# Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation relating the partial derivatives of an unknown function of several variables. A solution of the equation is a function satisfying this relation. The idea is to try to deduce information about an unknown function by first discovering a relationship between itself and its partial derivatives in the form of a PDE. The PDE can then be used to uncover information about the unknown function, and sometimes an explicit formula for the unknown function can be discovered.

A PDE usually has many (possibly infinitely many) solutions; a particular problem often requires additional boundary conditions which constrain 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 the solutions often are parametrized by functions (informally put, this means that the set of solutions is much larger).

Partial differential equations are ubiquitous in science and especially in physics, as physical laws can usually be written in form of PDEs. They describe phenomena such as fluid flow, the growth of crystals, diffusion, gravitation, and the behavior of 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.

## 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 ux, that is:

$\displaystyle u_x = {\part u \over \part x}$
$\displaystyle u_{xy} = {\part^2 u \over \part y\, \part x}$

Especially in (mathematical) physics, one often prefers use of the nabla operator $\displaystyle \nabla=(\part_x,\part_y,\part_z)$ for spatial derivatives and a dot ($\displaystyle \dot u$ ) for time derivatives, e.g. to write the wave equation (see below) as $\displaystyle \ddot u=c^2\nabla^2u$ .

### Laplace's equation

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

$\displaystyle u_{xx} + u_{yy} + u_{zz} = 0$

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

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

$\displaystyle u_{xx} + u_{yy} + u_{zz} = f$

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:

$\displaystyle u_{tt} = c^2( u_{xx} + u_{yy} + u_{zz} )$

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:

$\displaystyle u_t = k ( u_{xx} + u_{yy} + u_{zz} )$

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

### Euler-Tricomi equation

The Euler-Tricomi equation is used in the investigation of transonic flow. It is

$\displaystyle u_{xx}=xu_{yy}$

The advection equation describes the transport of a conserved scalar $\displaystyle \psi$ in a velocity field $\displaystyle {\bold u}=(u,v,w)$ . It is:

$\displaystyle \psi_t+(u\psi)_x+(v\psi)_y+(w\psi)_z=0.$

If the velocity field is solenoidal (that is, $\displaystyle \nabla\cdot{\bold u}=0$ ), then the equation may be simplified to

$\displaystyle \psi_t+\psi.u_x+\psi.u_y+\psi.w_z=0.$

The one dimensional steady flow advection equation $\displaystyle \psi_t+u.\psi_x=0$ (where $\displaystyle u$ is constant) is commonly referred to as the pigpen problem. If $\displaystyle u$ is not constant and equal to $\displaystyle \psi$ the equation is referred to as Burgers' equation.

### Ginzburg-Landau equation

The Ginzburg-Landau equation occurs in a wide variety of applications. It is

$\displaystyle iu_t+pu_{xx} +q|u|^2u=i\gamma u$

where $\displaystyle p,q\in\mathbb{C}$ and $\displaystyle \gamma\in\mathbb{R}$ are constants and $\displaystyle i$ is the imaginary unit.

### The Dym equation

The Dym equation is named for Harry Dym and occurs in the study of solitons. It is

$\displaystyle u_t = u^3u_{xxx}.$

### Other examples

The Schrödinger equation is a PDE at the heart of non-relativistic quantum mechanics. In the WKB approximation it is the Hamilton-Jacobi equation.

Except for Burgers' 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, when possible, 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. Still, existence and uniqueness results (such as the Cauchy-Kovalevskaya theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis).

Nevertheless, some techniques can be used for several types of equations. The h-principle is the most powerful method to solve underdetermined equations. The Riquier-Janet theory is an effective method for obtaining information about many analytic overdetermined systems.

The method of characteristics can be used in some very special 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. Alternatives 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

Second-order partial differential equations, and systems of second-order PDEs, can usually be classified as parabolic, hyperbolic or elliptic. This classification gives an intuitive insight into the behavior of the system itself. Assuming $\displaystyle u_{xy}=u_{yx},$ the general second-order PDE is of the form

$\displaystyle Au_{xx} + Bu_{xy} + Cu_{yy} + \cdots = 0,$

which looks remarkably similar to the equation for a conic section:

$\displaystyle Ax^2 + Bxy + Cy^2 + \cdots = 0.$

Just as one classifies conic sections into parabolic, hyperbolic, and elliptic based on the discriminant $\displaystyle B^2 - 4AC$ , the same can be done for a second-order PDE.

1. $\displaystyle B^2 - 4AC < 0$  : elliptic equations tend to smooth out any disturbances. A typical example is Laplace's equation. The motion of a fluid at sub-sonic speeds can be approximated with elliptic PDEs.
2. $\displaystyle B^2 - 4AC = 0$  : parabolic equations tend to smooth out any pre-existing disturbances in the data. A typical example is the heat equation.
3. $\displaystyle B^2 - 4AC > 0$  : hyperbolic equations tend to amplify any disturbances in the data. A typical example is the wave equation. The motion of a fluid at super-sonic speeds can be approximated with hyperbolic PDEs.

This method of classification can easily be extended to systems of partial differential equations by examining the eigenvalues of the coefficient matrix. In this situation, the classification scheme becomes:

1. Elliptic: The eigenvalues are all positive or all negative.
2. Parabolic : The eigenvalues are all positive or all negative, save one which is zero.
3. Hyperbolic : There is at least one negative and at least one positive eigenvalue, and none of the eigenvalues are zero.

This matches with positive-definite and negative-definite matrix analysis, of the sort that comes up during a discussion of maxima and minima.

### Examples

The matrix associated with the system

$\displaystyle u_t+2v_x=0$
$\displaystyle v_t-u_x=0$

has coefficients,

$\displaystyle \begin{bmatrix} 2 & 0 \\ 0 & -1\end{bmatrix}$

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

### Equations of mixed type

If a PDE has coefficients which are not constant, it is possible that it will not belong to any of these categories but rather be of mixed type. A simple but important example is the Euler-Tricomi equation

$\displaystyle u_{xx}=xu_{yy}$

which is called elliptic-hyperbolic because it is elliptic in the region x > 0, hyperbolic in the region x < 0, and degenerate parabolic on the line x = 0.