In
mathematics, a hyperbolic partial differential equation of order
is a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
(PDE) that, roughly speaking, has a well-posed
initial value problem
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or o ...
for the first
derivatives. More precisely, the
Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Eucl ...
. Many of the equations of
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects ...
are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
. In one spatial dimension, this is
:
The equation has the property that, if ''u'' and its first time derivative are arbitrarily specified initial data on the line (with sufficient smoothness properties), then there exists a solution for all time ''t''.
The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Relative to a fixed time coordinate, disturbances have a finite
propagation speed. They travel along the
characteristics of the equation. This feature qualitatively distinguishes hyperbolic equations from
elliptic partial differential equations and
parabolic partial differential equation
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivati ...
s. A perturbation of the initial (or boundary) data of an elliptic or parabolic equation is felt at once by essentially all points in the domain.
Although the definition of hyperbolicity is fundamentally a qualitative one, there are precise criteria that depend on the particular kind of differential equation under consideration. There is a well-developed theory for linear
differential operators
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
, due to
Lars Gårding, in the context of
microlocal analysis. Nonlinear differential equations are hyperbolic if their linearizations are hyperbolic in the sense of Gårding. There is a somewhat different theory for first order systems of equations coming from systems of
conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...
s.
Definition
A partial differential equation is hyperbolic at a point
provided that the
Cauchy problem is uniquely solvable in a neighborhood of
for any initial data given on a non-characteristic hypersurface passing through
.
Here the prescribed initial data consist of all (transverse) derivatives of the function on the surface up to one less than the order of the differential equation.
Examples
By a linear change of variables, any equation of the form
:
with
:
can be transformed to the
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
, apart from lower order terms which are inessential for the qualitative understanding of the equation.
This definition is analogous to the definition of a planar
hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, c ...
.
The one-dimensional
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
:
:
is an example of a hyperbolic equation. The two-dimensional and three-dimensional wave equations also fall into the category of hyperbolic PDE. This type of second-order hyperbolic partial differential equation may be transformed to a hyperbolic system of first-order differential equations.
Hyperbolic system of partial differential equations
The following is a system of
first order partial differential equations for
unknown
functions
,
, where
:
where
are once
continuously differentiable functions,
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
in general.
Next, for each
define the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
:
The system () is hyperbolic if for all
the matrix
has only
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (201 ...
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...
s and is
diagonalizable
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
.
If the matrix
has ''s'' ''distinct'' real eigenvalues, it follows that it is diagonalizable. In this case the system () is called strictly hyperbolic.
If the matrix
is symmetric, it follows that it is diagonalizable and the eigenvalues are real. In this case the system () is called symmetric hyperbolic.
Hyperbolic system and conservation laws
There is a connection between a hyperbolic system and a
conservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...
. Consider a hyperbolic system of one partial differential equation for one unknown function
. Then the system () has the form
Here,
can be interpreted as a quantity that moves around according to the
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
given by
. To see that the quantity
is conserved,
integrate () over a domain
:
If
and
are sufficiently smooth functions, we can use the
divergence theorem and change the order of the integration and
to get a conservation law for the quantity
in the general form
:
which means that the time rate of change of
in the domain
is equal to the net flux of
through its boundary
. Since this is an equality, it can be concluded that
is conserved within
.
See also
*
Elliptic partial differential equation
*
Hypoelliptic operator In the theory of partial differential equations, a partial differential operator P defined on an open subset
:U \subset^n
is called hypoelliptic if for every distribution u defined on an open subset V \subset U such that Pu is C^\infty ( smoot ...
*
Parabolic partial differential equation
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivati ...
References
Further reading
* A. D. Polyanin, ''Handbook of Linear Partial Differential Equations for Engineers and Scientists'', Chapman & Hall/CRC Press, Boca Raton, 2002.
External links
*
Linear Hyperbolic Equationsat EqWorld: The World of Mathematical Equations.
Nonlinear Hyperbolic Equationsat EqWorld: The World of Mathematical Equations.
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