I have the following fractional partial differetial equation
$$ \dfrac{\partial x(z,t)}{\partial t}=\dfrac{\partial ^\alpha x(z,t)}{\partial z^\alpha} $$ with $1\leq \alpha \leq 2$ and $0<z<1$. The Caputo definition is considered for the fractional derivative.
How to chose the space for $x(z,t)$?