Page 288 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 12 PARTIAL DIFFERENTIAL EQUATIONS 265

specified around a closed boundary, the equation describes a steady-state
condition.

Solving Elliptic Partial Differential Equations:
Replacing Derivatives with Finite Differences
In Chapter 6 we used the following approximation for a derivative
dF(x) - F(x + h) - F(x)
-- (12-5)
dx h
where h was a suitably small value. Equation 12-5 is the forward difference
equation. The corresponding backward difference equation is
dF(x) - F(x) - F(x - h)
-- ( 12-6)
dx h
For a partial derivative involving two independent variables, the finite
difference equation will involve suitable small differences in both x and y. We
will use h and k to represent these differences. The forward and backward
difference equations corresponding to 12-5 and 12-6 are
dF(x, Y) - F(x + h, Y) - F(x, Y) ( 12-7)
-
dx h
dm, r) - F(x, Y) - F(x - h, Y) (1 2-8)
-
dx h
and, for the partial derivative
dF(x + h, Y) - dF(x, Y)
-
a2F(x,Y) - dx dx (1 2-9)
ax2 h
Since we have used the forward difference equation 12-9 to calculate the partial
derivative, we can use backward differences for dF/dx in order to eliminate bias.
The result is
h,
+
h,
awx, Y) - ~(x Y) - WX, Y) + ~(x Y) (12-10)
-
-
ax2 h2
and in a similar fashion,
a2m, Y) - F(x, Y + k) - 2J-k Y) + F(x, Y - k) (12-1 1)
-
au2 k2
and
d2F(x,y) - F(x + h,y + k) - 2F(x,y) + F(x - h,y - k) (1 2- 12)
-
axay hk