First of all, thank you for taking the time to thoroughly go through this.
The contradictions that you found is what I was wondering about actually when I was going over the system by hand. I realized that when I was trying to better clarify the initial and boundary conditions for myself. I thought that maybe I was doing something wrong since my supervisor told me that it should work.
The next suggestion I got for the initial conditions was:
E(0,0) = 0;
I was unsure as to whether these would work or not because putting them into what I had for maple gave me the error that initial/boundary conditions must depend on one or two dependent variables. So I didn't see how that was going work.
I'm not sure but I was also given the suggestion to try discretizing in 'a' with an RKF method, or take the boundary conditions and make them homogeneous putting the current conditions into the system itself. I wonder if this will make a difference.
If you or any others have any other suggestions for how to proceed that would be much appreciated.
Thanks again for taking the time to thoroughly look over my code and find my issues.