9-DifferntEquations Page 262 Wednesday, February 4, 2004 12:51 PM
262 The Mathematics of Financial Modeling and Investment Management
with boundary conditions g(0) = g(l) = 0. From the above equations and
boundary conditions, it can be seen that bcan assume only the negative
values,
k^2 π^2
b= – ------------,k= 12 ,,...
l^2
while the functions gcan only be of the form
kπ
gx()= Bksin------x
l
Substituting for h, we obtain
^22
a k
2
π
()= Bk′exp (^) – ------------------t
l^2
ht
Therefore, we can see that there are denumerably infinite solutions of
the diffusion equation of the form
a^2 k^2 π^2 kπ
fk(tx, )= Ckexp– ------------------tsin ------x
l^2 l
All these solutions satisfy the boundary conditions f(t, 0 ) = f(t,l) = 0. By
linearity, we know that the infinite sum
∞ ∞
a^2 k^1 π^2 kπ
ftx ( , )= ∑fk(tx, ) = ∑Ckexp– ------------------tsin ------x
k= 1 k= 1 l^2 l
will satisfy the diffusion equation. Clearly f(t,x) satisfies the boundary
conditions f(t,0) = f(t,l) = 0. In order to satisfy the initial condition,
given that φ(x) is bounded and continuous and that φ(0) = φ(l) = 0, it can
be demonstrated that the coefficients Cs can be uniquely determined
through the following integrals, which are called the Fourier integrals: