Elliptic partial differential equation

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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. 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.

The simplest nontrivial examples of elliptic PDE's are the Laplace equation, \nabla ^2 u=u_{xx}+u_{yy}=0, and the Poisson equation, \nabla ^2 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]

Qualitative Behavior[edit]

Elliptic equations have no real characteristic curves, curves along which it is not possible to eliminate at least one second derivative of u from the conditions of the Cauchy problem.[1] Since characteristic curves are the only curves along with 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 u_t=\nabla^2 u by setting u_t=0. This means that Laplace's equation describes a steady state of 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[edit]

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

\xi =\xi (x,y) and \eta=\eta(x,y) .

If u(\xi, \eta)=u[\xi(x, y), \eta(x,y)], applying the chain rule once gives

u_{x}=u_\xi \xi_x+u_\eta \eta_x and u_{y}=u_\xi \xi_y+u_\eta \eta_y,

a second application gives

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},

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

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 \xi and \eta

au_{\xi\xi} + 2bu_{\xi\eta} + cu_{\eta} \text{ + (lower-order terms)}= 0,\,

where

a=A{\xi_x}^2+2B\xi_x\xi_y+C{\xi_y}^2,

b=2A\xi_x\eta_x+2B(\xi_x\eta_y+\xi_y\eta_x) +2C\xi_y\eta_y , and

c=A{\eta_x}^2+2B\eta_x\eta_y+C{\eta_y}^2.

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

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=2A\xi_x\eta_x+2B(\xi_x\eta_y+\xi_y\eta_x) +2C\xi_y\eta_y ,

Adding i times the second equation to the first and setting \phi=\xi+ i \eta gives the quadratic equation

A{\phi_x}^2+2B\phi_x\phi_y+C{\phi_y}^2=0.

Since the discriminant  B^2-AC<0, this equation has two distinct solutions,

{\phi_x},{\phi_y}=\frac{B\pm i\sqrt{AC-B^2}}{A}

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

Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu +G= 0,\,

into the canonical form

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

as desired.

In higher dimensions[edit]

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

\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\part^2 u}{\partial x_i \partial x_j} \quad \text{ + (lower-order terms)} =0.

This equation is considered elliptic if there are no characteristic surfaces, 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[edit]

References[edit]

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

External links[edit]