We consider the wave equation $$y_{tt}=y_{xx}+a(t,x)y, \text{ x$\in$(0,1)}, t\in (0,\infty).$$ with Dirichlet boundary conditions. I want to transform this equation to a hyperbolic system of the form $$z_t=Az_x+Bz.$$ So, I introduced the following substitutions: $z^1=y_t$, $z^2=y_x$, I obtained $$z^1_t=z^2_{x}+az^1+a_ty$$ $$z^2_t=z^1_{x}$$
The question here:
How I get rid of the $y$ in the $z^1$ formula? Is there more adequate substitution then this ? thank you.
You need to include $y$ in your $z$ vector. Set $z^1 =y, z^2 = y_t, z^3 = y_x$. Then $$ \partial_t \begin{pmatrix} y \\ y_t \\ y_x \end{pmatrix} = \begin{pmatrix} y_t \\ y_{xx} + a y \\ y_{xt} \end{pmatrix} = \begin{pmatrix} 0 & 0& 0\\ 0 & 0& 1\\ 0 & 1& 0\\ \end{pmatrix} \partial_x \begin{pmatrix} y \\ y_t \\ y_x \end{pmatrix} + \begin{pmatrix} 0 & 1& 0\\ a & 0& 0\\ 0 & 0& 0\\ \end{pmatrix} \begin{pmatrix} y \\ y_t \\ y_x \end{pmatrix}. $$ This is the form you want.