$u_t−ku_{xx}=0$
$u(t,0^+)=u(t,2\pi^−)$
$u_x (t,0^+)=u_x (t,2\pi^−)$
$u(0,x)=f(x)$
$k>0 , t\geq 0 , (t,x)\in[0,\infty)×[0,2\pi]$
$k$ is a constant, $u_t$ is the partial derivative with respect to $t$, $u_x$ is the partial derivative of $u$ with respect to $x$.
I have an ugly answer, that would probably be more of a hassle to read, since i am not sure how to use $\mathrm{\TeX}$ here.
I first used separation of variables to determine the time function is $C\exp(-\lambda kt)$ where $C$ is a constant. The space function has both sine and cosine functions, because of the boundary conditions if $\lambda > 0$. If $\lambda = 0$ then the space function is equal to a constant, and there are no non trivial solutions for $\lambda < 0$. Is this the right track so far? I am also unclear about hoe to determine the constants that appear in the solution of $u(x,t)$.
