Page 277 - Excel for Scientists and Engineers: Numerical Methods

254 EXCEL: NUMERICAL METHODS

Finite-Difference Methods

As described in the following, approximating the derivative of a function by
a finite difference quotient will allow us to reduce a boundary-value problem to a
system of simultaneous equations that can be solved by methods that have been
discussed in Chapter 9. Problems that are difficult or impossible to solve by the
shooting method may sometimes be solved by the finite-difference method.
Consider a two-point boundary value problem, where y is known at the ends
of the range and the expression for the second derivative y" is given. For a
differential equation of the general form
y" + ay = bx + c (1 1-9)
where a = F(x), we can replace the second derivative y" by the central difference
formula
1, - Yi+l - 2Yi + Yi-1
Y- (1 1-10)
h2

where h = Ax (equation 1 1-1 O assumes equally spaced x values) to obtain
(1 1-1 1)

where xI and yj represent the point at which the derivative is calculated.
Rearranging equation 1 1 - 1 1 yields
+
yl+1 + (h2a - 2)~, yI-l= h2 (bxj + C) (1 1-12)
We now divide the interval between the two boundary values into n equal parts to
yield n simultaneous equations obtained from equation 11-12. The procedure is
best illustrated by an example.

Solving a Second-Order Ordinary Differential Equation
by the Finite-Difference Method
We wish to solve the boundary value problem

(1 1-13)

with boundary values y = 2 at x = 1 and y = -1 at x = 3. The differential equation
is of the general form of equation 11-9 with a = -(0.15-x/2.3), b = 1 and c = 0.
For this simple example, we will subdivide the x interval, x = 1 to x = 3, into ten
subintervals; thus h = 0.2 and the x values defining the subintervals (sometimes
called the meshpoints) are x1 = 1 .O, x2 = 1.2, . . ., x11 = 3.0. We can now write an
equation of the form