Hyperbolic Partial Differential Equation
A partial differential equation of second-order, i.e., one of the form
| Au_(xx)+2Bu_(xy)+Cu_(yy)+Du_x+Eu_y+F=0, |
(1)
|
is called hyperbolic if the matrix
| [画像: Z=[A B; B C] ] |
(2)
|
![画像: Z=[A B; B C]](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/HyperbolicPartialDifferentialEquation/NumberedEquation2.svg){kind=link}
satisfies det(Z)<0. The wave equation is an example of a hyperbolic partial differential equation. Initial-boundary conditions are used to give
| u(x,y,t)=g(x,y,t) for x in partialOmega,t>0 |
(3)
|
| u(x,y,0)=v_0(x,y) in Omega |
(4)
|
| u_t(x,y,0)=v_1(x,y) in Omega, |
(5)
|
where
| u_(xy)=f(u_x,u_t,x,y) |
(6)
|
holds in Omega.
See also
Elliptic Partial Differential Equation, Parabolic Partial Differential Equation, Partial Differential EquationExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Hyperbolic Partial Differential Equation." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HyperbolicPartialDifferentialEquation.html