Write the call price as a function C(K, T) of strike K and expiry T. The square of the local volatility is the derivative with respect to expiry plus an interest adjustment, divided by K² / 2 times the second derivative with respect to strike. Dupire is said to have given a forward partial differential equation in strike and expiry in a January 1994 article in Risk.
The form of the formula and a worked example are in the article on local volatility models.