So far, we have developed finite difference equations only for the case where the coefficients c and are constants. In practice, these coefficients often depend on space, on time, or on the field variable . To indicate some of the many techniques used to deal with variable coefficient FDEs, we consider the following two cases.
Case 1---c and depend explicitly on x and t: In this case, if we return to the integral conservation law, Eq. 10, and proceed with similar arguments as in the constant coefficient case, we obtain an explicit FDE of the form
The formulations of the implicit and Crank--Nicolson methods for the case and are left as exercises.