Elliptic partial differential equation

An elliptic partial differential equation is a general partial differential equation of second order of the form

$Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+F=0\,$

that satisfies the condition

$B^{2}-AC<0.\$

(Assuming implicitly that $u_{xy}=u_{yx}$. )

Just as one classifies conic sections and quadratic forms based on the discriminant $B^{2}-4AC$, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by $B^{2}-AC,$ due to the convention (discussion and explanation here). The above form is analogous to the equation for a planar ellipse:

$Ax^{2}+2Bxy+Cy^{2}+\cdots =0$ , which becomes (for : $u_{xy}=u_{yx}=0$) :

$Au_{xx}+Cu_{yy}+Du_{x}+Eu_{y}+F=0$ , and $Ax^{2}+Cy^{2}+\cdots =0$ . This resembles the standard ellipse equation: ${x^{2} \over a^{2}}+{y^{2} \over b^{2}}-1=0.$

In general, if there are n independent variables x₁, x₂ , ..., x_n, a general linear partial differential equation of second order has the form

$Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\quad {\text{ + (lower-order terms)}}=0\,$, where L is an elliptic operator.

For example, in three dimensions (x,y,z) :

$a{\frac {\partial ^{2}u}{\partial x^{2}}}+b{\frac {\partial ^{2}u}{\partial x\partial y}}+c{\frac {\partial ^{2}u}{\partial y^{2}}}+d{\frac {\partial ^{2}u}{\partial y\partial z}}+e{\frac {\partial ^{2}u}{\partial z^{2}}}{\text{ + (lower-order terms)}}=0,$

which, for completely separable u (i.e. u(x,y,z)=u(x)u(y)u(z) ) gives

$a{\frac {\partial ^{2}u}{\partial x^{2}}}+c{\frac {\partial ^{2}u}{\partial y^{2}}}+e{\frac {\partial ^{2}u}{\partial z^{2}}}{\text{ + (lower-order terms)}}=0.$

This can be compared to the equation for an ellipsoid; ${x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1.$

See also

• Elliptic operator
• Hyperbolic partial differential equation
• Parabolic partial differential equation
• PDEs of second order, for fuller discussion

External links

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