For example I have a PDE with the dependent variable $u$ and the independent variables $x$ and $y$ . Suppose I have the change of variables that $v=u_x$ , I know the condition of $u(x,a)=f(x)$ will change to the condition of $v(x,a)=f_x(x)$ .
But when the condition is for example of the form $u(f(x),g(x))=h(x)$ , applying the change of variables that $v=u_x$ , the condition will change to what?