New initial condition when changing variables in uniform heat equation

Question

I will be very glad of your help if you explain to me what it follows from the equality to zero $\phi = 0$ in the initial condition (highlighted in red in the figure) after the indicated chaching of variables from v to $\phi$?

Answer 1

If $v(x,0)=0$ for all $x$, then $$ \dfrac{\partial v}{\partial x}(x,0) = \dfrac{d}{dx} \Bigl( v(x,0) \Bigr) = \dfrac{d}{dx} \Bigl( 0 \Bigr) =0 $$ for all $x$ too. So $$ \varphi(x,0) = v(x,0) - \dfrac{1}{h} \dfrac{\partial v}{\partial x}(x,0) = 0 $$ for all $x$.