differential operator

English

Noun

differential operator (plural differential operators)

(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.

Hyponyms

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

Related terms

partial differential operator
pseudodifferential operator

单词	differential operator
释义	differential operator English Noun differential operator (plural differential operators) (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. Hyponyms Laplace operator, Laplacian d'Alembert operator, d'Alembertian Related terms partial differential operator pseudodifferential operator
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