勒让德多项式在单模态反问题中的应用  

Application of Legendre Polynomials in Single-Mode Inverse Problem

作　　者：何敏 HE Min(School of Mathematics and Physics,Anqing Normal University,Anqing 246133,China)

机构地区：[1]安庆师范大学数理学院,安徽安庆246133

出　　处：《安庆师范大学学报（自然科学版）》2024年第2期58-61,共4页Journal of Anqing Normal University(Natural Science Edition)

基　　金：安徽省高等学校省级质量工程项目(2021kcszsfkc249,2023zygzts036)。

摘　　要：在所有研究杆和梁的单模态反问题的文献中,相关函数均被设为自变量的幂级数。鉴于勒让德多项式具有优良属性,本文讨论勒让德多项式在杆的一类单模态反问题中的应用。将两端弹性支承的杆的线密度函数、位移模态和轴向刚度函数均设为不同阶数勒让德多项式的线性组合,讨论杆的二阶振动方程的解析解,即当给定杆的位移模态和线密度函数时,如何确定杆的轴向刚度函数。该研究阐明了单模态反问题解的适定性,包括解的存在性、唯一性和稳定性,同时印证了将勒让德多项式作为基函数是行之有效的。In the existing literature on single-mode inverse problems of rods and beams,the correlation functions are set as power series of the independent variables.The excellent properties of Legendre polynomials are noted,and the application of Legendre polynomials to a kind of single mode inverse problem of rod was discussed.The linear density function,displacement mode and axial stiffness function of the rod with elastic supports at both ends are firstly set as a linear combination of Legendre polynomials of different orders,and the analytical expression of the second-order vibration equation of the rod was discussed.Specifically,it addresses the determination of the rod's axial stiffness function given its displacement mode and linear density function.The well-posedness of the solution to this inverse problem,including the existence,uniqueness,and stability of the solution was elucidated.At the same time,it is effective to use Legendre polynomials as the basis function is proved.

关 键 词：勒让德多项式 单模态 反问题 

分 类 号：O32[理学—一般力学与力学基础]