Elliptic partial differential equation information

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

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 $u_{x}={\frac {\partial u}{\partial x}}$ , $u_{xy}={\frac {\partial ^{2}u}{\partial x\partial y}}$ and similarly for $u_{xx},u_{y},u_{yy}$ . A PDE written in this form is elliptic if

B^{2}-AC<0,

with this naming convention inspired by the equation for a planar ellipse. Equations with $B^{2}-AC=0$ are termed parabolic while those with $B^{2}-AC>0$ are hyperbolic.

The simplest examples of elliptic PDEs are the Laplace equation, $\Delta u=u_{xx}+u_{yy}=0$ , and the Poisson equation, $\Delta u=u_{xx}+u_{yy}=f(x,y).$ 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

u_{xx}+u_{yy}+{\text{ (lower-order terms)}}=0

through a change of variables.^[1]^[2]

^ Pinchover, Yehuda; Rubinstein, Jacob (2005). An Introduction to Partial Differential Equations. Cambridge: Cambridge University Press. ISBN 978-0-521-84886-2.
^ Zauderer, Erich (1989). Partial Differential Equations of Applied Mathematics. New York: John Wiley&Sons. ISBN 0-471-61298-7.

[pinchover-1] Pinchover, Yehuda; Rubinstein, Jacob (2005). An Introduction to Partial Differential Equations. Cambridge: Cambridge University Press. ISBN 978-0-521-84886-2.

[Zauderer-2] Zauderer, Erich (1989). Partial Differential Equations of Applied Mathematics. New York: John Wiley&Sons. ISBN 0-471-61298-7.

Elliptic partial differential equation information

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