Dear Professor Dulhaine,
I am reading this great book" Fourier Series and Orthogonal Functions" by Harry Davis in order to resolve some promising leads in a problem I worked on 26 years ago. I was able to solve
D c(x,t)' ' - u c(x,t)' -k c(x,t) = c'(x,t)
Where the primes on the left are derivatives with respect to "x" and the derivative prime on the right of the equal sign is the derivative of c(x,t) with respect to " t " the heat equation with some whistles.
Where the boundary at x=0 is Co Exp[- b t] = C(0,t)
And the boundary conditions are dC(L,t)/dx + h C(L,t) = 0 (solution is Exp-ht]??)like the one you just mentioned
and C(0,t) = 0
where D, u, k, b, L are constants. The first boundary condition is what you helped me with. I never dreamed that it had so much beauty in it and I also noted the fact that it is the ODE for an exponential function h Exp[-h x] where h is positive.
My question for you is: is it worth my time ie does it have any application in the real world
The only appliication I have is a groundwater equation for a system undergoing Dispersion Advection Reaction l and I know of someone who gave a very elegant solution with c only a function of x but not t.
I can get a very nice solution with t if I let h = b which are constants in the BC.
I could go on and on about this old equation and I was able to solve the equation for a limited domain. Do you think it is possible to solve this equation for Exp[-b t}= C(o,t) and c(infinity, t) = 0 and C(t,o)=0?
I don't think so because of the "sink" in the equation would give me a negative C(x,t) eventually.
I would be very interested to know if I am wasting your and my time.
Thanks so much for your help. I promise not to bug you with this differential equation except of course for now.
Thanks some more for your recent reply! I will look into it.
Mike Bernthal