Page 539 - Modelling in Transport Phenomena A Conceptual Approach

A.8. METHODS OF LNTEGRATION 519

where the roots and weight factors for n = 1,2,3, and 4 axe given in Table A.3.

Table A.3 Roots and weight factors for Gauss-Legendre quadrature (Abramowitz
and Stegun, 1970).

n Roots (~i) Weight Factors (wi)
1 f0.57735 02691 89626 1.00000 00000 00000
0.00000 00000 00000 0.88888 88888 88889
f0.77459 66692 41483 0.55555 55555 55556
f0.33998 10435 84856 0.65214 51548 62546
f0.86113 63115 94053 0.34785 48451 37454
0.00000 00000 00000 0.56888 88888 88889
4 f0.53846 93101 05683 0.47862 86704 99366
f0.90617 98459 38664 0.23692 68850 56189
~~

Example A.7 Evaluate

I=JI”&dx

wing the five-point (n = 4) Gauss-Legendre quadrature formula and compare with
the analytical solution.

Solution

Since b = 2 and a = 1, from Eq. (A.8-13)

Then
r)
1
I
I
,5
,5
F(u)
=
F(u) = (y)
=-
=-
__ I
u+7
(y) u+7
+2
+2
The five-point quadrature is given by
The values of wi and F(ui) are given in the table below: