Can Fourier Series Solve a Discontinuous Heat Problem on a One-Dimensional Rod?

[nicksauce]

The question is:

Write a short paragraph that physical problem modeled by the equation:

$$\frac{\partial{U}}{\partial{t}} = \frac{1}{4}\frac{\partial^2{U}}{\partial^2{x}} -12[U - 8x]$$

Subject to
IC: U(x,0) = 3x
BCs: U(0,t) =0, u(2,t) = 2t

Okay so clearly, the physical problem is some kind of heat problem with a one dimensional rod and a heat source term. However, the problem is that the IC and the BCs are conflicting, since U(2,0) = 6, from the IC, but U(2,0) = 0 from the BC. Any advice to go about answering this?

 

[HallsofIvy]

What that means is that your function is not continuous. That's alright if you are using integration techniques which will "smooth" the discontinuity. I would recommend using a Fourier series solution. In fact, I would write V(x,t)= U(x,t)- 2t and solve the problme for V so that the boundary conditions are "homogeneous" and you can use a Fourier sine series.