Example 5 Consider in polar coordinates, i.e.,
(See Figure 1.7.)
Then we have
Example 6 In cylindrical coordinates the PDE can be written
We can set up a grid system as shown with , and represent derivatives by either central or forward differences. How do we deal with the point ?
Therefore at we use the PDE in Cartesian form, i.e. (1.5)
Thus in 2D at we have
However, the and axes are two perpendicular axes and the generalisation of equation (1.6) gives
where is the mean value of along , i.e., on . When , equation (1.4) becomes
and the question arises as to how do we deal with the term at ?
The answer is that we use l'Hopital's rule, i.e.
(N.B. at if problem is symmetrical)