7.4 Potential Equation 415
Figure 1 Pointion a square mesh and its four neighbors.Figure 2 Numbering for mesh points, and values on boundary.shown, we can write down the equations to be solved:
u 2 +u 4 +^12 − 4 u 1 = 0
u 1 +u 3 +u 5 + 1 − 4 u 2 = 0
u 2 +u 6 +^12 − 4 u 3 = 0
u 1 +u 5 +u 7 − 4 u 4 = 0
u 2 +u 4 +u 6 +u 8 − 4 u 5 = 0
u 3 +u 5 +u 9 − 4 u 6 = 0
u 4 +u 8 +^12 − 4 u 7 = 0
u 5 +u 7 +u 9 + 1 − 4 u 8 = 0
u 6 +u 8 +^12 − 4 u 9 = 0.(8)
This is simply a system of simultaneous equations. It can be solved by elim-
ination to obtain the results shown in Fig. 3. In this particular case, there are
numerous symmetries in the problem, sou 1 =u 3 =u 7 =u 9 ,u 2 =u 8 ,and
u 4 =u 6. Thus, onlyu 1 ,u 2 ,u 4 ,andu 5 need to be found. The system can be
reduced to four equations in these four unknowns, which can even be solved
manually.