# Diffusion Equations and Boundary Conditions

Hello, I have a test coming up soon and I don't understand how to solve the kind of problem as given below. Please provide some explanation in the solution. Thanks in advance,

Eriko

Consider
the
diffusion
equation

€

φt = a

2

φxx + x

3

, −L ≤ x ≤ L

with
initial
condition

€

φ(x,0) =1

(i) Suppose
the
boundary
conditions
are

€

φ(−L,t) = 0 φ(L,t) = 0

Does
an
equilibrium
solution
exist?
If so,
solve
for
it.

(ii) Suppose
the
boundary
conditions
are

€

φ(−L,t) = 0 φx (L,t) = 0

Does
an
equilibrium
solution
exist?
If
so,
solve
for
it.

(iii) Suppose
the
boundary
conditions
are

€

φ(−L,t) =1 φx (L,t) = 0

Does
an
equilibrium
solution exist?
If
so,
solve
for
it.

#### Solution Preview

...is a function that only depends on the coordinate x.

Thus, our equation

(1.2)

Becomes a simple second order ordinary differential equation:

(1.3)

The homogenous equation is:

(1.4)

Which leads to the simple homogenous solution:

(1.5)

For the particular solution, we see that the second derivative should have a term of , so we "guess" a solution in the form:

(1.6)

Substituting ...