Can an integral equation always be rewritten as a differential equation? Given an integral equation is there always a differential equation which has the same (say smooth) solutions?
It seems like not but can one prove this in some example?
Edit: Naively I'm hoping for some algorithm which takes an integral equation and applies some operations like taking derivatives, substituting variables for some new ones, adding additional differential equations etc... such that after this procedure you have made all integral signs vanish and obtained a differential equation which has the same solutions as the integral eqution. (maybe similarly to how one can transform any system of PDEs into a system of first order equations)
 A: While I second Deane's comment that the author should be a bit more specific about the kind of equations he is interested in, in general the answer is no for integral and, more broadly,integro-differential equations. However, the latter can be reduced to functional-differential equations rather than to purely differential ones. For more details, see Section 6.6 of the book Symmetries and Conservation Laws for Differential Equations of Mathematical Physics
(unfortunately the relevant pages appear to be excluded from Google preview).
For instance, I greatly doubt that one could reduce the Smoluchowski coagulation equation from Example 6.5 of the above book to a differential (as opposed to functional-differential) equation or system thereof. 
A: With Charles Matthews comments in perspective, these are some notes I made sometime ago on this topic.  I dont have the books in front of me so I can't look up the details right now.
1) In Zabreyko's book Integral equations (902860393X), there is the method based on Green's functions in Appendix A.  
2) Bellman in Perturbation techniques Sec 10 points out that the other way (ODE to integral equation) is actually better

Conversion of differential eqn to integral equation
  is one of the powerful devices in
  approximation theory.  Its potency is
  due to the fact that integration is a
  smoothing op, while differentiation
  accentuates small variations.  If u(t)
  and v(t) are close together, then
  ∫u(s)ds and ∫v(s)ds will be comparable
  in value, but du/dt and dv/dt may be
  arbitrarily far apart.  Consequently,
  when carrying out successive
  approximations, we prefer integral
  operators to differential operators. 
  On the other hand, in numerical
  solutions, we prefer differential
  operators to integral operators.

3) You can also look up Handbook of Integral Equations by Polyanin.
Sec 8.4.5, Sec 9.7 and sec 9.3.3 are three situations where the method reduces a specific integral equation to an ODE
A: No, when it comes to stochastic differential equations these are only a shortcut - there is only a meaning to the integral representation, the paths are non-differentiable.
See e.g. here: http://en.wikipedia.org/wiki/Ito_calculus
A: Well, the answer, if there is any, must be subtle. Take the following example. Let $H$ be the Hilbert transform. Then
$$\frac{d}{dx}Hu=f$$
is an integral equation, where the kernel is the Fourier transform of $\xi\mapsto|\xi|$. It cannot be rewritten as a set of differential equations (ODEs) within $\mathbb R$. Nevertheless, it can be recast in terms of PDEs, using an harmonic extension in the upper half-plane:
$$-\Delta_{x,y}\phi=0\hbox{ in }\{y>0\},\qquad \frac{\partial\phi}{\partial y}(x,0)=f(x),\quad u(x)=\phi(x,0).$$
A: In general, no. An integral equation can be non-local, whereas a differential equation is local (in the sense that it can be described by a function over the jet-bundle). As an illustration
Let $K(x) = \delta_0(x) + \delta_1(x)$ be an integral kernel, where $\delta_i$ are the Dirac delta's supported at $i$. Consider the integral equation, for some fixed smooth $f$
$$ f(x) = \int K(x-y) \phi(y) dy $$
for the unknown $\phi$. The equation reduces to $\phi(x) + \phi(x+1) = f(x)$. Any continuous function $g(x)$ on $[0,1]$ satisfying $g(0) + g(1) = f(0)$ generates a continuous solution of the equation. I challenge you to find a differential equation whose solution set can be thus generated. 
