In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
hyperbolic partial differential equation of order
n is a
partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
(PDE) that, roughly speaking, has a wellposed
initial value problemIn mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...
for the first
n−1 derivatives. More precisely, the
Cauchy problemA Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain. Cauchy problems are an extension of initial value problems and are to be contrasted with boundary value problems...
can be locally solved for arbitrary initial data along any noncharacteristic hypersurface. Many of the equations of
mechanicsMechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the
wave equationThe wave equation is an important secondorder linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
. 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 initial line
t = 0 (with sufficient smoothness properties), then there exists a solution for all time.
The solutions of hyperbolic equations are "wavelike." 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
characteristicsIn mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to firstorder equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation...
of the equation. This feature qualitatively distinguishes hyperbolic equations from elliptic partial differential equations and
parabolic partial differential equationA parabolic partial differential equation is a type of secondorder partial differential equation , describing a wide family of problems in science including heat diffusion, ocean acoustic propagation, in physical or mathematical systems with a time variable, and which behave essentially like heat...
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 welldeveloped theory for linear differential operators, due to
Lars GårdingLars Gårding is a Swedish mathematician. He has made notable contributions to the study of partial differential operators. He is a professor emeritus of mathematics at Lund University in Sweden...
, in the context of
microlocal analysisIn mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variablecoefficientslinear and nonlinear partial differential equations...
. 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 lawIn physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
s.
Definition
A partial differential equation is hyperbolic at a point
P provided that the
Cauchy problemA Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain. Cauchy problems are an extension of initial value problems and are to be contrasted with boundary value problems...
is uniquely solvable in a neighborhood of
P for any initial data given on a noncharacteristic hypersurface passing through
P. Here the prescribed initial data consists 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, 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.
The onedimensional
wave equationThe wave equation is an important secondorder linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
:
is an example of a hyperbolic equation. The twodimensional and threedimensional wave equations also fall into the category of hyperbolic PDE.
This type of secondorder hyperbolic partial differential equation may be transformed to a hyperbolic system of firstorder differential equations.
Hyperbolic system of partial differential equations
Consider the following system of
first order partial differential equations for
unknown
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s
,
, where
are once
continuouslyIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
differentiableIn calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a nonvertical tangent line at each point in its domain...
functions, nonlinear in general.
Now define for each
a
matrixIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
We say that the system
is
hyperbolic if for all
the matrix
has only
realIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
eigenvalues and is
diagonalizableIn linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
.
If the matrix
has
distinct real eigenvalues, it follows that it's diagonalizable. In this case the system
is called
strictly hyperbolic.
Hyperbolic system and conservation laws
There is a connection between a hyperbolic system and a
conservation lawIn physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....
. Consider a hyperbolic system of one partial differential equation for one unknown function
. Then the system
has the form
Now
can be some quantity with a
fluxIn the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
. To show that this quantity is conserved,
integrateIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
over a domain
If
and
are sufficiently smooth functions, we can use the
divergence theoremIn vector calculus, the divergence theorem, also known as Gauss' theorem , Ostrogradsky's theorem , or Gauss–Ostrogradsky theorem is a result that relates the flow of a vector field through a surface to the behavior of the vector field inside the surface.More precisely, 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
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highestorder derivatives be positive, which implies the key property that the principal symbol is...
 Parabolic partial differential equation
A parabolic partial differential equation is a type of secondorder partial differential equation , describing a wide family of problems in science including heat diffusion, ocean acoustic propagation, in physical or mathematical systems with a time variable, and which behave essentially like heat...
 Hypoelliptic operator
 Relativistic heat conduction
The theory of relativistic heat conduction claims to be the only model for heat conduction that is compatible with the theory of special relativity, the second law of thermodynamics, electrodynamics, and quantum mechanics, simultaneously...
External links