数理方程：线性非齐次方程在齐次边界条件下的解法

Posted 2020-06-30 羽夜

更新：28 MAR 2016

以波动方程为例

\(\dfrac{\partial^2u}{\partial t^2}=a^2\dfrac{\partial^2 u}{\partial x^2}+f(x,t),\qquad 0<x<l,\quad t>0\)

边界条件：齐次

\(u|_{x=0}=u|_{x=l}=0,\qquad t>0\)

初始条件：任意（最后用到Fourier变换）

\(u|_{t=0}=\varphi(x),\ \left.\dfrac{\partial u}{\partial t}\right|_{t=0}=\psi(x),\qquad 0 \leqslant x \leqslant l\)

解法：分解待求函数\(u(x,t)\)。设

\(u(x,t)=v(x,t)+w(x,t)\)

将方程非齐次项归结到\(v(x,t)\)，将初始条件归结到\(w(x,t)\)，即

对于\(v(x,t)\)

\(\dfrac{\partial^2v}{\partial t^2}=a^2\dfrac{\partial^2 v}{\partial x^2}+f(x,t),\qquad 0<x<l,\quad t>0\)

\(v|_{x=0}=v|_{x=l}=0,\qquad t>0\)

\(v|_{t=0}=0,\ \left.\dfrac{\partial v}{\partial t}\right|_{t=0}=0,\qquad 0 \leqslant x \leqslant l\)

对于\(w(x,t)\)

\(\dfrac{\partial^2w}{\partial t^2}=a^2\dfrac{\partial^2 w}{\partial x^2}+f(x,t),\qquad 0<x<l,\quad t>0\)

\(w|_{x=0}=w|_{x=l}=0,\qquad t>0\)

\(w|_{t=0}=\varphi(x),\ \left.\dfrac{\partial w}{\partial t}\right|_{t=0}=\psi(x),\qquad 0 \leqslant x \leqslant l\)