Fourier method for non-uniform heat diffusion

[squaremeplz]

Homework Statement

The Fourier method specifies that

change of
heat energy of = heat out from left boundary - heat in from right boundary
segment in time Δt


If vars such as c,p.. are constant in the equation, and T(-L,L) = T_a, does the the aforementioned Fourier method also work for bodies that are in equilibrium. (I.e) if there is no source/sink, can you can work through the Fourier method to get u(x,t) = c, where c is the temperature for all x_n in [-L,L] at any time t?

 

[lippyka]

Homework Equations The Fourier method:u(x,t) = 1/2 T_a + 1/L \sum_{n=1}^{\infty} [A_n cos(n\pi x/L) + B_n sin(n\pi x/L)] exp(-n^2 \pi^2 c p t/L^2)The Attempt at a SolutionYes, it would still work. Since there is no source/sink, the heat energy of the segment is 0, meaning that the heat out from the left boundary and the heat in from the right boundary are the same. This means that when you solve the equation above, the A_n and B_n coefficients will be 0. Therefore u(x,t) = 1/2 T_a, which would be constant for all x_n in [-L,L], since T_a is constant.