
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 ?
and in Cartesian coordinates it may be written We write and At a general point e.g. 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) Therefore, at and Therefore
[You can now do Question 1 of the Examples]
