Assume a solution fo the form to the modified diffusion equation . First show that the equation seperates and find th general solution for X(x) and T(t). Next assuming that D>0, L>0, solve the boundary value problem

for all

for all

Finally characterise the difference between the solutions for and

So

This gives us:

(1)

(2)

Now we need to evaluate these for different values of C: (C=0,C>0,C<0)

C=0

Equation (1) gives...

(3)

Let

So (3) becomes

Therefore,

Then

hence

For C=0, T(t) = constant. Because if C=0, then T'=Ct ==> T'=0

Moving to try the next step:

if

setting

clearly

setting

clearly

Thus

as before

So here we get the trivial solution for C>0

so the last part is the bit I am unsure on:

if

setting

clearly

setting

clearly

Thus

as before

which is the same as C = 0

Finally it says:

Finally characterise the difference between the solutions for and

only turns in the equation for T if then the function will tend to grow exponentially if then the function will tend to be sinusoidal due to the component if it will decay exponentially.

Is what I've said here sufficient (and correct?)