3  Properties of the Simple Diffusion Equation

It is instructive to consider the properties of the simple diffusion equation
u(x,t)
t
 = 2 u(x,t)
x2
   
(6)
to illuminate the nature of the diffusion operator K and its inverse. The equation is the one-dimensional form of the parabolic equation (2) with D º 1 and the first derivative terms dropped. In the forward problem f is given and g is unknown and in the inverse problem g is given and f is unknown.

3.1  Fourier analysis

Let u(x,t) in the model problem (6) be expanded as a Fourier series. Let us consider one component of the Fourier series for f of angular frequency N p
fN(x) = ei N px.
Substituting this into the model problem gives
gN(x) = e-N2 p2 T e i N px
and hence that
gN = kN fN
where kN = e-N2 p2 T.

3.2  First Kind Equation Analogy

For the model equation the equation (1) can also be written in the form of a Fredholm integral equation of the first kind
g(x) = 1
2
Ö
 
pT
 
ó
õ 		¥

-¥ 
 exp{ -(x-y)2
4 T
 } f(y) dy   .
(7)
Discretising the equation (7) using an integral equation method such as collocation results in a matrix-vector equation of the form
g » K f  
(8)
where gi = g(xi), fi = f(xi).
The solution g of (7) for the direct problem is simply the evaluation of an integral at each x. However for the inverse problem the solution f of (7) entails the solution on an integral equation of the first kind. Such equations are notoriously difficult to solve, especially for integral operators with smooth kernels, such as the Gaussian kernel in (7). The matrix K in (8) will generally be severely ill-conditioned. An outline of the difficulties associated with first kind equations and methods for their solution is given in Miller (1974), for example.

3.3  Discussion

The Fourier analysis and the first kind equation (7) provide important information on some of the properties of the diffusion operator. For example the eigenvalues of the model problem (6) are e(-N2 p2 T) for N=1,2,..., hence the eigenvalues cluster and tend to zero as N ® ¥. It follows from the Fourier analysis that fN = eN2 p2 T gN and so that the inverse problem effects a growth factor of e(N2 p2 T) on the sinusoidal function gN.
Equation (7) demonstrates other interesting properties of the diffusion operator K. In particular, if f is a delta function then g is a Gaussian and if f is step function then the discontinuity is smoothed in g to an error function. The general conclusion that can be made from this is that g is a C¥ function whereas f may belong to the space of C0 or simply L2
The danger of non-existence of a solution to the inverse problem can easily be demonstrated. For example if f is a delta distribution then it becomes a Gaussian distribution of greater and greater spread with T. If the result is observed at T then the precise form of the Gaussian distribution is given by (7). It might be thought that with g being a smooth function there should not be any problem in returning a solution. However there is clearly no inverse solution for reverse times of T1 > T. Hence it is easily the case that even an apparently benign function g can have no inverse.