NationMaster - Encyclopedia: Hyperbolic partial differential equation

The wave equation: The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ...

is such a hyperbolic equation. A physical interpretation is that local changes in u take time to propagate.

This type of second-order hyperbolic partial differential equation may be transformed to a hyperbolic system of first-order differential equations.

1 Hyperbolic system of partial differential equations
2 Hyperbolic system and conservation laws
3 See also
4 External links
5 Bibliography

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 $vec x in mathbb{R}^d$ In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... Partial plot of a function f. ...

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

$vec {f^j} in C^1(mathbb{R}^s, mathbb{R}^s), j = 1, ldots, d$ are once continuously differentiable functions, nonlinear in general. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ...

Now define for each $vec {f^j}$ a matrix $s times s$ In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

, for each .

We say that the system $( * )$ is hyperbolic if for all $alpha_1, ldots, alpha_d in mathbb{R}$ the matrix $A := alpha_1 A^1 + cdots + alpha_d A^d$ has only real eigenvalues and is diagonalizable. In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...

If the matrix $A$ has distinct real eigenvalues, it follows 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 In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

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

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 $Ω$ In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks. ... In calculus, the integral of a function is an extension of the concept of a sum. ...

If $u$ and $vec f$ are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and and we get a conservation law for the quantity $u$ in a common form In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradskyâ€“Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ...

External links

Linear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.
Nonlinear Hyperbolic Equations at EqWorld: The World of Mathematical Equations.

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

Category: Hyperbolic partial differential equations

Results from FactBites:

partial differential equation - definition of partial differential equation in Encyclopedia (800 words)
	In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function.
	Where ordinary differential equations have solutions that are families with each solution characterized by the values of some parameters, for a PDE it is more helpful to think that the parameters are function data (informally put, this means that the set of solutions is much larger).
	Partial differential equations are ubiquitous in science, as they describe phenomena such as fluid flow, gravitational fields, and electromagnetic fields.

Partial differential equation - Wikipedia, the free encyclopedia (3001 words)
	Partial differential equations are used to formulate and solve problems that involve unknown functions of several variables, such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, elasticity, or more generally any process that is distributed in space, or distributed in space and time.
	A solution of a partial differential equation is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined.
	In the WKB approximation it is the Hamilton-Jacobi equation.

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Contents

Hyperbolic system and conservation laws

See also

External links

Bibliography