: Mathematics: hyperbolic partial differential equation

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hyperbolic partial differential equation

In mathematics, a hyperbolic partial differential equation of order ${\\displaystyle n}$ is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first ${\\displaystyle n-1}$ 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 ${\\displaystyle {\ rac {\\partial ^{2}u}{\\partial t^{2}}}=c^{2}{\ rac {\\partial ^{2}u}{\\partial x^{2}}}}$ The equation has the property that, if $u$ and its first time derivative are arbitrarily specified initial data on the line $t = 0$ (with sufficient smoothness properties), then there exists a solution for all time $t$ .
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Hyperbolic_partial_differential_equation)

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