽F(x), we prove that our homotopy equations have a bounded solution curve. The numerical tests confirm the efficiency of our proposed method. 相似文献
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A class of nonlinear boundary-value problems containing a parameter is studied analytically and numerically. It is shown that under certain circumstances there are two families of solutions when the parameter tends to zero; one family comprises small solutions and is obtained by regular perturbations, while the other family comprises finite solutions incorporating boundary and interior layers. It is shown by numerical integration that the two families are smooth continuations of each other when the parameter passes through finite values. 相似文献
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罗炯兴 《数学的实践与认识》2019,(9)
研究了结合变量替换应用同伦分析方法,去求解二阶非线性微分方程的两点边值问题,并得到了逼近解析解的函数级数形式.给出了应用同伦分析方法求解二阶非线性问题的三个实例,显示了同伦分析方法可以比较有效地求解非线性问题. 相似文献
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In this paper we present a homotopy continuation method for finding the Karush-Kuhn-Tucker point of a class of nonlinear non-convex programming problems. Two numerical examples are given to show that this method is effective. It should be pointed out that we extend the results of Lin et al. (see Appl. Math. Comput., 80(1996), 209-224) to a broader class of non-convex programming problems. 相似文献
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奇异非线性Sturm-Liouville问题的正解 总被引:12,自引:0,他引:12
本文研究奇异非线性Sturm-Liouville问题其中(Lφ)(x)=(p(x)φ′(x))′+g(x)φ(x),并且允许h(x)在x=0和x=1奇异。应用锥理论和不动点指数方法,在与相应的线性算子第一特征值有关的条件下获得了正解的存在性结果,本质地推广和改进了文献[1-9]中的主要结论。 相似文献
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利用M(o)nch不动点定理,在更广泛的的条件下,证明了Banach空间中Sturm-Liouville问题解的存在性. 相似文献
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Zhong-Hua Yang 《计算数学(英文版)》1988,6(1):1-6
The nonlinear singular problem $f(u)=0$ is considered. Here $f$ is a $C^3$ mapping from $E^n$ to $E^n$. The Jacobian matrix $f'(u)$ is singular at the solution $u^*$ of $f(u)=0$. A new acceleration method in the homotopy Newton's continuation is proposed. The quadratic convergence of the new algorithm is proved. A numerical example is given. 相似文献
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非线性奇异边值问题的正解 总被引:1,自引:0,他引:1
路慧芹 《应用泛函分析学报》2002,4(3):217-221
利用锥映射的不动点指数定量,研究了一类非线性奇异边值问题多个正解的存在性问题。在构造的解的存在条件之下,证明了奇异边值问题至少有两个正解的存在性定理。 相似文献
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In this work, we consider the second-order discontinuous equation in the real line,with \({k > 0}\) and \({f : \mathbb{R}^{3} \rightarrow \mathbb{R}}\) an \({L^{1}}\)-Carathéodory function. The existence of homoclinic solutions in presence of not necessarily ordered lower and upper solutions is proved, without periodicity assumptions or asymptotic conditions. Some applications to Duffing-like equations are presented in last section.
相似文献
$$u^{\prime \prime}(t)-ku(t) = f( t, u(t), u^{\prime}(t)), \quad a.e.t \in \mathbb {R},$$
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In an arbitrary bounded 2‐D domain, a singular perturbation approach is developed to analyze the asymptotic behavior of several biharmonic linear and nonlinear eigenvalue problems for which the solution exhibits a concentration behavior either due to a hole in the domain, or as a result of a nonlinearity that is nonnegligible only in some localized region in the domain. The specific form for the biharmonic nonlinear eigenvalue problem is motivated by the study of the steady‐state deflection of one of the two surfaces in a Micro‐Electro‐Mechanical System capacitor. The linear eigenvalue problem that is considered is to calculate the spectrum of the biharmonic operator in a domain with an interior hole of asymptotically small radius. One key novel feature in the analysis of our singularly perturbed biharmonic problems, which is absent in related second‐order elliptic problems, is that a point constraint must be imposed on the leading order outer solution to asymptotically match inner and outer representations of the solution. Our asymptotic analysis also relies heavily on the use of logarithmic switchback terms, notorious in the study of Low Reynolds number fluid flow, and on detailed properties of the biharmonic Green’s function and its associated regular part near the singularity. For a few simple domains, full numerical solutions to the biharmonic problems are computed to verify the asymptotic results obtained from the analysis. 相似文献
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Based on the homotopy analysis method, the nonlinear vibration of porous functionally graded material (FGM) conveying pipes under generalized boundary conditions was studied. Based on the power-law distribution of the FGM and the Voigt model, the physical properties of the porous pipe material were described. Under the Euler-Bernoulli beam theory and the von Kármán nonlinear theory, and by means of Hamilton’s variational principle, the dynamic control equations and generalized boundary conditions for porous FGM conveying pipes were established. The homotopy analysis method was used to solve the nonlinear vibration characteristics of the porous FGM conveying pipe under generalized boundary conditions. The numerical results show that, the translation spring has little effect on the critical velocity of instability, while the rotation spring increases the critical velocity of instability, making the system more stable; in the nonlinear system, the viscoelastic coefficient does not change the critical velocity; the pipe length, the power-law exponent and the porosity all influence the nonlinear free vibration of the porous FGM conveying pipe. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved. 相似文献
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In this paper we consider parametric nonlinear elliptic problems driven by the p-Laplacian differential operator and with the parameter ?? near ?? 1, the principal eigenvalue of the negative Dirichlet p-Laplacian (near resonance). We consider both cases when ???<??? 1 (near resonance from the left) and when ???>??? 1 (near resonance from the right). Our approach combines variational methods based on the critical point theory, together with truncation techniques and Morse theory. 相似文献
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We classify global Lipschitz solutions to two-phase free boundary problems governed by concave fully nonlinear equations as either two-plane solutions or solutions to a one-phase problem. © 2019 Wiley Periodicals, Inc. 相似文献
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《数学物理学报(A辑)》2010,(6)
考察了非线性三阶周期边值问题的正解.在适当的变换下此问题被转化为一个积分方程.利用锥上的Guo-Lakshmikantham不动点指数定理建立了单个或多重正解的存在性. 相似文献