220
4. Series-Expansion Methods
:
1.0
I:
0.5
0.0
:
:
I
1.0
0.0
(X)
I:
:
1\ :I
j - 3
j-2
j - I
j
j + l
j + 2
j + 3
FIGURE 4.7. Quadratic expansion functions for rpi, an endpoint node centered at grid
point j , and rpj+I' amidpoint node centered at j + I. The x -axis is labeled in units of Sx .
function assurnes the values al and a3 at the endpoint nodes XI and X3. and assumes the value bz at the midpoint node X2 . The quadratic Lagrange interpolating
polynomial that assurnes these values at the nodes is
0.5
On the interval XI ::: X ::: X3 . the preceding is algebraically identical to
where rpi is the endpoint quadratic expansion function
3
I (X _x .)2 if Ix - xjl ::: 2ßx.
rpi(x) =
1 -
Ix -x'l
2ßx
2
'
(4.95)
I
J + - __ J
ßx
0,
otherwise,
and rpj is the midpoint quadratic expansion function
X- Xj_l) (2- X-X j_I) ,
rpj(x) =
( ßx
{ 0,
ßx
(4.96)
otherwise.
These expansion functions are plotted in Fig. 4.7. The jth endpoint expansion
function is zero outside an interval of length 4ßx centered at the node x j; it is
unity at X j and is zero at every other node . The jth midpoint expansion function
is zero outside an interval of length 2ßx centered at x j; it is equal to unity at x j
and zero at the other nodes . As was the case with chapeau expansion functions,
the coefficient of the jth quadratic expansion function is also the value of the
approximate solution at the jth node .