Partial differential equations (PDEs)
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
(ODEs) 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 ux, that is:
Especially in (mathematical) physics, one often prefers use of the nabla operator
for spatial derivatives and a dot ( ) for time derivatives, e.g. to
write the wave equation (see below) as
.
Laplace's equation
A very important and basic PDE is Laplace's equation:uxx + uyy + uzz = 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:uxx + uyy + uzz = 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:
utt = c2(uxx + uyy + uzz)
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:
ut = k(uxx + uyy + uzz)
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
uxx = xuyy
Advection equation
The advection equation describes the transport of a conserved scalar function ψ in a
velocity field
. It is:
ψt + (uψ)x + (vψ)y + (wψ)z = 0.
If the velocity field is solenoidal (that is,
simplified to
), then the equation may be
ψt + ψ.ux + ψ.uy + ψ.wz = 0.
The one dimensional steady flow advection equation ψt + u.ψx = 0 (where u is
constant) is commonly referred to as the pigpen problem. If u is not constant and
equal to ψ the equation is referred to as Burgers' equation.
Ginzburg-Landau equation
The Ginzburg-Landau equation occurs in a wide variety of applications. It is
iut + puxx + q | u | 2u = iγu
where
and
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
ut = u3uxxx.
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 hprinciple 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 ranging from finite difference schemes to multigrid,
finite element and finite volume 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. This classification gives an
intuitive insight into the behaviour of the system itself. The general second-order PDE
is of the form
which looks remarkably similar to the equation for a conic section:
The reason B has a coefficient of 2 is due to the assumed commutativity of partial
derivatives in the first case (recall that mixed derivatives which are continuous do not
depend on the order of taking the partial derivatives in the different variables!) , and
the commutativity of multiplication in the second. Just as one classifies conic sections
into parabolic, hyperbolic, and elliptic based on the discriminant B2 − AC, the same
can be done for a second-order PDE.
1. B2 − AC < 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. B2 − AC = 0 : parabolic equations tend to smooth out any pre-existing
disturbances in the data. A typical example is the heat equation.
3. B2 − AC > 0 : hyperbolic equations tend to amplify any disturbances in the
data. A typical example is the wave equation. The motion of a fluid at supersonic 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. Moreover, using the
concepts of positive-definiteness and negative-definiteness, it is possible to extend
this classification to PDEs and systems of PDEs which are of order higher than 2 (as
well as for systems of PDEs of 1st order).
Examples
The matrix associated with the system
ut + 2vx = 0
vt − ux = 0
has coefficients,
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
uxx = xuyy
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.
External links
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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
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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 1-58488355-3
A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order
Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-41527267-X
D. Zwillinger, Handbook of Differential Equations (3rd edition), Academic
Press, Boston, 1997.