  Differential (Ore) type polynomials with "approximate" polynomial
coefficients are introduced. These provide an effective notion of approximate
differential operators, with a strong algebraic structure. We introduce the
approximate Greatest Common Right Divisor Problem (GCRD) of differential
polynomials, as a non-commutative generalization of the well-studied
approximate GCD problem.
  Given two differential polynomials, we present an algorithm to find nearby
differential polynomials with a non-trivial GCRD, where nearby is defined with
respect to a suitable coefficient norm. Intuitively, given two linear
differential polynomials as input, the (approximate) GCRD problem corresponds
to finding the (approximate) differential polynomial whose solution space is
the intersection of the solution spaces of the two inputs.
  The approximate GCRD problem is proven to be locally well-posed. A method
based on the singular value decomposition of a differential Sylvester matrix is
developed to produce an initial approximation of the GCRD. With a sufficiently
good initial approximation, Newton iteration is shown to converge quadratically
to an optimal solution. Finally, sufficient conditions for existence of a
solution to the global problem are presented along with examples demonstrating
that no solution exists when these conditions are not satisfied.
