Hi I am trying to calculate $E(\phi(X(1))$ with $X(t)$ satisfies the following 

$$d(X(t))=\sigma(X(t))dW(t)$$

$$X(0)=x_0$$

where $\phi$ and $\sigma$ are arbitrary functions and $W(t)$ is Brownian motion. I think I should apply Feynman–Kac formula but not sure about how to deal with terminal conditions. Is the following right?

$$U_t+1/2\;\sigma^2 U_{xx}=0$$

$$U(x,1)=\phi(x) $$

Is the terminal condition supposed to be a function? Say, if $\phi(x)=log(x)$ then $U(x,1)=log(x)$.  I am confused...