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Topic: Hyperbolic partial differential equation

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Partial differential equation

Elliptic partial differential equation

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	Partial differential equation - Wikipedia, the free encyclopedia
		Partial differential equations are used to formulate and solve problems that involve unknown functions of several variables, such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, elasticity, or more generally any process that is distributed in space, or distributed in space and time.
		A solution of a partial differential equation is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined.
		In the WKB approximation it is the Hamilton-Jacobi equation.
	en.wikipedia.org /wiki/Partial_differential_equation (3001 words)

	Hyperbolic partial differential equation - Wikipedia, the free encyclopedia
		A hyperbolic partial differential equation is usually a second-order partial differential equation of the form
		This type of second-order hyperbolic partial differential equation may be transformed to a hyperbolic system of first-order differential equations.
		There is a connection between a hyperbolic system and a conservation law.
	en.wikipedia.org /wiki/Hyperbolic_partial_differential_equation (253 words)

	Partial differential equation Article, Partialdifferentialequation Information (Site not responding. Last check: 2007-11-04)
		In mathematics, and in particular calculus, a partial differential equation (PDE) is an equation involving partial derivatives ofan unknown function.
		Where ordinary differential equations have solutionsthat are families with each solution characterized by the values of some parameters, for a PDE it is more helpful to think thatthe parameters are function data (informally put,this means that the set of solutions is much larger).
		Partial differential equations are ubiquitous in science, as they describe phenomena such as fluid flow, gravitational fields,and electromagnetic fields.
	www.anoca.org /equations/solutions/partial_differential_equation.html (719 words)

	Partial Differential Equation (Site not responding. Last check: 2007-11-04)
		In mathematics, and in particular calculus, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function.
		The Schrödinger equation is a PDE at the heart of quantum mechanics.
		A single partial differential equation, or even a system of partial differential equations, may be classified as parabolic, hyperbolic or elliptic.
	www.wikiverse.org /partial-differential-equation (721 words)

	ipedia.com: Partial differential equation Article (Site not responding. Last check: 2007-11-04)
		In mathematics, and in particular calculus, a partial differential equation is an equation involving partial derivatives of an unknown function.
		The advection equation describes the transport of a conserved scalar in a velocity field.
		Linear PDEs are generally solved by decomposing the equation according to a set of basis functions, solving those individually and using superposition to find the solution corresponding to the boundary conditions.
	www.ipedia.com /partial_differential_equation.html (770 words)

	Using Mathematica to Enhance Learning of Oceanographic Process: Breaking of Waves and Burger's Equation -- from ...
		This is especially the case in the treatment of the solutions of linear partial differential equations, where by using normal modes and Fourier series one takes advantage of the student's familiarities with eigenvalues and eigenvectors and the principles of superposition to build the solutions of such equations.
		A notable exception is the method of characteristics for hyperbolic differential equations whose effectiveness is documented for both linear and nonlinear equations.
		Because hyperbolic equations are the primary examples of equations that support wave propagation, these equations enjoy a special status in oceanography, especially in the context of underwater acoustics and wave formation in shallow coastal waters.
	library.wolfram.com /infocenter/Articles/3465 (260 words)

	Language and Classification
		The order of the equation is the order of the highest partial derivative of u, the number of variables is simply the number of independent variables for u in the equation, and the equation has constant coefficients if u and the coefficients of all the partial derivatives of u are constant.
		In thinking of partial differential equations, it is a common practice to carry over the language that has been used for matrix or ordinary differential equations in as far as possible.
		) = 0 is a parabolic partial differential equation,
	www.mathphysics.com /pde/herod/jvh10.html (1342 words)

	Propagating fronts (Site not responding. Last check: 2007-11-04)
		Numerical solutions to a model equation that describes cell population dynamics are presented and analyzed.
		A distinctive feature of the model equation (a hyperbolic partial differential equation) is the presence of delayed arguments in the time (t) and maturation (x) variables due to the non-zero length of the cell cycle.
		This nonlinear, nonlocal, and delayed kinetic term is also shown to be responsible for the existence of a Hopf bifurcation and subsequent period doublings to apparent ``chaos" along the characteristics of this hyperbolic partial differential equation.
	www.amath.washington.edu /~crabb/propagating.html (300 words)

	Classification
		An example hyperbolic partial differential equation is the one governing flow into a packed bed in which a species contained in the inlet can be absorbed onto the packing.
		The example equation is a partial differential equation because the dependent variable (concentration) is a function of two independent variables, time and spatial position.
		This equation is elliptic, and the primary distinguishing characteristic is that the solution at any point is influenced by the boundary conditions on the entire surface.
	www.mindspring.com /~ravenna/Nonlinear/class/class.htm (1609 words)

	Elliptic operator - Wikipedia, the free encyclopedia
		In mathematics, an elliptic operator is one of the major types of differential operator P.
		The usual partial differential equations involving time, such as the heat equation and the Schrödinger equation, also contain elliptic operators involving the spacial variables, as well time derivatives.
		Their solutions (harmonic functions of a general kind) tend to be smooth functions (if the coefficients in the operator are continuous).
	en.wikipedia.org /wiki/Elliptic_partial_differential_equation (245 words)

	XVIII. Partial differential operators.
		Notice that this transformation to achieve an equation lacking the first derivative with respect to x is generally possible when the coefficient on the second derivative with respect to x is not zero, and is otherwise impossible.
		We choose the constants as the coefficients for the partial differential equations
		For ordinary differential equations boundary value problems, the dot product came with the problem in a sense: it was an integral over an appropriate interval on which the functions were defined.
	www.mathphysics.com /pde/ch18wr.html (4173 words)

	Hypercomplex Electromagnetic Theory (Site not responding. Last check: 2007-11-04)
		The second equation is the reformulated Lorentz condition which, in conjunction with the vectorial Maxwell's equations, assures continuity.
		The Maxwell's equations in a non-vacuum medium are inhomogeneous wave equations.
		This approach is doubly attractive because the boundary conditions are often given as gradients (first-order partial differentials) that are normal to the boundary.
	home.usit.net /~cmdaven/eelectro.htm (1014 words)

	CFL Condition (Site not responding. Last check: 2007-11-04)
		The domain of dependence of a hyperbolic partial differential equation (PDE) for a given point in the problem domain is that portion of the problem domain that influences the value of the solution at the given point.
		For the wave equation, the domain of dependence for a given point is an isosceles triangle having its apex at the given point, one edge on the spatial axis, and remaining edges with slopes
		The standard finite difference scheme for the wave equation uses centered, second-order accurate differences in both space and time (see Wave Equation module), whose stencil also yields a triangular-shaped domain of dependence as the numerical scheme marches forward in time.
	www.cse.uiuc.edu /eot/modules/pde/wavecfl (387 words)

	Upcoming Events \| Department of Computational and Applied Mathematics \| Rice University (Site not responding. Last check: 2007-11-04)
		We consider the initial value problem for a strictly hyperbolic partial differential equation on the circle.
		The transformation involves applying differential operators, solving an elliptic differential equation, and applying a coordinate transformation involving the characteristics, which can be done at cost O(N).
		The resulting ODE is solved using a multiscale time-stepping method, which results in an algorithm of complexity O(N) for the original hyperbolic equation.
	dacnet.rice.edu /~caam/calendar/index.cfm?EventRecord=5982&TimeFrame=8 (89 words)

	An partial differential equation (Site not responding. Last check: 2007-11-04)
		In general, partial differential equations are much more difficult to solve analytically than are ordinary differential equations.
		Fortunately, partial differential equations of second-order are often amenable to analytical solution.
		The wave equation is an example of a hyperbolic partial differential equation.
	www.tdbasics.net /pde.html (158 words)

	Encyclopedia article on Aerodynamics [EncycloZine] (Site not responding. Last check: 2007-11-04)
		Mathematically, supersonic flow is described by a hyperbolic partial differential equation while subsonic flow is described by an elliptic partial differential equation.
		Another example of the difference between supersonic and subsonic flow is the behaviour in a convergent duct (known as a nozzle in subsonic flow and a diffuser in supersonic flow).
		The curved shock waves chemically alter the surrounding air or gas, creating a partially ionized plasma with their high temperatures (caused in part by significant aerodynamic heating of the body).
	encyclozine.com /Aerodynamics (1279 words)

	[No title]
		Analytical and numerical techniques for solving differential systems with either a single independent variable or multiple independent variables are used.
		Obtain the differential equations from the mathematical modeling of various physical and other systems.
		Solve the hyperbolic partial differential equation by the method of characteristics.
	www.csupomona.edu /~tknguyen/egr509/O-509-02.doc (328 words)

	Mathematics of characteristics (Site not responding. Last check: 2007-11-04)
		If the initial data are given on a curve not transverse to the characteristics, such as on a characteristics itself, then the equation does not necessarily have a solution.
		We define a characteristic curve to be a curve that if data are given on that curve, the differential equation does not enable us to determine the solution at any point not on the characteristic.
		In the general case of anisentropic flow, the equations of motion cannot be written in the form of differential operators, Eq.
	astron.berkeley.edu /~jrg/ay202/node181.html (746 words)

	Partial differential operators.
		We will also assume that the boundary of the region is piece-wise smooth, and denote this boundary by \boundary D. Just as in ordinary differential equations, in partial differential equations some boundary conditions will be needed to solve the equations.
		Before we pursue the idea of rescaling and translating in second order partial differential equations in order to come up with standard forms, we need to recall that there is also the troublesome need to rotate the axis in order to get some quadratic forms into the standard one.
		The techniques will change these equations into the standard forms for elliptic, hyperbolic, or parabolic partial differential equations.
	www.mathphysics.com /pde/green/g16.html (4669 words)

	Citebase - A scalar hyperbolic equation with GR-type non-linearity (Site not responding. Last check: 2007-11-04)
		Authors: Khokhlov, A. Novikov, I. We study a scalar hyperbolic partial differential equation with non-linear terms similar to those of the equations of general relativity.
		We study the stability of three-dimensional numerical evolutions of the Einstein equations, comparing the standard ADM formulation to variations on a family of formulations that separate out the conformal and traceless parts of the system.
		In the 3+1 framework of the Einstein equations for the case of vanishing shift vector and arbitrary lapse, we calculate explicitly the four boundary equations arising from the vanishing of the projection of the Einstein tensor along the normal to the boundary surface of the initial-boundary value...
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:gr-qc/0303063 (865 words)

	Classification of Partial Differential Equations
		Examples of hyperbolic, parabolic, and elliptic equations are the classical wave equation, the diffusion equation, and Laplace's equation, representing the application of PDEs to sound and light, heat, and electrostatic phenomena, respectively.
		Elliptic equations produce stationary and energy-minimizing solutions, parabolic equations produce a smooth-spreading flow of an initial disturbance, and hyperbolic equations produce a propagating disturbance.
		Another useful classification of partial differential equations is based on a property called linearity.
	webphysics.davidson.edu /Faculty/wc/WaveHTML/node6.html (337 words)

	BFS 2002 Contributed Talks Abstracts (Site not responding. Last check: 2007-11-04)
		We approximate the partial differential operator on this grid by appealing to the SDE representation of the stock process and computing the logarithm of the transition probability matrix of an approximating Markov chain.
		Similarly as transition densities and stochastic differential equations, this notion describes the local behaviour of a stochastic process.
		Furthermore, we partially extend this approach to model the joint distribution of returns with a certain class of operator stable laws with diagonal exponent, thus allowing each return to have a different index of tail thickness.
	www.ma.utexas.edu /Bachelier2002/BFS2002.Contributed_abstracts.html (16794 words)

	Citations: estimates for Galerkin methods for second order hyperbolic equations - Dupont (ResearchIndex)
		These conventional numerical methods use an Eulerian approach in which one uses a static spatial mesh and discretizes the governing equations in time instead of along the characteristics, and do not utilize the facts that second order hyperbolic equations have two families of characteristics....
		Traditionally, for non hereditary problems of type P or H, the approach to such problems is to semidiscretize in space, using the finite element method, to arrive at a time continuous system, and then to discretize this system using classical methods, see e.g.
		Galerkin methods of solving a system of ODE s are given in [23,36] and adaptive schemes are developed and discussed with comparisons to standard ODE techniques in [39,30] Also, in [3] the implementation of the hp version of the finite element (in time) method for parabolic problems is....
	citeseer.ist.psu.edu /context/7809/0 (1832 words)

	Computing optimal control with a hyperbolic partial differential equation (Site not responding. Last check: 2007-11-04)
		We present a method for solving a class of optimal control problems involving hyperbolic partial differential equations.
		A numerical integration method for the solution of a general linear second-order hyperbolic partial differential equation representing the type of dynamics under consideration is given.
		Fast automatic differentiation is applied to calculate the exact gradient of the discretized problem so that existing optimization algorithms may be imposed on the problem.
	anziamj.austms.org.au /V40/part2/Rehbock.html (163 words)

	Stochastic partial differential equations
		Here is a list of research papers related to stochastic partial differential equations (SPDEs) and their applications.
		On a class of stochastic partial differential equations related to turbulent transport Probab Theory Relat Fields 111 (1998) 1, 101-122.
		Semi-discretization of stochastic partial differential equations on R by a finite-difference method, Mathematics of Computation, 69 (2000), 653-666..
	www.cmap.polytechnique.fr /~rama/spde/articles.htm (1032 words)

	Pearson Education - Friendly Introduction to Numerical Analysis, A (Site not responding. Last check: 2007-11-04)
		The multigrid method and irregular domains for elliptic partial differential equations.
		Source and decay terms, polar coordinates and problems in two space dimensions for parabolic partial differential equations.
		Numerical dispersion and diffusion and the convection-diffusion equation.
	www.pearsoned.co.uk /Bookshop/detail.asp?item=144019 (1066 words)

	Tutorial (Partial Differential Equation Toolbox)
		The basic equation of the PDE Toolbox is the PDE
		Analogously, we shall use the terms parabolic equation and hyperbolic equation for equations with spatial operators like the one above, and first and second order time derivatives, respectively.
		For the parabolic and hyperbolic PDE the coefficients c, a, f, and d can depend on time.
	www-rohan.sdsu.edu /doc/matlab/toolbox/pde/1tut3.html (309 words)

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