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elliptic partial differential equation

Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form
${\\displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu+G=0,\\,}$
where $A$ , $B$ , $C$ , $D$ , $E$ , $F$ , and $G$ are functions of $x$ and $y$ and where ${\\displaystyle u_{x}={\ rac {\\partial u}{\\partial x}}}$ , ${\\displaystyle u_{xy}={\ rac {\\partial ^{2}u}{\\partial x\\partial y}}}$ and similarly for ${\\displaystyle u_{xx},u_{y},u_{yy}}$ . A PDE written in this form is elliptic if
${\\displaystyle B^{2}-AC<0,}$
with this naming convention inspired by the equation for a planar ellipse. The simplest examples of elliptic PDE's are the Laplace equation, ${\\displaystyle \\Delta u=u_{xx}+u_{yy}=0}$ , and the Poisson equation, ${\\displaystyle \\Delta u=u_{xx}+u_{yy}=f(x,y).}$ In a sense, any other elliptic PDE in two variables can be considered to be a generalization of one of these equations, as it can always be put into the canonical form
${\\displaystyle u_{xx}+u_{yy}+{\ ext{ (lower-order terms)}}=0}$
through a change of variables.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Elliptic_partial_differential_equation)

partial differential equation

équation aux dérivées partielles elliptique

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