The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 261 Wednesday, February 4, 2004 12:51 PM

Differential Equations and Difference Equations 261

any x and for t ≥0 a continuous and bounded function f(t,x) that satis-

fies the diffusion equation and which, for t = 0, is equal to a continuous

and bounded function f(0,x) = φ(x), ∀x.

Solution of the Diffusion Equation

The first boundary value problem of the diffusion equation can be

solved exactly. We illustrate here a widely used method based on the

separation of variables which is applicable if the boundary conditions

on the vertical sides vanish (that is, if f 1 (t) = f 2 (t) = 0). The method

involves looking for a tentative solution in the form of a product of two

functions, one that depends only on t and the other that depends only

on x: f(t,x) = h(t)g(x).

If we substitute the previous tentative solution in the diffusion equation

∂f 2 ∂^2 f

-----= a ---------

∂t ∂x^2

we obtain an equation where the left side depends only on t while the

right side depends only on x:

dh t-------------()-gx()= a (^2) ht()-----------------d ()
(^2) gx
dt dx^2
() 1 2 d
2
dh t gx() 1
-------------- ---------- = a ----------------------------
dt ht() dx^2 gx()
This condition can be satisfied only if the two sides are equal to a con-
stant. The original diffusion equation is therefore transformed into two
ordinary differential equations:
1 dh t()
----- -------------- = bh t()
a^2 dt
d
2
gx()
----------------- = bg x()
dx
2