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Wikipedia, In mathematics, a hyperbolic partial differential equation is usually a secondorder partial differential equation (PDE) of the form with
This definition is analogous to the definition of a planar hyperbola. The onedimensional wave equation: 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 equationsConsider the following system of s first order partial differential equations for s unknown functions , , where are once continuously differentiable functions, nonlinear in general. Now define for each a matrix We say that the system ( * ) is hyperbolic if for all the matrix has only real eigenvalues and is diagonalizable. 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 lawsThere 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 Now u can be some quantity with a flux . To show that this quantity is conserved, integrate ( * * ) over a domain Ω 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. See also
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Published  July 2009


