In mathematics, a hyperbolic partial differential equation of order
is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first
[citation needed] More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface.
The equation has the property that, if u and its first time derivative are arbitrarily specified initial data on the line t = 0 (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.
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, due to Lars Gårding, in the context of microlocal analysis.
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
[1] 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.
By a linear change of variables, any equation of the form
(lower order derivative terms)
{\displaystyle A{\frac {\partial ^{2}u}{\partial x^{2}}}+2B{\frac {\partial ^{2}u}{\partial x\partial y}}+C{\frac {\partial ^{2}u}{\partial y^{2}}}+{\text{(lower order derivative terms)}}=0}
can be transformed to the wave equation, apart from lower order terms which are inessential for the qualitative understanding of the 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.
[2]: 402 The following is a system of first-order partial differential equations for
are once continuously differentiable functions, nonlinear in general.
has s distinct real eigenvalues, it follows that it is diagonalizable.
In this case the system (∗) is called strictly hyperbolic.
is symmetric, it follows that it is diagonalizable and the eigenvalues are real.
In this case the system (∗) is called symmetric hyperbolic.
There is a connection between a hyperbolic system and a conservation law.
Consider a hyperbolic system of one partial differential equation for one unknown function
can be interpreted as a quantity that moves around according to the flux given by
are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and
which means that the time rate of change of