How Does the Convection Diffusion Equation Model Temperature Near a Heat Source?

[liquidFuzz]

I'm tinkering with the Convection Diffusion Equation (a second order differential equation) to model a temperature behavior in proximity to a heat source in a water bath. Just to get going I solved the system for some arbitrarily chosen boundary conditions. The result is that the temperature at the ambient boundary has a derivative that isn't zero. This is sort of okay in my solution as it is only meant to show a likely temperature behavior. But how do I tackle this problem if I like to solve it with a boundary condition such as u'(1)=0, i.e. x=1 is at ambient temperature. The problem I have with this is that if I chose an exponential solution such as u(x) = C1 + C2 exp(x) I get C2 = 0 when I try to find the constants C.

 

[RUber]

I believe this is usually handled by letting your C2 by a polynomial and solving for the boundary conditions.

 

[liquidFuzz]

Okay, so C2 equals some function f(x)? A linear function..?