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In mathematics, a hyperbolic partial differential equation of order is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first derivatives. [ citation needed ] More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface .
In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation.
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena in, i.a., engineering science, quantum mechanics and financial mathematics. Examples include the heat equation, time-dependent Schrödinger equation and the Black–Scholes ...
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.. The function is often thought of as an "unknown" that solves the equation, similar to how x is thought of as an unknown number solving, e.g., an algebraic equation like x 2 − 3x + 2 = 0.
For hyperbolic partial differential equations, the linear test problem is the constant coefficient advection equation, as opposed to the heat equation (or diffusion equation), which is the correct choice for a parabolic differential equation.
In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling , elliptic PDEs are frequently used to model steady states , unlike parabolic PDE and hyperbolic PDE which generally model phenomena that change in time.
The shallow-water equations (SWE) are a set of hyperbolic partial differential equations (or parabolic if viscous shear is considered) that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). [1]
In applied mathematics, discontinuous Galerkin methods (DG methods) form a class of numerical methods for solving differential equations.They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications.