 parabolic equation

any of a class of partial differential equations arising in the mathematical analysis of diffusion phenomena, as in the heating of a slab. The simplest such equation in one dimension, u_{xx} = u_{t}, governs the temperature distribution at the various points along a thin rod from moment to moment. The solutions to even this simple problem are complicated, but they are constructed largely from a function called the fundamental solution of the equation, given by an exponential function, exp [(−x^{2}/4t)/t^{1/2}]. To determine the complete solution to this type of problem, the initial temperature distribution along the rod and the manner in which the temperature at the ends of the rod is changing must also be known. These additional conditions are called initial values and boundary values (boundary value), respectively, and together are sometimes called auxiliary conditions.In the analogous two and threedimensional problems, the initial temperature distribution throughout the region must be known, as well as the temperature distribution along the boundary from moment to moment. The differential equation in two dimensions is, in the simplest case,u_{xx} + u_{yy} = u_{t},with an additional u_{zz} term added for the threedimensional case. These equations are appropriate only if the medium is of uniform composition throughout, while, for problems of nonuniform composition or for some other diffusiontype problems, more complicated equations may arise. These equations are also called parabolic in the given region if they can be written in the simpler form described above by using a different coordinate system. An equation in one dimension the higherorder terms of which areau_{xx} + bu_{xt} + cu_{tt}can be so transformed if b^{2} − 4ac = 0. If the coefficients a, b, c depend on the values of x, the equation will be parabolic in a region if b^{2} − 4ac = 0 at each point of the region.
* * *
Universalium. 2010.
Look at other dictionaries:
Parabolic cylindrical coordinates — are a three dimensional orthogonal coordinate system that results from projecting the two dimensional parabolic coordinate system in theperpendicular z direction. Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic… … Wikipedia
Parabolic coordinates — are a two dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three dimensional version of parabolic coordinates is obtained by rotating the two dimensional system about the symmetry axis of the… … Wikipedia
Parabolic antenna — A parabolic satellite communications antenna at Erdfunkstelle Raisting, the biggest facility for satellite communication in the world, in Raisting, Bavaria, Germany. It has a Cassegrain type feed. A parabolic antenna is an antenna that uses a… … Wikipedia
Parabolic partial differential equation — A parabolic partial differential equation is a type of second order partial differential equation, describing a wide family of problems in science including heat diffusion and stock option pricing. These problems, also known as evolution problems … Wikipedia
Parabolic trajectory — In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit.Under standard assumptions a body… … Wikipedia
Parabolic cylinder function — In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation:frac{d^2f}{dz^2} + left(az^2+bz+c ight)f=0.This equation is found, for example, when the technique of separation of variables … Wikipedia
Équation aux dérivées partielles — En mathématiques, plus précisément en calcul différentiel, une équation aux dérivées partielles (EDP) est une équation dont les solutions sont les fonctions inconnues vérifiant certaines conditions concernant leurs dérivées partielles. Une EDP a… … Wikipédia en Français
Parabolic constant — In mathematics, the ratio of the arc length of the parabolic segment formed by the latus rectum of any parabola to its focal parameter is a constant, denoted ! P 2 .In the diagram, the latus rectum is pictured in blue, the parabolic segment that… … Wikipedia
parabolic — 1. adjective a) Of, or pertaining to, or in the shape of a parabola or paraboloid b) Of or pertaining to a parable 2. noun A parabolic function, equation etc … Wiktionary
Partial differential equation — A visualisation of a solution to the heat equation on a two dimensional plane In mathematics, partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several… … Wikipedia