Hyperbolic partial differential equation
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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 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is
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
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
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 .[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.
Examples
By a linear change of variables, any equation of the form
The one-dimensional wave equation:
Hyperbolic system of partial differential equations
The following is a system of first order partial differential equations for unknown functions , , where :
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(∗)
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where are once continuously differentiable functions, nonlinear[disambiguation needed] in general.
Next, for each define the
The system (∗) is hyperbolic if for all the matrix has only
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
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(∗∗)
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Here, can be interpreted as a quantity that moves around according to the flux 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
See also
- Elliptic partial differential equation
- Hypoelliptic operator
- Parabolic partial differential equation
References
- ^ Rozhdestvenskii, B.L. (2001) [1994], "Hyperbolic partial differential equation", Encyclopedia of Mathematics, EMS Press
- ^ OCLC 465190110
Further reading
- A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
External links
- "Hyperbolic partial differential equation, numerical methods", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Linear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.
- Nonlinear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.