Page 178 - Numerical methods for chemical engineering

164 4 Initial value problems

weighted integral
b
'
I w = w(x) f (x)dx (4.58)
a
[a, b] may be finite or infinite ( e.g. [0, ∞], [−∞, ∞]), and w(x) is a nonnegative weight
function, w(x) ≥ 0, all of whose moments are finite,
' b
k
µ k = x w(x)dx k = 0, 1, 2,. . . (4.59)
a
As examples of weighted integrals, consider
b
'
Cartesian f (x)dx w(x) = 1 (4.60)
a
' R
radial in two dimensions f (r)2πrdr w(r) = 2πr (4.61)
0
' R
2
radial in three dimensions f (r)4πr dr w(r) = 4πr 2 (4.62)
0
We now examine the optimal placement of support points for the rule
' b N

I w = w(x) f (x)dx ≈ w j f (x j ) (4.63)
a j=0
Here for brevity, we discuss only the major results. The supplemental material in the accom-
panying website presents a more extensive discussion, with proofs of all theorems.

Preliminary definitions
Let us define the following sets of polynomials:

j ≡{p(x)|degree(p) ≤ j}= all polynomials of degree j or less
(4.64)
j
j−1
1 j ≡{p(x)|p(x) = x + a 1 x + ··· + a j }
That is, 1 j is the set of polynomials that have degree j exactly, and that also have a leading
coefficient of 1. Thus, 1 j ⊂ j .
Definition A function f (x) is said to be square-integrable over [a, b] for w(x) if the
following definite integral exists and is finite:
' b
2 2
f = w(x)[ f (x)] dx ≥ 0 (4.65)
2
a
Definition The scalar product of two real square-integrable functions is
' b
f |g ≡ w(x) f (x)g(x)dx (4.66)
a
Definition Two square-integrable functions are said to be orthogonal if their scalar product
is zero,
' b
f |g = w(x) f (x)g(x)dx = 0 ⇒ f ⊥ g (4.67)
a