共查询到20条相似文献,搜索用时 62 毫秒
1.
John P. Boyd 《高等学校计算数学学报(英文版)》2011,4(2):142-157
The three-dimensional spherical polytropic Lane-Emden problem is
$y_{rr}+(2/r) y_{r} + y^{m}=0, y(0)=1, y_{r}(0)=0$ where $m
\in [0, 5]$ is a constant parameter. The domain is $r \in [0, \xi]$
where $\xi$ is the first root of $y(r)$. We recast this as a
nonlinear eigenproblem, with three boundary conditions and $\xi$ as
the eigenvalue allowing imposition of the extra boundary condition,
by making the change of coordinate $x \equiv r/\xi$: $y_{xx}+(2/x)
y_{x}+ \xi^{2} y^{m}=0, y(0)=1, y_{x}(0)=0,$ $y(1)=0$. We find
that a Newton-Kantorovich iteration always converges from an
$m$-independent starting point $y^{(0)}(x)=\cos([\pi/2] x),
\xi^{(0)}=3$. We apply a Chebyshev pseudospectral method to
discretize $x$. The Lane-Emden equation has branch point
singularities at the endpoint $x=1$ whenever $m$ is not an integer;
we show that the Chebyshev coefficients are $a_{n} \sim
constant/n^{2m+5}$ as $n \rightarrow \infty$. However, a
Chebyshev truncation of $N=100$ always gives at least ten decimal
places of accuracy — much more accuracy when $m$ is an integer.
The numerical algorithm is so simple that the complete code (in
Maple) is given as a one page table. 相似文献
2.
By using some NCP functions, we reformulate the extended linear complementarity problem as a nonsmooth equation. Then we propose
a self-adaptive trust region algorithm for solving this nonsmooth equation. The novelty of this method is that the trust region
radius is controlled by the objective function value which can be adjusted automatically according to the algorithm. The global
convergence is obtained under mild conditions and the local superlinear convergence rate is also established under strict
complementarity conditions.
This work is supported by National Natural Science Foundation of China (No. 10671126) and Shanghai Leading Academic Discipline
Project (S30501). 相似文献
3.
It is well known that a linear complementarity problem (LCP) can be formulated as a system of nonsmooth equations F(x) = 0, where F is a map from Rninto itself. Using the aggregate function, we construct a smooth Newton homotopy H(x,t) = 0. Under certain assumptions, we prove the existence of a smooth path defined by the Newton homotopy which leads to a solution of the original problem, and study limiting properties of the homotopy path. 相似文献
4.
Jihua Yang 《Journal of Nonlinear Modeling and Analysis》2020,2(3):431-445
This paper is devoted to study the following complete hyper-elliptic integral of the first kind
$$J(h)=\oint\limits_{\Gamma_h}\frac{\alpha_0+\alpha_1x+\alpha_2x^2+\alpha_3x^3}{y}dx,$$
where $\alpha_i\in\mathbb{R}$, $\Gamma_h$ is an oval contained in the level set $\{H(x,y)=h, h\in(-\frac{5}{36},0)\}$ and $H(x,y)=\frac{1}{2}y^2-\frac{1}{4}x^4+\frac{1}{9}x^9$. We show that the 3-dimensional real vector spaces of these integrals are Chebyshev for $\alpha_0=0$ and Chebyshev with accuracy one for $\alpha_i=0\ (i=1,2,3)$. 相似文献
5.
Zhong ChenWei Jiang 《Applied mathematics and computation》2011,217(19):7790-7798
The purpose of this paper is to obtain the approximation solution of linear and strong nonlinear weakly singular Volterra integral equation of the second kind, especially for such a situation that the equation is of nonsmooth solution and the situation that the problem is a strong nonlinear problem. For this purpose, we firstly make a transform to the equation such that the solution of the new equation is as smooth as we like. Through modifying homotopy perturbation method, an algorithm is successfully established to solve the linear and nonlinear weakly singular Volterra integral equation of the second kind. And the convergence of the algorithm is proved strictly. Comparisons are made between our method and other methods, and the results reveal that the modified homotopy perturbation is effective. 相似文献
6.
Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem 下载免费PDF全文
Elsayed M. E. Zayed 《偏微分方程(英文版)》2016,29(1):59-70
In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator\begin{equation*}Au(x)=-\Delta \Delta u(x)+V(x)u(x),\end{equation*}for all $x\in R^{n}$, in the Hilbert space $H=L_{2}(R^{n},H_{1})$ with the operator potential $V(x)\in C^{1}(R^{n},L(H_{1}))$, where $L(H_{1})$ is the space of all bounded linear operators on the Hilbert space $H_{1}$, while $\Delta \Delta u$\ is the biharmonic differential operator and\begin{equation*}\Delta u{=-}\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det g}}\frac{\partial }{{\partial x_{i}}}\left[ \sqrt{\det g}g^{-1}(x)\frac{\partial u}{{\partial x}_{j}}\right]\end{equation*}is the Laplace-Beltrami differential operator in $R^{n}$. Here $g(x)=(g_{ij}(x))$ is the Riemannian matrix, while $g^{-1}(x)$ is the inverse of the matrix $g(x)$. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation $Au=-\Delta \Delta u+V(x)u(x)=f(x)$ in the Hilbert space $H$ where $f(x)\in H$ as an application of the separation approach. 相似文献
7.
A superlinearly convergent ODE-type trust region algorithm for nonsmooth nonlinear equations 总被引:1,自引:0,他引:1
Ou Yigui 《Journal of Applied Mathematics and Computing》2006,22(3):371-380
This paper presents a new trust region algorithm for solving nonsmooth nonlinear equation problems which posses the smooth plus non-smooth decomposition. At each iteration, this method obtains a trial step by solving a system of linear equations, hence avoiding the need for solving a quadratic programming subproblem with a trust region bound. From a computational point of view, this approach may reduce computational effort and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and locally super-linearly convergent. Some numerical examples are reported. 相似文献
8.
9.
应用锥压缩锥拉伸不动点定理和Leray-Schauder 抉择定理研究了一类具有P-Laplace算子的奇异离散边值问题$$\left\{\begin{array}{l}\Delta[\phi (\Delta x(i-1))]+ q_{1}(i)f_{1}(i,x(i),y(i))=0, ~~~i\in \{1,2,...,T\}\\\Delta[\phi (\Delta y(i-1))]+ q_{2}(i)f_{2}(i,x(i),y(i))=0,\\x(0)=x(T+1)=y(0)=y(T+1)=0,\end{array}\right.$$的单一和多重正解的存在性,其中$\phi(s) = |s|^{p-2}s, ~p>1$,非线性项$f_{k}(i,x,y)(k=1,2)$在$(x,y)=(0,0)$具有奇性. 相似文献
10.
In this article, we investigate a nonlinear system of differential equations with two parameters
$$\left\{ \begin{array}{l} x"(t)=a(t)x(t)-\lambda f(t, x(t), y(t)),\y"(t)=-b(t)y(t)+\mu g(t, x(t), y(t)),\end{array}\right.$$
where $a,b \in C(\textbf{R},\textbf{R}_+)$ are $\omega-$periodic for some period $\omega > 0$, $a,b \not\equiv 0$, $f,g \in C(\textbf{R} \times \textbf{R}_+ \times \textbf{R}_+ ,\textbf{R}_+)$ are $\omega-$periodic functions in $t$, $\lambda$ and $\mu$ are positive parameters. Based upon a new fixed point theorem, we establish sufficient conditions for the existence and uniqueness of positive periodic solutions to this system for any fixed $\lambda,\mu>0$. Finally, we give a simple example to illustrate our main result. 相似文献
11.
This paper presents a new trust region algorithm for solving a class of composite nonsmooth optimizations. It is distinguished by the fact that this method does not enforce strict monotonicity of the objective function values at successive iterates and that this method extends the existing results for this type of nonlinear optimization with smooth, or piecewise smooth, or convex objective functions or their composition. It is proved that this algorithm is globally convergent under certain conditions. Finally, some numerical results for several optimization problems are reported which show that the nonmonotonic trust region method is competitive with the usual trust region method. 相似文献
12.
This paper preasents a new trust region algorithm for solving a class of composite nonsmooth optimizations.It is distinguished by the fact that this method does not enforce strict monotonicity of the objective function values at successive itereates and that this method extends the existing results for this type of nonlinear optimization with smooth ,or piecewis somooth,or convex objective functions or their composition It is pyoved that this algorithm is globally convergent under certain conditions.Finally,some numerical results for several optimization problems are reported which show that the nonmonotonic trust region method is competitive with the usual trust region method. 相似文献
13.
The authors study oscillatory property of nonlinear functional differential equation
$L_nx(t)+p(t)f(x(t),x(g(t)))=r(t)$(1)
where L_nx(t) is an n-th order linear differential operator defined by
$L_0x(t)=x(t)$,
$L_kx(t)=\frac{d}{dt}(a_k-1(t)L_k-1x(t)),k=1,2,\cdots,n.$
Sufficient conditions are obtained which guarantee that all continuable solutions of (1) are oscillatory or tend to zero as t\rightarrow \infinity. 相似文献
14.
Changfeng Ma 《Journal of Global Optimization》2010,48(2):241-261
The nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the solution of a nonsmooth system of equations. In this paper, we propose a new smoothing and regularization
Newton method for solving nonlinear complementarity problem with P
0-function (P
0-NCP). Without requiring strict complementarity assumption at the P
0-NCP solution, the proposed algorithm is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore,
the algorithm has local quadratic convergence under mild conditions. Numerical experiments indicate that the proposed method
is quite effective. In addition, in this paper, the regularization parameter ε in our algorithm is viewed as an independent variable, hence, our algorithm seems to be simpler and more easily implemented
compared to many previous methods. 相似文献
15.
Methods are developed and analyzed for estimating the distance to a local minimizer of a nonlinear programming problem. One estimate, based on the solution of a constrained convex quadratic program, can be used when strict complementary slackness and the second-order sufficient optimality conditions hold. A second estimate, based on the solution of an unconstrained nonconvex, nonsmooth optimization problem, is valid even when strict complementary slackness is violated. Both estimates are valid in a neighborhood of a local minimizer. An active set algorithm is developed for computing a stationary point of the nonsmooth error estimator. Each iteration of the algorithm requires the solution of a symmetric, positive semidefinite linear system, followed by a line search. Convergence is achieved in a finite number of iterations. The error bounds are based on stability properties for nonlinear programs. The theory is illustrated by some numerical examples. 相似文献
16.
Mandeep Singh Amit K Verm Ravi P Agarwal 《Journal of Applied Analysis & Computation》2019,9(4):1242-1260
In this article, we propose a novel modification to Quasi-Newton method, which is now a days popularly known as variation iteration method (VIM) and use it to solve the following class of nonlinear singular differential equations which arises in chemistry $-y''(x)-\frac{\alpha}{x}y''(x)=f(x,y),~x\in(0,1),$ where $\alpha\geq1$, subject to certain two point and three point boundary conditions. We compute the relaxation parameter as a function of Bessel and the modified Bessel functions. Since rate of convergence of solutions to the iterative scheme depends on the relaxation parameter, thus we can have faster convergence. We validate our results for two point and three point boundary conditions. We allow $\partial f/\partial y$ to take both positive and negative values. 相似文献
17.
提出了求解一类带一般凸约束的复合非光滑优化的信赖域算法 .和通常的信赖域方法不同的是 :该方法在每一步迭代时不是迫使目标函数严格单调递减 ,而是采用非单调策略 .由于光滑函数、逐段光滑函数、凸函数以及它们的复合都是局部Lipschitz函数 ,故本文所提方法是已有的处理同类型问题 ,包括带界约束的非线性最优化问题的方法的一般化 ,从而使得信赖域方法的适用范围扩大了 .同时 ,在一定条件下 ,该算法还是整体收敛的 .数值实验结果表明 :从计算的角度来看 ,非单调策略对高度非线性优化问题的求解非常有效 相似文献
18.
Zhong-Hua Yang 《计算数学(英文版)》1984,2(3):201-209
In this paper we consider the nonlinear operator equation $\lambda x=Lx+G(\lambda,x)$ where $L$ is a closed linear operator of $X-›X, X$ is a real Banach Space, with a simple eigenvalue $\lambda_0\neq 0$. We discretize its Liapunov-Schmidt bifurcation equation instead of the original nonlinear operator equation and estimate the approximating order of our approximate solution to the genuine solution. Our method is more convenient and more accurate. Meanwhile we put forward several abstract Newton-type iterative schemes, which are more efficient for practical computation, and get the result of their super-linear convergence. 相似文献
19.
Antal Járai 《Aequationes Mathematicae》2003,65(3):236-266
Summary. We prove that - under certain conditions - measurable solutions $f$ of the functional equation $f(x)=h(x,y,f(g_{1}(x,y)),\ldots,f(g_{n}(x,y))),\quad(x,y)\in D \subset \mathbb{R}^{s} \times \mathbb{R}^{l}$ are continuous, even if $1\le l\le s$. As a tool we introduce new classes of functions which - roughly speaking - interpolate between continuous and Lebesgue measurable functions. Connection between these classes are also investigated. 相似文献
20.
Chuang-yin Dang 《计算数学(英文版)》2006,24(6):711-718
A mapping f:Z~n→R~n is said to possess the direction preserving property if fi(x)>0implies fi(y)≥0 for any integer points x and y with ‖x-y‖∞≤1.In this paper,a simplicial algorithm is developed for computing an integer zero point of a mappingwith the direction preserving property.We assume that there is an integer point x~0 withc≤x~0≤d satisfying that max_(1≤i≤n)(x_i-x_i~0)fi(x)>0 for any integer point x withf(x)≠0 on the boundary of H={x∈R~n|c-e≤x≤d e},where c and d are twofinite integer points with c≤d and e=(1,1,…1)~∈R~n.This assumption is impliedby one of two conditions for the existence of an integer zero point of a mapping with thepreserving property in van der Laan et al.(2004).Under this assumption, starting at x~0,the algorithm follows a finite simplicial path and terminates at an integer zero point ofthe mapping.This result has applications in general economic equilibrium models withindivisible commodities. 相似文献