# 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.

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#### 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 ...