Functional vs Differential Equations?BackgroundI'm under the impression that a differential equation has locality hardwired into it (which is why we see them more in physics). However, if I wanted to write something non-local I'd use a functional equation. I am aware there are cases where the differential equation has a functional equation form as well.Question- Is my initial assumption true: "differential equation has locality hardwired into it"?- Is the cardinality of the set of functional equations more than that of the differential equations?- Is there any functional equation which does not have any differential equation equivalent?- Is there any differential equation which does not have any functional equation equivalent?CaveatWhile different theories of physics have different definitions of locality (for example bell locality in quantum mechanics and micro-causality in QFT) they all agree one cannot send information (non-random message) from point A to point B without a mediator (which is what I mean by locality in the above in the above).

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Functional vs Differential Equations?BackgroundI&#039;m under the impression that a differential equation has locality hardwired into it (which is why we see them more in physics). However, if I wanted to write something non-local I&#039;d use a functional equation. I am aware there are cases where the differential equation has a functional equation form as well.Question- Is my initial assumption true: &quot;differential equation has locality hardwired into it&quot;?- Is the cardinality of the set of functional equations more than that of the differential equations?- Is there any functional equation which does not have any differential equation equivalent?- Is there any differential equation which does not have any functional equation equivalent?CaveatWhile different theories of physics have different definitions of locality (for example bell locality in quantum mechanics and micro-causality in QFT) they all agree one cannot send information (non-random message) from point A to point B without a mediator (which is what I mean by locality in the above in the above).

Phillip Fletcher · Accepted Answer

Step 1Functional equations include the usual differential equations as a specialized subclass.One of my favorite functional equations is  y  =  f  (  x  )        y    ′    =  sin  ⁡  (  f  (  y  )It is clear that   f  (  x  )  =  n  π for any integer n is a solution; it is much less clear whether any other solutions exist. This equation has no differential equation equivalent.Differential equations only have locallity built in if you are considering the class of well-posed initial value problems. Boundary value problems do not necessarily exhibit locality (although I guess that is a cheat because the boundary condition itself is a non-local matter). Thus some of the most important equations in physics, including the steady-state Schroedinger equation in spherical or cylindrical coordinates, are non-local when the wave function is considered as a function of the angle about some axis.Step 2In between functional equations (which when non-trivial are often impossibly difficult to work with) and differential equations (which often can be attacked by perturbation theory, and a host of other cool tools) I offer differential difference equations. For example, for some given initial value curve   P  (  t  )  :  [  0  ,  1  )  ↦      R   and som initial velocity value v,                                                d            2                    x          (          t          )                          d                      t            2                              |              t      =              t        0              =  −  x  (      t    0    −  1  )    t  ∈  [  0  ,  1  )    ⟹    x  (  t  )  =  P  (  t  )                            d                  d          t                    x      (      t      )      |              t      =      1        =  vDifferential difference equation problems can have many of the same features as ordinary eigenvalue problems, yet can exhibit some of the same headaches as full-blown non-trivial functional equation problems.The cardinality issue is not easy and may be subtle. I believe the cardinality of the set of all differential equations is the same as the cardinality of the functions, but when you expand to functional equations, that may change.

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