ABSTRACT
Let's evaluate the second integral in Eq. (12.79), a/(i)I aYi, to illustrate the procedure. The corresponding result for the first integral in Eq. (12.79), a/(i-I)layi, will be presented later. Substituting Eq. (12.81), withji(x) approximated by y(x), into the second integral in Eq. (12.79) and differentiating with respect to Yi gives
a/(i) a JX;+I a JXi+1~ = ::1" G(x, Y, y') dx = ::1,'. [(y')2 - Ql + 2Fy] dx ~, vy, Xi uy, Xi
Equation (12.83) requires the functions y(i)(x), a[y(i) (x)]laYi, y'(x) = d[y(i)(x)]jdx, and a[y'(x)]laYi' Recall Eq. (12.70) for y(i)(x):
y(i)(x) = Yi ( _ x ~;+1) +Yi+1 (x~;i) Differentiating Eq. (12.84) with respect to Yi yields
(12.84)
(12.85)
Differentiating Eq. (12.84) with respect to x gives (i) _
y'(x) = d[y (x)] = _li.+Yi+1 =Yi+1 Yi dx lui lui lui
Differentiating Eq. (12.86) with respect to Yi gives
Substituting Eqs. (12.84) to (12.87) into Eq. (12.83) yields
a/(i) JX;+I (Yi+1 - Yi) ( 1) JXi+1 ( X - Xi+1) -=2 -- dx+2 F - dx aYi x; lui lui Xi lui
_ 2[+1 Q~i( _ X~;+1) +Yi+1 (x~;)] (_ X~;+I) dx
(12.86)
(12.87)
(12.88)
(12.89)
where the order of the second and third terms in Eq. (12.83) has been interchanged. Simplifying the integrals in Eq. (12.88) gives
a/(i) = _ 2JX;+ I (Yi+1 ~ y;) dx _ 2Jxi+1 F(x -Xi+1) dx all. lu· X lui'/'1 Xi I i
J
X;+I Qy. - 2 ----1(2-- 2xi+IX +~+J)dx
J X;+I QYi+1 . 2
+ 2 M (x-- XiX - xi+1 X+ XiXi+l) dx Xi I
