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

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.