Heun方程、正交多项式和Toda链

发布时间：2018-06-21 07:44

本文选题：椭圆函数 + Heun方程　；参考：《河南大学》2016年硕士论文

【摘要】：为了研究黑洞中微分方程解的性质,本文主要以椭圆函数为例,利用微分方程、正交多项式、三阶递推关系和Toda链之间的关系来得到特殊函数的解。全文共分三章。在第一章,我们简要介绍微分方程在数学物理中的应用。在第二章,我们介绍已知的Fuchsian和Heun方程的一些性质。在第三章,我们以椭圆函数为例,由椭圆函数与正交多项式之间的关系,得到椭圆函数的三阶递推关系式,进而化为谐振子方程,最后将复杂的微分方程求解转化为了简单的代数求解;再由每一个三阶递推关系式都对应一个半无穷Toda链,通过求解半无穷Toda链,最终得到微分方程解的结构可由其对应的三阶递推关系式解的结构来得到。我们将其运用到了黑洞中的Lame方程和超几何方程,得到这两个特殊方程的简化式。
[Abstract]:In order to study the properties of solutions of differential equations in black holes, the solutions of special functions are obtained by using the relations among differential equations, orthogonal polynomials, third-order recursion relations and Toda chains, taking elliptic functions as an example. The full text is divided into three chapters. In the first chapter, we briefly introduce the application of differential equations in mathematics and physics. In chapter 2, we introduce some properties of the known Fuchsian and Heun equations. In the third chapter, we take elliptic function as an example, from the relation between elliptic function and orthogonal polynomial, we obtain the third order recursive relation of elliptic function, and then transform it into harmonic oscillator equation. Finally, the solution of complex differential equations is transformed into a simple algebraic solution, and then each third-order recursive relation corresponds to a semi-infinite Toda chain, and by solving the semi-infinite Toda chain, Finally, the structure of the solution of the differential equation can be obtained by the structure of the corresponding third-order recursive relation solution. We apply it to the Lame equation and the hypergeometric equation in black hole, and obtain the simplified formulas of these two special equations.
【学位授予单位】：河南大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

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