I am trying to solve a 1-D wave equation PDE. My first step was to find all the initial conditions I could which is proving more difficult than I thought. The problem is below.
The initial conditions I found were: $$\frac{\partial}{\partial x}u(-\infty - 12 , 0) = 0$$ $$\frac{\partial}{\partial x}u(12 - 17, 0) = \frac{8}{5}$$ $$\frac{\partial}{\partial x}u(17 - 22, 0) = -\frac{8}{5}$$ $$\frac{\partial}{\partial x}u(22 - \infty, 0) = 0$$
Are these correct/are there anymore I have missed? And how would I use this in an equation since they all are within a "range", can I just pick any $x$ value between $12$ and $17$ and write the derivative at that point is $\frac{8}{5}$ and use it to find a particular solution?
example: $$\frac{\partial}{\partial x}u(13, 0) = \frac{8}{5}$$
You need to define the entire function at $t=0$, not just the slopes.
The initial condition is piece-wise linear and has this general form
$$ u(x,0) = A(x) = \begin{cases} 0, & x < 12 \\ ax + b, & 12 < x < 17 \\ cx + d, & 17 < x < 22 \\ 0, & x > 22 \end{cases} $$
You can find the constants $a,b,c,d$ using the given function values at $x=12,17,22$
In fact, you've already determined that $a=8/5$ and $c=-8/5$