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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

 Z=[A B; B C]
(2)

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 Equation

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Cite this as:

Weisstein, Eric W. "Hyperbolic Partial Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicPartialDifferentialEquation.html

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