On solvibility of heat equation with time-space reversed

On solvibility of heat equation with time-space reversed

Consider the heat equation $$u_t=u_xx, (t,x)\in \mathbb{R}^2$$ with
initial condition on $x=0$: \begin{cases} u(0,t)=1+\sin t\\
u_x(0,t)=\sin(\pi/4+t) \end{cases} and with periodic boundary condition on
$t: u(x,t)=u(x,t+2\pi)$.
It possibly does not work by simple using the Fourier mode analysis by
testing $$u(x,t)=\sum_n e^{int}c_n e^{{1 \over 2}(1-i)nx}$$ since this
does not give any feasible solution for given initial condition. By
reversing $x,t$ it is quite easy to see that the solution uniquely exists
by C-K theorem, but how can I construct such a solution?