The Partial differential equation reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Partial differential equation

In mathematics, and in particular calculus, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function.

A PDE usually has many solutions; a problem often includes additional boundary conditions which restrict the solution set. Where ordinary differential equations have solutions that are families with each solution characterized by the values of some parameters, for a PDE it is more helpful to think that the parameters are function data (informally put, this means that the set of solutions is much larger). That is true fairly generally, unless the equations are heavily over-determined.

Partial differential equations are ubiquitous in science, as they describe phenomena such as fluid flow, gravitational fields, and electromagnetic fields. They are important in fields such as aircraft simulation, computer graphics, and weather prediction. The central equations of general relativity and quantum mechanics are also partial differential equations.

Table of contents

1 Notation and examples 1.1 Laplace's equation 1.2 Wave equation 1.3 Heat equation 1.4 Advection equation 2 Methods to solve PDEs 3 Classification 3.5 Examples

Notation and examples

In PDEs, it is common to write the unknown function as u and its partial derivative with respect to the variable x as u_x, that is:

Laplace's equation

A very important and basic PDE is Laplace's equation:-

for the unknown function u(x,y,z). Solutions to this equation, known as harmonic functions, serve as the potentials of vector field in physics, such as the gravitational or electrostatic fields.

A generalization of Laplace's equation is Poisson's equation:-

where f(x,y,z) is a given function. The solutions to this equation describe potentials of gravitational and electrostatic fields in the presence of masses or electrical charges, respectively.

Wave equation

The wave equation is an equation for an unknown function u(x,y,z,t) (where we think of t as a time variable) which reads:-

Its solutions describe waves such as sound or light waves; c is a number which represents the speed of the wave. In lower dimensions, this equation describes the vibration of a string or drum. Solutions will typically be combinations of oscillating sine waves.

Heat equation

The heat equation describes the temperature in a given region over time. It is:-

Solutions will typically "even out" over time. The number k describes the thermal conductivity of the material.

Advection equation

The advection equation describes the transport of a conserved scalar in a velocity field . It is:

If the velocity field is solenoidal (that is, ), then the equation my be simplified to

The one dimensional steady flow advection equation (where is constant) is commonly referred to as the pigpen problem. If is not constant and equal to the equation is referred to as Burger's equation

The Schrödinger equation is a PDE at the heart of quantum mechanics.

Except for Burger's equation, all the above equations are linear in the sense that they can be written in the form Au = f for a given linear operator A and a given function f. Other important non-linear equations include the Navier-Stokes equations describing the flow of fluids and Einstein's field equations of general relativity.

Methods to solve PDEs

Linear PDEs are generally solved by decomposing the equation according to a set of basis functions, solving those individually and using superposition to find the solution corresponding to the boundary conditions. The method of separation of variables has many important particular applications.

There are no generally applicable methods to solve non-linear PDEs; indeed, many PDEs cannot be solved analytically at all. Nevertheless, some techniques can be used for several types of equations. The h-principle is the most powerful method to solve underdetermined equations.

The method of characteristics can be used in some cases to solve partial differential equations.

In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. An alternative are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.

Classification

A single partial differential equation, or even a system of partial differential equations, may be classified as parabolic, hyperbolic or elliptic. Given a system of partial differential equations, there is an associated matrix. Properties of the eigensystem of the associated matrix determine the classification of the system.

Examples

The matrix associated with the system

has coefficients,

The eigenvectors are (0,1) and (1,0) with eigenvalues 2 and -1. Thus, the system is hyperbolic.