降线问题的阿贝尔积分方程的求解  

Method for Solving the Abel Integral Equation of the Downline Problem

作　　者：邢家省[1,2] XING Jiasheng(School of Mathematics,Beihang University,Beijing 100191,China;LMIB of the Ministry of Education,Beihang University,Beijing 100191,China)

机构地区：[1]北京航空航天大学数学科学学院,北京100191 [2]北京航空航天大学“数学、信息与行为”教育部重点实验室,北京100191

出　　处：《吉首大学学报（自然科学版）》2022年第6期6-10,79,共6页Journal of Jishou University(Natural Sciences Edition)

基　　金：国家自然科学基金资助项目(12171022);北京航空航天大学校级重大教改项目(凡舟教育基金团队建设202109—202412)。

摘　　要：考虑给定下降时间函数的降线问题的求解,将降线问题转化为阿贝尔积分方程求解问题.对于无限区间上的积分方程,介绍了阿贝尔运用拉普拉斯变换求解积分方程的过程,给出了求解公式;对于有限区间上的积分方程,采用阿贝尔积分变换法进行求解,运用累次积分交换积分次序,由一个定积分的恒等式得出求解公式,并将积分方程的求解公式应用于等时降线问题的求解,通过求解等时降线问题的微分方程,证明了等时降线是一条倒摆线.For the solution of the downline problem with a given descent time function,the problem is transformed into an Abel integral equation solution problem.For the integral equation on infinite interval,this paper introduces Abel's method and the process of solving integral equation with Laplace transformation,and gives the solution formula.For the limited range of integral equation,this paper uses the Abel integral transformation method to solve the integral equation,uses the iterated integral to exchange integral sequence,and obtains the solution formula from a definite integral identity.The solution formula of the integral equation is applied to the isochronous curve problem.The isochron is proved be an inverted cycloid with the solution of the differential equation of the isochronous curve problem.

关 键 词：降线问题 阿贝尔积分方程 阿贝尔积分变换 等时降线 微分方程 摆线 

分 类 号：O175.29[理学—数学]