From this equation
and as
if follows that
Please read here (example 4):
Pauls Online Notes : Differential Equations - Solving the Heat Equation
After some calculations in a PDE I have the following equations:
For the first equation, there are 3 possible cases:
(i) is positive
(ii) is negative
(iii) equals 0.
I also have the boundary conditions
I was hoping someone could answer some general questions I have about these cases.
First of all, by starting with case (iii) we have:
Then with BC's I get
so is a solution.
Also,
so ......my question here is if leads to a solution as just shown, does this automatically mean that need to be included in my fourier series solution to the PDE?
Now for case (i):
leads to solutions
so with BC's
Now leads to a trivial solution, so
for
and
so
For the third case,
Now I get
Apparently this should be a trivial solution, but I don't understand how it is different from case (ii). I would get
so
What am I missing here? How can I have two values for ?
Thanks to anyone who can help me understand this.
From this equation
and as
if follows that
Please read here (example 4):
Pauls Online Notes : Differential Equations - Solving the Heat Equation
I have really confused myself now....
First of all, I made a mistake. I should have had:
So why can't ?and as
if follows that
The way I'm thinking is that for case 1, we have
and then we can have for because we are looking for non-trivial solutions.
And then in case 2, we have
Now , and they only difference I can see is that is positive in case 1 and negative in case 2...
so why can't you still end up with the possibility that ? Is it because that is negative?
Is it because perhaps ?