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I have a differential equation depending on $z$, $x$ and $t$ in a 2d-system which looks like $$\partial_z A = \partial_x A+\partial_x^2 A+A+f(t)\cdot A$$ with $$\frac{df(t)}{dt}=f(A)$$ The formula describes the propagation of light (as the complex function $A$) through material in $z$-direction. Thus I can relate the position of $A$ with the time $t$.
But how can I couple both, so that I can solve the equation? If $f$ is constant, the solution is easy. But when $f$ depends on the time (and thus on the position of the current iteration) I do not know how to proceed.

Is that problem solvable, or do I have to use a numeric approach?

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    I'm confused, you say the differential equations depends on $x, z$ and $t$ but then you have $\partial_{r} A$ and $\partial_{r}^{2} A$? – 2017-01-12
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    My fault, fix that now. Thanks! – 2017-01-12

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