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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 numerical analysis, the FTCS (forward time-centered space) method is a finite difference method used for numerically solving the heat equation and similar parabolic partial differential equations. [1] It is a first-order method in time, explicit in time, and is conditionally stable when applied to the heat equation.
The Lax–Friedrichs method, named after Peter Lax and Kurt O. Friedrichs, is a numerical method for the solution of hyperbolic partial differential equations based on finite differences. The method can be described as the FTCS (forward in time, centered in space) scheme with a numerical dissipation term of 1/2.
In mathematics, the convergence condition by Courant–Friedrichs–Lewy (CFL) is a necessary condition for convergence while solving certain partial differential equations (usually hyperbolic PDEs) numerically. It arises in the numerical analysis of explicit time integration schemes, when these are used for the numerical solution.
Hyperbolic partial differential equations (10 P) I. Integrable systems (2 C, 41 P) M. ... Parabolic partial differential equations (17 P) PDE theorists (142 P)
The most important implication of the hyperbolic equation is that by switching from a parabolic (dissipative) to a hyperbolic (includes a conservative term) partial differential equation, there is the possibility of phenomena such as thermal resonance [12] [13] [14] and thermal shock waves. [15]