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671.
A modification of Lagrange interpolation based on the zeros of the Chebyshev polynomial of the second kind is constructed, which interpolates at many ofgiven data. Thus, for this node-system the main result gives an affimative answer to a problem suggested by Bernstein in 1930. Moreover, our modification has a Timan-Gopengauz type approximation rate. 相似文献
672.
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
673.
1IntroductionLetL:=L:l-1,11bethefunctionspacewhichconsist8offunctionsfsatisfyingwherep(t)isaweightfunction,denotethenormthenwecanknowfroml11that.hereexistsanorthogonalsetofpolynodrialsyto,yt1,...,ytnt...andEspeciallyweknowfrom[1]thatifweightfunctionp(f)maybechosenasw1(t)=(1-t')-t,wz(t)=(1-f')i,ws(t)=(1-t)t(1 t)-1,w`(t)=(1-f)-t(1 t)i,thecorrespondingrespectiveorthogonalpolynondalsareTJ')(t)=cosno,Tj')(t)=sin(n 1)o/sino,T1')(t)=sinngo/sint,T;')(f)=cosnge/cos9,wheret=coso,thenby(1.3)theyarethed… 相似文献
674.
In this paper we study Banach spaces that admit weighted Chebyshev centres for finite sets. Such spaces have been extensively studied recently by Veselý using the approach of finitely intersecting balls. Following his approach we exhibit large classes of Banach spaces that have this property. Certain stability results for spaces of vector valued continuous and Bochner integrable functions are also obtained. 相似文献
675.
The magnetohydrodynamics (MHD) Falkner-Skan flow of the Maxwell fluid is studied. Suitable transform reduces the partial differential equation into a nonlinear three order boundary value problem over a semi-infinite interval. An efficient approach based on the rational Chebyshev collocation method is performed to find the solution to the proposed boundary value problem. The rational Chebyshev collocation method is equipped with the orthogonal rational Chebyshev function which solves the problem on the semi-infinite domain without truncating it to a finite domain. The obtained results are presented through the illustrative graphs and tables which demonstrate the affectivity, stability, and convergence of the rational Chebyshev collocation method. To check the accuracy of the obtained results, a numerical method is applied for solving the problem. The variations of various embedded parameters into the problem are examined. 相似文献
676.
Raed S. Batahan 《Linear algebra and its applications》2006,419(1):82-92
In this paper, an extension of the Hermite matrix polynomials is introduced. Some relevant matrix functions appear in terms of the two-variable Hermite matrix polynomials. Furthermore, in order to give qualitative properties of this family of matrix polynomials, the Chebyshev matrix polynomials of the second kind are introduced. 相似文献
677.
Mehdi Dehghan 《Numerical Methods for Partial Differential Equations》2006,22(1):220-257
Certain problems arising in engineering are modeled by nonstandard parabolic initial‐boundary value problems in one space variable, which involve an integral term over the spatial domain of a function of the desired solution. Hence, in the past few years interest has substantially increased in the solutions of these problems. As a result numerous research papers have also been devoted to the subject. Although considerable amount of work has been done in the past, there is still a lack of a completely satisfactory computational scheme. Also, there are some cases that have not been studied numerically yet. In the current article several approaches for the numerical solution of the one‐dimensional parabolic equation subject to the specification of mass, which have been considered in the literature, are reported. Finite difference methods have been proposed for the numerical solution of the new nonclassic boundary value problem. To investigate the performance of the proposed algorithm, we consider solving a test problem. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006 相似文献
678.
Yafang Gong 《分析论及其应用》2006,22(4)
Based on Bernstein's Theorem, Kalandia's Lemma describes the error estimate and the smoothness of the remainder under the second part of Holder norm when a Holder function is approximated by its best polynomial approximation. In this paper, Kalandia's Lemma is generalized to the cases that the best polynomial is replaced by one of its four kinds of Chebyshev polynomial expansions, the error estimates of the remainder are given out under Holder norm or the weighted Holder norms. 相似文献
679.
Zinoviy Grinshpun 《Journal of Mathematical Analysis and Applications》2002,272(1):349-361
The paper considers the mutual relationship of oscillations of the Bernstein-Szegö orthogonal polynomials of different kinds in the boundaries that are determined by the weight functions with indication of Chebyshev's alternances. 相似文献
680.
刘静 《数学的实践与认识》2006,36(10):111-118
在sm a le点估计理论引导下,利用优序列方法,研究γ-条件下,变形chebyshev迭代方法在求解Banach空间中非线性方程F(x)=0时的收敛性问题,并给出了误差估计,而且通过一个积分方程实例比较了它和N ew ton法,导数超前计值的变形N ew ton法,避免导数求逆的变形N ew ton法的每步误差. 相似文献