arXiv 0909.11502009-09-07math-phmath.MP

一个精确的解析解

An accurate analytic solution to the Thomas-Fermi equation

M. Turkyilmazoglu

备注：: 11 pages, 3 figures

摘要：

Liao（Appl.Math.Comp.144，（2003））通过一种新的分析技术，即同伦分析方法，得到了Thomas Fermi方程的显式解析解。然而，基函数和辅助线性微分算子的选择使得收敛到精确解的速度相当慢。本文的主要工作是构造新的基函数和辅助线性算子，以形成更好的同伦。为了减少计算量，适当地选择基函数和辅助算子对获得精确解是非常重要的。本文提出的同伦不仅大大减少了计算工作量，至少使同伦级数的收敛性提高了一倍，而且与Liao的同伦级数相比，扩大了同伦级数的收敛区域. Pade逼近所获得的解决方案增加的准确性，甚至到更高的程度。为了支持这一点，使用所提出的方法得到的显式的解析表达式进行了比较，数值计算和廖。

英文摘要：

The explicit analytic solution of the Thomas Fermi equation thorough a new kind of analytic technique, namely the homotopy analysis method, was employed by Liao (Appl. Math. Comp. 144, (2003)). However, the base functions and the auxiliary linear differential operator chosen were such that the convergence to the exact solution was fairly slow. New base functions and auxiliary linear operator to form a better homotopy are the main concern of the present paper. It is known that proper choice of base functions and auxiliary operator is extremely significant in gaining the exact solution in order to reduce the computational cost. The proposed homotopy here not only greatly reduces the computational efforts by at least doubling the convergence of the homotopy series, but also enlarges the convergence region of the homotopy series as compared with that of Liao. Pade approximants to the obtained solutions increase the accuracy even to a higher degree. To support this, the explicit analytical expressions obtained using the proposed approach are compared with the numerically computed ones and those of Liao.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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