Partial Differential Equation

A partial differential equation (PDE) is an equation involving functions and their Partial Derivatives; for example, the Wave Equation

$${\partial^2\psi\over\partial x^2}+{\partial^2\psi\over\parti... ...rtial z^2} = {1\over v^2} {\partial^2\psi\over \partial t^2}.$$ (1)

In general, partial differential equations are much more difficult to solve analytically than are Ordinary Differential Equations. They may sometimes be solved using a Bäcklund Transformation, Characteristic, Green's Function, Integral Transform, Lax Pair, Separation of Variables, or--when all else fails (which it frequently does)--numerical methods.

Fortunately, partial differential equations of second-order are often amenable to analytical solution. Such PDEs are of the form

$$Au_{xx} + 2B u_{xy} + C u_{yy} + Du_x + Eu_y + F = 0.$$ (2)

Second-order PDEs are then classified according to the properties of the Matrix

$${\hbox{\sf Z}} \equiv \left[{\matrix{A & B\cr B & C\cr}}\right]$$ (3)

as Elliptic, Hyperbolic, or Parabolic.

If ${\hbox{\sf Z}}$ is a Positive Definite Matrix, i.e., det $({\hbox{\sf Z}})> 0$, the PDE is said to be Elliptic. Laplace's Equation and Poisson's Equation are examples. Boundary conditions are used to give the constraint $u(x,y) = g(x,y)$ on $\partial \Omega$, where

$$u_{xx} + u_{yy} = f(u_x,u_y,u,x,y)$$ (4)

holds in $\Omega$.

If det $({\hbox{\sf Z}})<0$, the PDE is said to be Hyperbolic. The Wave Equation is an example of a hyperbolic partial differential equation. Initial-boundary conditions are used to give

$$u(x,y,t) = g(x,y,t) \quad\hbox{for }x \in \partial \Omega, t > 0$$ (5)

$$u(x,y,0) = v_0(x,y) \quad\hbox{in } \Omega$$ (6)

$$u_t(x,y,0)=v_1(x,y) \quad\hbox{in } \Omega,$$ (7)

where

$$u_{xy} = f(u_x,u_t,x,y)$$ (8)

holds in $\Omega$.

If det $({\hbox{\sf Z}})=0$, the PDE is said to be parabolic. The Heat Conduction Equation equation and other diffusion equations are examples. Initial-boundary conditions are used to give

$$u(x,t) = g(x,t) \quad\hbox{for } x \in \partial \Omega, t>0$$ (9)

$$u(x,0) = v(x) \quad\hbox{for } x \in\Omega,$$ (10)

where

$$u_{xx} = f(u_x,u_y,u,x,y)$$ (11)

holds in $\Omega$.

See also Bäcklund Transformation, Boundary Conditions, Characteristic (Partial Differential Equation), Elliptic Partial Differential Equation, Green's Function, Hyperbolic Partial Differential Equation, Integral Transform, Johnson's Equation, Lax Pair, Monge-Ampère Differential Equation, Parabolic Partial Differential Equation, Separation of Variables

References

Partial Differential Equations

Arfken, G. ``Partial Differential Equations of Theoretical Physics.'' §8.1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 437-440, 1985.

Bateman, H. Partial Differential Equations of Mathematical Physics. New York: Dover, 1944.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Partial Differential Equations.'' Ch. 19 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 818-880, 1992.

Sobolev, S. L. Partial Differential Equations of Mathematical Physics. New York: Dover, 1989.

Sommerfeld, A. Partial Differential Equations in Physics. New York: Academic Press, 1964.

Webster, A. G. Partial Differential Equations of Mathematical Physics, 2nd corr. ed. New York: Dover, 1955.