Hyperbolic partial differential equation


Hyperbolic partial differential equation

In mathematics, a hyperbolic partial differential equation is usually a second-order partial differential equation (PDE) of the form

:A u_{xx} + 2 B u_{xy} + C u_{yy} + D u_x + E u_y + F = 0

with

: det egin{pmatrix} A & B \ B & C end{pmatrix} = A C - B^2 < 0.

The one-dimensional wave equation:

:frac{partial^2 u}{partial t^2} - c^2frac{partial^2 u}{partial x^2} = 0

is an example of 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 vec u = (u_1, ldots, u_s) , vec u =vec u (vec x,t), where vec x in mathbb{R}^d

:(*) quad frac{partial vec u}{partial t} + sum_{j=1}^d frac{partial}{partial x_j} vec {f^j} (vec u) = 0,

vec {f^j} in C^1(mathbb{R}^s, mathbb{R}^s), j = 1, ldots, d are once continuously differentiable functions, nonlinear in general.

Now define for each vec {f^j} a matrix s imes s

:A^j:=egin{pmatrix} frac{partial f_1^j}{partial u_1} & cdots & frac{partial f_1^j}{partial u_s} \ vdots & ddots & vdots \ frac{partial f_s^j}{partial u_1} & cdots &frac{partial f_s^j}{partial u_s}end{pmatrix}, ext{ for }j = 1, ldots, d.

We say that the system (*) is hyperbolic if for all alpha_1, ldots, alpha_d in mathbb{R} the matrix A := alpha_1 A^1 + cdots + alpha_d A^dhas 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 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 u = u(vec x, t). Then the system (*) has the form

:(**) quad frac{partial u}{partial t} + sum_{j=1}^d frac{partial}{partial x_j} {f^j} (u) = 0,

Now u can be some quantity with a flux vec f = (f^1, ldots, f^d). To show that this quantity is conserved, integrate (**) over a domain Omega

:int_{Omega} frac{partial u}{partial t} dOmega + int_{Omega} abla cdot vec f(u) dOmega = 0.

If u and vec f are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and partial / partial t to get a conservation law for the quantity u in the general form

:frac{d}{dt} int_{Omega} u dOmega + int_{Gamma} vec f(u) cdot vec n dGamma = 0,which means that the time rate of change of u in the domain Omega is equal to the net flux of u through its boundary Gamma. Since this is an equality, it can be concluded that u is conserved within Omega.

See also

* Elliptic partial differential equation
* Parabolic partial differential equation
* Hypoelliptic operator

Bibliography

* 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

* [http://eqworld.ipmnet.ru/en/solutions/lpde/lpdetoc2.pdf Linear Hyperbolic Equations] at EqWorld: The World of Mathematical Equations.
* [http://eqworld.ipmnet.ru/en/solutions/npde/npde-toc2.pdf Nonlinear Hyperbolic Equations] at EqWorld: The World of Mathematical Equations.


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