傅里叶级数的若干应用

Some Applications of Fourier Series

通过几类经典实例,探讨傅里叶级数在数项级数求和、常微分方程和波动方程中的应用。通过选取合理的函数,将其展开成傅里叶级数,傅里叶级数在某个特殊点的值求得数项级数的和。考虑二阶常微分方程的通解的结构具有傅里叶级数的形式,通过待定系数法,求得微分方程的通解。对于具有初边值问题的波方程,通过变量替换法,得出具有傅里叶级数的非平凡的特解,利用逐项求导和积分的方法,得出傅里叶系数,从而得出该波方程的特解。

By using several classic examples,this paper discusses the application of Fourier series in the summation of Multinomial series,ordinary differential equations and wave equations.By choosing a reasonable function and expanding it into a Fourier series,the value of the Fourier series at a particular point is found as a sum of several terms of the series.The general solution of a second-order ordinary differential equation is considered as the form of a Fourier series,by the method of coefficients to be determined,it is derived.For a wave equation with an initial margin problem,a nontrivial special solution with a Fourier series is derived by the variable substitution method,and the Fourier coefficients are derived by utilizing a term-by-term method of derivatives and integrals to arrive at a special solution of this wave equation.