The Mathematics of Financial Modelingand Investment Management

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– ------------------tsin ------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– ------------------tsin ------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: