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Second-order
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the 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_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, where , , , , , , and are functions of and and where u_x=\frac, u_=\frac and similarly for u_,u_y,u_. 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 examples of elliptic PDE's are the
Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \n ...
, \Delta u=u_+u_=0, and the
Poisson equation Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with ...
, \Delta u=u_+u_=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_+u_+\text=0 through a change of variables.


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 from the conditions of the Cauchy problem. 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 u_t=\Delta u by setting u_t=0. This means that Laplace's equation describes a steady state of the heat equation. 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.


Derivation of canonical form

We derive the canonical form for elliptic equations in two variables, u_+u_+u_+\text=0 . :\xi =\xi (x,y) and \eta=\eta(x,y) . If u(\xi, \eta)=u xi(x, y), \eta(x,y)/math>, applying the chain rule once gives :u_=u_\xi \xi_x+u_\eta \eta_x and u_=u_\xi \xi_y+u_\eta \eta_y, a second application gives :u_=u_ _x+u_ _x+2u_\xi_x\eta_x+u_\xi_+u_\eta_, :u_=u_ _y+u_ _y+2u_\xi_y\eta_y+u_\xi_+u_\eta_, and :u_=u_ \xi_x\xi_y+u_ \eta_x\eta_y+u_(\xi_x\eta_y+\xi_y\eta_x)+u_\xi_+u_\eta_. We can replace our PDE in x and y with an equivalent equation in \xi and \eta :au_ + 2bu_ + cu_ \text= 0,\, where :a=A^2+2B\xi_x\xi_y+C^2, :b=2A\xi_x\eta_x+2B(\xi_x\eta_y+\xi_y\eta_x) +2C\xi_y\eta_y , and :c=A^2+2B\eta_x\eta_y+C^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(^2-^2)+2B(\xi_x\xi_y-\eta_x\eta_y)+C(^2-^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^2+2B\phi_x\phi_y+C^2=0. Since the discriminant B^2-AC<0, this equation has two distinct solutions, :,=\frac which are complex conjugates. Choosing either solution, we can solve for \phi(x,y), and recover \xi and \eta with the transformations \xi=\operatorname \phi and \eta=\operatorname\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_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, into the canonical form :u_+u_+\text=0, as desired.


In higher dimensions

A general second-order partial differential equation in variables takes the form :\sum_^n\sum_^n a_ \frac \; \text = 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 from the conditions of the Cauchy problem. Unlike the two-dimensional case, this equation cannot in general be reduced to a simple canonical form.


See also

* Elliptic operator * Hyperbolic partial differential equation * Parabolic partial differential equation * PDEs of second order (for fuller discussion)


References


External links

* * *{{MathWorld, title=Elliptic Partial Differential Equation, id=EllipticPartialDifferentialEquation Partial differential equations