differential operator

differential operator (plural differential operators)

1. (mathematics, mathematical analysis) An operator defined as a function of the differentiation operator (the operator which maps functions to their derivatives).

   Define ${\displaystyle P(D)}$ to be the differential operator given by ${\displaystyle P(D)=D^{2}+5D+6}$, where ${\displaystyle D}$ is the differentiation operator. Then ${\displaystyle P(D)(\sin(x))=-\sin(x)+5\cos(x)+6}$.

   • 2000, S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators, page 328:

     […] these conditions appeared in the very first papers on ordinary differential operators.

   • 2012, Youri Egorov, Bert-Wolfgang Schulze, Pseudo-Differential Operators, Singularities, Applications, page 27:

     A differential operator ${\displaystyle P\left(D\right)}$ is hypoelliptic if for any domain ⊂ ${\displaystyle \mathbb {R} ^{n}}$ any solution ${\displaystyle u}$ of the equation ${\displaystyle P\left(D\right)u=0}$ from the class ${\displaystyle {\mathfrak {D}}'\left(\Omega \right)}$ is a function from ${\displaystyle C^{\infty }}$(ω) for any open set ω ⊂⊂ ${\displaystyle \Omega }$.
     A complete algebraic description of all hypoelliptic differential operators has been obtained by Hörmander in [H1].

   • 1998, L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis‎, page 251:

     Secondly, differential operators and to some extent their fundamental solutions are local even with respect to the wave front set.

• Laplace operator, Laplacian
• d'Alembert operator, d'Alembertian

• partial differential operator
• pseudodifferential operator