7.3.1 Homogeneous equation and boundary conditions

Consider the boundary-value problem

Our goal is to find the solution of ( MDH) using the method of separation of variables or Fourier method. A separable solution is a solution of the form

to the problem

Plugging this into the diffusion equation, we get

or

In order for the last relation to be an equality each side must be identically equal to a constant:

By the boundary conditions

it follows

so that X( x) satisfies the following eigenvalue (Sturm-Liouville) problem

(7.46)

while T( t) satisfies the equation

(7.47)

We are looking for the values of ? which lead to nontrivial solutions. Consider the following three cases:

1. Let ? = ? ² > 0, ? > 0. Then the equation (7.46) has the general solution

   
   

   By the boundary conditions it follows

   
   

   and c ₁ = c ₂ = 0 because

   
   

2. If ? = 0, X( x) has the form

   
   

   It follows again c ₁ = c ₂ = 0 .

   So in the first two cases the problem (7.46) admits the trivial solution only.

3. If ? = ? ? ² < 0, ? > 0, then the equation (7.46) has the general solution

   
   

By the boundary conditions it follows that for a nontivial solution

Then

So the only nontrivial solution of (7.46) appears when

and has the...