This definition is analogous to the definition of a planar hyperbola.
The one-dimensional wave equation:
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
Consider the following system of s first order partial differential equations for s unknown functions , , where
Now define for each a matrix
If the matrix A 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 law. Consider a hyperbolic system of one partial differential equation for one unknown function . Then the system ( * ) has the form
If u 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 u in the general form
which means that the time rate of change of u in the domain Ω is equal to the net flux of u through its boundary Γ. Since this is an equality, it can be concluded that u is conserved within Ω.
Published - July 2009
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