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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.

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Can you explain your notation? What is $p_j$, $p_{new}$, $q_j$ and $u_j$? What do you mean by the notation $dt/dx$? For continuous PDEs I would just tell you to do a change of variable $t \to (T-t)$ and solve the initial value problem, which may or may not be well-posed (think backwards heat equation). The discrete case things are slightly different, so please explain the notation. –  Willie Wong Sep 3 '11 at 14:18
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Where is $p_{new}$ evaluated? The right hand sides are all scalars, and by your definition $p_{new}$ should be a function. Is $q$ a prescribed function? is $u$? What is $f$? What is $f'$? what is $a$? –  Willie Wong Sep 3 '11 at 14:56
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