Suppose i have wave equation and its BC and IC
\begin{align}
u_{tt}-c^2u_{xx}=0,\quad 0\lt x \lt \pi\\
\text{Neumann BC :}\\
u_x(0,t)=\sin t\\
u_x(\pi, t)=e^{-t^2}\\
\text{IC :}\\
u(x,0)=\sin x\\
u_t(x,0)=0
\end{align}
In my opinion, it's hard to solve this pde with eigenfunction expansion. I have to use another tools like green function, or integral transform. But, the instructor in the book forced me to solve this with eigenfunction expansion.
I know that, for inhomogeneous Dirichlet on heat equation we can use steady state concept to make them homogeneous BC. But my problem is wave equation with neuman condition. Is it mean i can solve my problem with the eigenfunction and i have to make the BC homogeneous first?
Sorry for my newbie question, cz i have condition to be self study without teacher. I feel difficult for this.
Best Answer
Find a new equation for $$ v(x,t) = -\frac{(x-\pi)^2}{2\pi}\sin t - \frac{x^2}{2\pi} e^{-t^2}+u(x,t). $$ This function satisfies homogeneous endpoint equations in $x$: $$ v_x(0,t) = -\sin t+u_x(0,t)=0 \\ v_x(\pi,t) = -e^{-t^2}+u_x(\pi,t)= 0 $$ This $v$ will satisfy an inhomogeneous equation $$ v_{tt}-c^2v_{xx}=(\cdots)+u_{tt}-c^2u_{xx} = (\cdots) $$ And the initial conditions will change as well. However, the equation in $v$ can be solved using eigenfunctions because $v_x(0,t)=v_x(\pi,t)=0$.