# Heat Equation with Period Boundary Condition.

Given the heat equation $u_t$=$u_{xx}$ with period boundary conditions u(x) = u(x+2$\pi$), and u(x,0) = $\sum_{n=1}^{10}$cos(nx),

1. Find an expression for the maximum value of u at time t > 0?

2. What happens to the solution as t → ∞?

So I attempted separation of variables and got u(x,0) = $\sum_{k=1}^{\infty}$$B_ksin(\frac{xk\pi}{l}), but I have no idea how to proceed. Any hints/links would be helpful. ## 1 Answer As you mentioned by separation of variables u(x,t)=X(x)T(t), the heat equations decouples to \dot{T} = -ET and X''=-EX with E an unknown positive real number (for now) and at this point we have no idea what it is. We might end up finding multiple E's satisfying our conditions (and indeed this is the case). So assuming there is some E like that:$$T_E(t)= e^{-Et}, \quad X_E=A_E \cos (\sqrt{E} x) + B_E \sin (\sqrt{E} x)$$note that I chose E positive since E negative does not have any periodic solution. Let say a solution corresponding to a value E is "good", if u(x+2\pi,t)=u(x,t). In other words we say a solution is "good" if it is indeed consistent with periodic boundary condition. Hence must have X_E(x+2\pi)=X_E(x) for a good solution. This means$$ \cos (\sqrt{E}(x+2\pi))=\cos (\sqrt{E}x), \quad \sin (\sqrt{E}(x+2\pi))=\sin (\sqrt{E}x) $$This can only happen if \sqrt{E}=n for some integer n, or if E =n^2 for "good" solutions. The full solution, before applying initial conditions, would be linear combination of all "good" solutions:$$ u(x,t)=\frac{A_0}{2}+\sum_{n=1}^\infty [A_n \cos (nx)+B_n\sin (nx)]e^{-n^2t}$$Now comes the initial conditions that$u(x,0)=\sum_{n=1}^{10}\cos (nx)$. By matching this to the solution above, can you finish the job? • How did you get the value for$E_n$? Also, I attempted setting the u(x,0) equal to each other, I decided that$B_n$= 0 and$A_n$= 1 , and$A_0$= 2 as well (?). I'm not sure if my steps were logical or not. – Tesuji Apr 7 '17 at 3:26 • Yes$B_n=0$for all$n$,$A_0=0$(note that your sum runs starts$n=1$) and for$n=1,\cdots, 10$you have$A_n=1$, for the rest of$n$, you have$A_n=0$. – Hamed Apr 7 '17 at 3:31 • Whoops oh yea. How did you determine$E_n$I tried comparing the U(x+2$\pi$) and u(x) but the sin functions and cos functions just didn't change at all. – Tesuji Apr 7 '17 at 3:34 • I edited the solution slightly to make it clearer where the values of$E\$ are coming from. – Hamed Apr 7 '17 at 3:39
• Ok that makes a lot of sense. Thank you so much. – Tesuji Apr 7 '17 at 3:47