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My question is:

Is there a duality between a solution of an ODE,PDE,SDE or integral equations with their analog counterpart in the discrete domain?

I mean if I know a solution to the difference equation will that mean that I can find a solution to the differential equation and vice versa?

In the case of well behaved, analytic solutions I believe the answer is yes, cause I can insert a power series as the solution and look for a difference equation which suits the differential equation, but what happens when we can't guess for an analytic solution is the continuous domain, can we still find a discrete counterpart to our DE?

There's one book which I found at my university library that I lend but still didn't find the time to read it thoroughly, it's called "Differential-Difference Equations" by Richard Bellman and Kenneth Cooke, I'll start reading next week, hopefully I'll have time for it.

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Difference equations are much more difficult to handle, but can be treated numerically. Numerics of differential equations (ODE and PDE) can be viewed as approximating differential equations by suitable difference equations. – Peter Michor Jul 7 '13 at 9:05

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