文献类型：会议论文

英文题名：Solving Ordinary Differential Equations by Simplex Integrals

作者：Zhou, Yongxiong[1,2];Xiang, Shuhuang[2]

机构：[1]Guangdong Ocean Univ, Dept Math, Zhanjiang 524088, Guangdong, Peoples R China;[2]Cent So Univ Technol, Dept Appl Math, Changsha 410083, Peoples R China

会议论文集：4th International Conference on Numerical Analysis and Applications

会议日期：JUN 16-20, 2008

会议地点：Univ Rousse, Dept Numer Anal & Statist, Lozenetz, BULGARIA

主办单位：Univ Rousse, Dept Numer Anal & Statist

语种：英文

外文关键词：Polynomial approximation - Numerical methods

外文摘要：This paper is devoted to the proper discrete solution for ordinary differential equations, especially to oscillating solution. In contrast to Lipschitz condition, we define a new condition following that integral(t1)(t0) f(t)dt vertical bar <= R max(xi 1,xi 2 is an element of[t0,t1]) vertical bar f(xi(1)) - f (xi(2)) vertical bar with small R for all t(0), t(1) in the correlative intervals. Under the assumption of this new condition, we obtain a new asymptotic formula phi(nu)(t) - Q(nu-1)(t) = O((Rh)(nu)), where simplex integral phi(nu)(t) denotes integral(t)(t0) center dot center dot center dot integral(xi nu-1)(t0) integral(xi nu-2)(t0) f(xi(nu))d xi(nu)d xi(nu-1) center dot center dot center dot xi(nu-1) and the nu - 1-th polynomials Q(nu-1)(t) in which coefficient correspond to simplex integrals phi(nk)(t) with n(k) > v, k = 1: 2, ... , nu. In other words, the accuracy for approximation increasing rapidly as the integrable functions oscillate rapidly or for small step h while it's difficult for us to pursuit a polynomial to approximate a highly oscillatory function. Applying this idea of approximation to ODE, this paper surveys the algorithmic issues. If ODE has the form P(n)y((n)) + Pn-1 y((n-1)) + center dot center dot center dot + P(1)y' + P(0)y = g(t), where P-n((t)), P-n-1((t)), ... , P-0((t)) are arbitrary degree polynomials, then we can solve it by the recursive relation about simplex integrals altogether with approximate relation. Finally, numerical examples about Airy and Bessel equations illustrate the efficiency of this technique.