416 Chapter 7 Numerical Methods
Figure 3 Numerical solution of Eqs. (3)–(6).Example.
Set up the replacement equations for the problem
∂^2 u
∂x^2 +∂^2 u
∂y^2 =^16 (u−^1 ),^0 <x<^1 ,^0 <y<^1 , (9)
u(x, 0 )= 0 , u(x, 1 )= 0 , 0 <x< 1 , (10)
u( 0 ,y)= 0 , u( 1 ,y)= 0 , 0 <y< 1. (11)We may use the same numbering as in the first example (Fig. 2). At each mesh
point, the replacement is
uN+uS+uE+uW− 4 ui
( x)^2= 16 (ui− 1 ). (12)Because x= 1 /4,( 1 / x)^2 =16, and the typical replacement equation be-
comes
uN+uS+uE+uW− 4 ui=ui− 1 ,or
uN+uS+uE+uW− 5 ui=− 1. (13)Finally, we may write out the equations to be solved. The first four of the nine
equations, corresponding to Eq. (13) withi= 1 , 2 , 3 ,4, are
u 2 +u 4 − 5 u 1 =− 1
u 1 +u 3 +u 5 − 5 u 2 =− 1
u 2 +u 6 − 5 u 3 =− 1
u 1 +u 5 +u 7 − 5 u 4 =− 1.(14)
The solution of this problem is left as an exercise.
On more complicated regions, the replacement for the Laplacian operator
hasexactlythesameform,sincewestillusethe“graph-papermesh.”Thesys-