I need to solve a partial differential equation backwards in time. In other words, I have the discrete partial differential equation \begin{align} p_\text{new} & = p_j + (dt/dx)q_jf'(u_j) + 0.5(dt/dx)(q_{j+1} - q_{j-1}) \\ & {} \quad {} + 0.5(dt/dx)a(p_{j+1} -2p_j + p_{j-1}). \end{align} and a final condition at some terminal time $T\;$, but no initial condition. How do I do this?
where $ p_j,p_{j-1},p_{j+1}, q_j, q_{j-1}, q_{j+1}, u_j $ are values of $p$, $q$ and $u$ at grid points $j, j-1, j+1 $ and $j$ respectively. $dt$ is time interval and $dx$ is space interval. $p_\text{new}$ is the updated value of $p$ in every iteration.
Any suggestion is welcome and appreciated.