Page 582 - Modelling in Transport Phenomena A Conceptual Approach

562 APPENDIX B. SOLUTIONS OF DIFFERENTIAL EQUATIONS

the solutions, i.e., ClO, + C202, also satisfies the partial differential equation and
the boundary conditions. Therefore, the complete solution is
00
e = c,, sin( nxc) (B.3-52)
n=l
The unknown coefficients Cn can be determined by using the initial condition. The
use of Eq. (B.3-36) results in

00
1 = C, sin(nx0 (B.3-53)
n=l
Since the eigenfunctions are simply orthogonal, multiplication of Eq. (B.3-53) by
sinmxt and integration from = 0 to = 1 gives

(B.3-54)

Note that the integral on the right side of Eq. (B.3-54) is zero when m # n and
nonzero when m = n. Therefore, when m = n the summation drops out and Q.
(B.3-54) reduces to the form
I'
Jd' sin(n?rt) d( = Cn sin2(nrJ) d( (B.3-55)

Evaluation of the integrals show that

2
cn = - [l - (- l)n] (B.3-56)
xn
The coefficients C,, take the following values depending on the value of n:
0 n = 2,4,6, ...
Cn={ - n = 1,3,5, ... (B .3- 57)
4
xn
Therefore, the solution becomes

(B.3-58)

Replacing n by 2k + 1 gives

(B .3- 59)