Consider the 1D heat equation [ frac{partial u}{partial t}=k frac{partial^{2} u}{partial x^{2}} ] with insulated boundary conditions such that
?u/?x (x=0,t)= ?u/?x(x=L,t)=0. Assume the thin rod as an initial temperature profile
u(x,0)=f(x).
Find u(x,t) when [ f(x)=1+cos (pi x / L) text { for } 0 leq x leq L ]