How can I change the dependent variable to turn this nonlinear equation to the heat equation?

Question

I have this equation $\frac{\partial V}{\partial t}=\frac{\partial^2 V}{\partial x^2}+(\frac{\partial V}{\partial x})^2$ and i want to use the method of variable change to arrive to this equation $\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}. $The initial conditions are $V=0,t=0,0<=x<=1 $ and the boundary conditions are $ \frac{\partial V}{\partial x}=1,x=0,t>0 $ and $V=0,x=1,t>0.$ How can i approach the problem?

Answer 1

If you put $u=e^V$, then $$ u_t = e^V V_t $$ and $$ u_{xx} = (V_x e^V)_x = (V_{xx}+V_x^2)e^{V}, $$ so $$ V_t = V_{xx}+V_x^2 \implies u_t = u_{xx}. $$