In the chapter 5 various finite difference approximations to ordinary
differential equations have been generated by making use of Taylor
series expansion of functions at some point say x0.
The same can be extended to higher dimensions in the following manner.
Descritisation of a rectangular domain
Consider a rectangular domain D: 
and 
.
Draw straight lines parallel to x-axis and y-axis as shown in the figure 6.3.1 such that 
for i = 1, 2, 3 ......n-1 and 
for j = 1, 2, 3..........m-1 where 
and 
are small positive steplengths obtained by =a/n
and =b/m.
Figure 6.3.1
Let 
be any point in the region D then the co-ordinates
xi and yj
can be obtained by
and
........(6.3.2)
where (x0,y0) is the coordinates
of the left bottom most point of the rectangle, that is (0, 0) in
the present problem . If u(x,y) is any continuous function with
all necessary derivatives exisit in D then
.............(6.3.3)
From (6.3.3) partial derivatives can be approximated
as
.............(6.3.4)
Making use of these approximations to replace partial
derivatives, the partial differential equations are converted into
difference equations and the resultant system of algebraic equations
are solved using any direct or iterative methods. Since the analytical
methods for finding solution of second order partial differential
equations depend on the type of PDE, the numerical schemes also
depend on the type of PDE.
