Page 581 - Modelling in Transport Phenomena A Conceptual Approach

B.3. SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS 561

The choice of a negative constant is due to the fact that the solution will decay to
zero as time increases. The choice of a positive constant would give a solution that
becomes infinite as time increases.
Note that Eq. (B.3-41) results in two ordinary differential equations. The
equation for F is given by
+
dF
- X2F = 0 (B.3-42)
dr
The solution of Eq. (B.3-42) is

F(T) = e--x2T (B.3-43)
On the other hand, the equation for G is

d2G
-+X2G=0 (B.3-44)
dt2
subject to the boundary conditions

at <=0 G=O (B.3-45)
at <=1 G=O (B .3-46)
Note that Eq. (B.3-44) is a Sturm-Liouville equation with a weight function of
unity. The solution of Eq. (B.3-44) is

G(<) = Asin(X<) + Bcos(X<) (B .3-47)
where A and B are constants. The use of the boundary condition defined by Eq.
(B.3-45) implies B = 0. Application of the boundary condition defined by Eq.
(B.3-46) gives
AsinX = 0 (B.3-48)
For a nontrivial solution, the eigenvalues are given by
sinX=O * &=nr n=1,2,3, ... (B 3-49)

The corresponding eigenfunctions are

Gn(<) = sin(nn<) (B.3-50)

Note that each of the product functions
22
e&, <) = e-n ?r sin(n.lr<) n = 1,2,3, ... (B.3-51)
is a solution of Eq. (B.3-35) and satisfies the initial and boundary conditions, Eqs.
(B.3-36)-( B.3-38).
If 61 and 62 are the solutions satisfying the linear and homogeneous partial
differential equation and the boundary conditions, then the linear combination of