Rdxfbgrtj

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

Auxx+2Buxy+Cuyy+Dux+Euy+Fu+G=0,{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 ux=∂u∂x{displaystyle u_{x}={frac {partial u}{partial x}}} and similarly for uxx,uy,uyy,uxy{displaystyle u_{xx},u_{y},u_{yy},u_{xy}}. A PDE written in this form is elliptic if

B2−AC<0,{displaystyle B^{2}-AC<0,}

with this naming convention inspired by the equation for a planar ellipse.

The simplest nontrivial examples of elliptic PDE's are the Laplace equation, Δu=uxx+uyy=0{displaystyle Delta u=u_{xx}+u_{yy}=0}, and the Poisson equation, Δu=uxx+uyy=f(x,y).{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

uxx+uyy+ (lower-order terms)=0{displaystyle u_{xx}+u_{yy}+{text{ (lower-order terms)}}=0}

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

Qualitative behavior

Elliptic equations have no real characteristic curves, curves along which it is not possible to eliminate at least one second derivative of u{displaystyle u} from the conditions of the Cauchy problem.^[1] Since characteristic curves are the only curves along which solutions to partial differential equations with smooth parameters can have discontinuous derivatives, solutions to elliptic equations cannot have discontinuous derivatives anywhere. This means elliptic equations are well suited to describe equilibrium states, where any discontinuities have already been smoothed out. For instance, we can obtain Laplace's equation from the heat equation ut=∇2u{displaystyle u_{t}=nabla ^{2}u} by setting ut=0{displaystyle u_{t}=0}. This means that Laplace's equation describes a steady state of the heat equation.^[2]

In parabolic and hyperbolic equations, characteristics describe lines along which information about the initial data travels. Since elliptic equations have no real characteristic curves, there is no meaningful sense of information propagation for elliptic equations. This makes elliptic equations better suited to describe static, rather than dynamic, processes.^[2]

Derivation of canonical form

We derive the canonical form for elliptic equations in two variables, uxx+uxy+uyy+ (lower-order terms)=0{displaystyle u_{xx}+u_{xy}+u_{yy}+{text{ (lower-order terms)}}=0}.

ξ=ξ(x,y){displaystyle xi =xi (x,y)} and η=η(x,y){displaystyle eta =eta (x,y)}.

If u(ξ,η)=u[ξ(x,y),η(x,y)]{displaystyle u(xi ,eta )=u[xi (x,y),eta (x,y)]}, applying the chain rule once gives

ux=uξξx+uηηx{displaystyle u_{x}=u_{xi }xi _{x}+u_{eta }eta _{x}} and uy=uξξy+uηηy{displaystyle u_{y}=u_{xi }xi _{y}+u_{eta }eta _{y}},

a second application gives

uxx=uξξξ2x+uηηη2x+2uξηξxηx+uξξxx+uηηxx,{displaystyle u_{xx}=u_{xi xi }{xi ^{2}}_{x}+u_{eta eta }{eta ^{2}}_{x}+2u_{xi eta }xi _{x}eta _{x}+u_{xi }xi _{xx}+u_{eta }eta _{xx},}

uyy=uξξξ2y+uηηη2y+2uξηξyηy+uξξyy+uηηyy,{displaystyle u_{yy}=u_{xi xi }{xi ^{2}}_{y}+u_{eta eta }{eta ^{2}}_{y}+2u_{xi eta }xi _{y}eta _{y}+u_{xi }xi _{yy}+u_{eta }eta _{yy},} and

uxy=uξξξxξy+uηηηxηy+uξη(ξxηy+ξyηx)+uξξxy+uηηxy.{displaystyle u_{xy}=u_{xi xi }xi _{x}xi _{y}+u_{eta eta }eta _{x}eta _{y}+u_{xi eta }(xi _{x}eta _{y}+xi _{y}eta _{x})+u_{xi }xi _{xy}+u_{eta }eta _{xy}.}

We can replace our PDE in x and y with an equivalent equation in ξ{displaystyle xi } and η{displaystyle eta }

auξξ+2buξη+cuηη + (lower-order terms)=0,{displaystyle au_{xi xi }+2bu_{xi eta }+cu_{eta eta }{text{ + (lower-order terms)}}=0,,}

where

a=Aξx2+2Bξxξy+Cξy2,{displaystyle a=A{xi _{x}}^{2}+2Bxi _{x}xi _{y}+C{xi _{y}}^{2},}

b=2Aξxηx+2B(ξxηy+ξyηx)+2Cξyηy,{displaystyle b=2Axi _{x}eta _{x}+2B(xi _{x}eta _{y}+xi _{y}eta _{x})+2Cxi _{y}eta _{y},} and

c=Aηx2+2Bηxηy+Cηy2.{displaystyle c=A{eta _{x}}^{2}+2Beta _{x}eta _{y}+C{eta _{y}}^{2}.}

To transform our PDE into the desired canonical form, we seek ξ{displaystyle xi } and η{displaystyle eta } such that a=c{displaystyle a=c} and b=0{displaystyle b=0}. This gives us the system of equations

a−c=A(ξx2−ηx2)+2B(ξxξy−ηxηy)+C(ξy2−ηy2)=0{displaystyle a-c=A({xi _{x}}^{2}-{eta _{x}}^{2})+2B(xi _{x}xi _{y}-eta _{x}eta _{y})+C({xi _{y}}^{2}-{eta _{y}}^{2})=0}

b=0=2Aξxηx+2B(ξxηy+ξyηx)+2Cξyηy,{displaystyle b=0=2Axi _{x}eta _{x}+2B(xi _{x}eta _{y}+xi _{y}eta _{x})+2Cxi _{y}eta _{y},}

Adding i{displaystyle i} times the second equation to the first and setting ϕ=ξ+iη{displaystyle phi =xi +ieta } gives the quadratic equation

Aϕx2+2Bϕxϕy+Cϕy2=0.{displaystyle A{phi _{x}}^{2}+2Bphi _{x}phi _{y}+C{phi _{y}}^{2}=0.}

Since the discriminant B2−AC<0{displaystyle B^{2}-AC<0}, this equation has two distinct solutions,

ϕx,ϕy=B±iAC−B2A{displaystyle {phi _{x}},{phi _{y}}={frac {Bpm i{sqrt {AC-B^{2}}}}{A}}}

which are complex conjugates. Choosing either solution, we can solve for ϕ(x,y){displaystyle phi (x,y)}, and recover ξ{displaystyle xi } and η{displaystyle eta } with the transformations ξ=Re⁡ϕ{displaystyle xi =operatorname {Re} phi } and η=Im⁡ϕ{displaystyle eta =operatorname {Im} phi }. Since η{displaystyle eta } and ξ{displaystyle xi } will satisfy a−c=0{displaystyle a-c=0} and b=0{displaystyle b=0}, so with a change of variables from x and y to η{displaystyle eta } and ξ{displaystyle xi } will transform the PDE

Auxx+2Buxy+Cuyy+Dux+Euy+Fu+G=0,{displaystyle Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu+G=0,,}

into the canonical form

uξξ+uηη+ (lower-order terms)=0,{displaystyle u_{xi xi }+u_{eta eta }+{text{ (lower-order terms)}}=0,}

as desired.

In higher dimensions

A general second order partial differential equation in n variables takes the form

∑i=1n∑j=1nai,j∂2u∂xi∂xj + (lower-order terms)=0.{displaystyle sum _{i=1}^{n}sum _{j=1}^{n}a_{i,j}{frac {partial ^{2}u}{partial x_{i}partial x_{j}}};{text{ + (lower-order terms)}}=0.}

This equation is considered elliptic if there are no characteristic surfaces, i.e. surfaces along which it is not possible to eliminate at least one second derivative of u from the conditions of the Cauchy problem.^[1]

Unlike the two dimensional case, this equation cannot in general be reduced to a simple canonical form.^[2]

See also

• Elliptic operator


• Hyperbolic partial differential equation


• Parabolic partial differential equation


• 
  PDEs of second order (for fuller discussion)

References

External links

• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Elliptic partial differential equation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Elliptic partial differential equation, numerical methods", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• Weisstein, Eric W. "Elliptic Partial Differential Equation". MathWorld.