Partial differential equation

In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. The idea is to describe a function indirectly by a relation between itself and its partial derivatives, rather than writing down a function explicitly. The relation should be local - it should connect the function and its derivatives in the same point. A solution of the equation is any function satisfying this relation.

A PDE usually has several solutions; a problem often includes 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 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.

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:

$u_{x}={\partial u \over \partial x}$

$u_{xy}={\partial ^{2}u \over \partial y\,\partial x}$

Especially in (mathematical) physics, one often prefers use of the nabla operator $\nabla =(\partial _{x},\partial _{y},\partial _{z})$ for spatial derivatives and a dot (${\dot {u}}$) for time derivatives, e.g. to write the wave equation (see below) as ${\ddot {u}}=c^{2}\nabla ^{2}u$.

Laplace's equation

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

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

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:-

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

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:

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

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:

$u_{t}=k(u_{xx}+u_{yy}+u_{zz})$

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

Euler-Tricomi equation

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

$u_{xx}=xu_{yy}$

Advection equation

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

$\psi _{t}+(u\psi )_{x}+(v\psi )_{y}+(w\psi )_{z}=0.$

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

$\psi _{t}+\psi .u_{x}+\psi .u_{y}+\psi .w_{z}=0.$

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

Ginzburg-Landau equation

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

$iu_{t}+pu_{xx}+q|u|^{2}u=i\gamma u$

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

The Dym equation

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

$u_{t}=u^{3}u_{xxx}.$

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. However, most problems in science and engineering are tackled using scientific computing rather than numerical analysis, as usually it is not known whether the numerical methods used produce solutions close to the true ones.

Classification

Second-order partial differential equations, and systems of second-order PDEs, can usually be classified as parabolic, hyperbolic or elliptic. These classifications give an insight into the intuitive behavior of the system itself. The general second-order PDE has equation

$Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots =0$

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

$Ax^{2}+2Bxy+Cy^{2}+\cdots =0$

The reason $B$ has a coefficient of 2 is due to the assumed commutativity of partial derivatives in the first case, and the commutativity of multiplication in the second. Just as one classifies conic sections into parabolic, hyperbolic, and elliptic based on the discriminant $B^{2}-2AC$, the same can be done for a second-order PDE.

1. $B^{2}-2AC<0$ : elliptic equations tend to smooth out any disturbances. Air resistance at sub-sonic speeds can be modeled with elliptic PDEs.
2. $B^{2}-2AC=0$ : parabolic equations tend to preserve any pre-existing disturbances in the data, but do not amplify the disturbance. Air resistance at mach 1 is representative of parabolic PDEs.
3. $B^{2}-2AC>0$ : hyperbolic equations tend to amplify any disturbances in the data. Air resistance at super-sonic speeds can be modeled 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

$u_{t}+2v_{x}=0$

$v_{t}-u_{x}=0$

has coefficients,

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

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

External links

• PDE example problems at exampleproblems.com
• Partial Differential Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
• Partial Differential Equations: Index at EqWorld: The World of Mathematical Equations.
• Partial Differential Equations: Methods at EqWorld: The World of Mathematical Equations.

References

• A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
• A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton, 2004. ISBN 1-58488-355-3
• A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
• D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic Press, Boston, 1997.