Page 577 - Modelling in Transport Phenomena A Conceptual Approach

B.3. SECONDORDER PARTW DLE'FERENTZAL EQUATIONS 557

Application of the boundary condition at x = 0 implies that B = 0. On the other
hand, the use of the boundary condition at x = r gives
0
Asin hr = 0 (3)

In order to have a nontrivial solution
0
sin fir =O + fir=nr n= 1,2,3, ... (4)

or,
fi=n + ~ ~ n=l,2,3, ... (5)
~
n
=
Equation (5) represents the eigenvalues of the problem. The corresponding eigen-
functions are
yn = An sin(nx) n = 1,2,3, ... (6)
where An is an arbitrury non-zero constant.
Since the eigenfunctwns are orthogonal to each other with respect to the weight
function w(x), it is possible to write
r=
sin(nz) sin(mx) dz = 0 n # m (7)

B.3.4.1 The method of Stodola and Vianello

The method of Stodola and Vianello (Bird et al., 1987; Hildebrand, 1976) is an
iterative procedure which makes use of successive approximation to estimate X
value in the following differential equation

(B.3-19)

with appropriate homogeneous boundary conditions at x = a and x = b.
The procedure is as follows:

1. Assume a trial function for yl(x) which satisfies the boundary conditions
x = a and x = b.
2. On the right side of l3q. (B.3-19), replace y(x) by yl(z).

3. Solve the resulting differential equation and express the solution in the form
Y(X) = fl(X) (B.3-20)

4. Repeat step (2) with a second trial function yz(x) defined by
(B.3-21)