 partial differential equation

Math.a differential equation containing partial derivatives. Cf. ordinary differential equation.[188590]
* * *
In mathematics, an equation that contains partial derivatives, expressing a process of change that depends on more than one independent variable.It can be read as a statement about how a process evolves without specifying the formula defining the process. Given the initial state of the process (such as its size at time zero) and a description of how it is changing (i.e., the partial differential equation), its defining formula can be found by various methods, most based on integration. Important partial differential equations include the heat equation, the wave equation, and Laplace's equation, which are central to mathematical physics.* * *
in mathematics, equation relating a function of several variables to its partial derivatives (derivative). A partial derivative of a function of several variables expresses how fast the function changes when one of its variables is changed, the others being held constant (compare ordinary differential equation). The partial derivative of a function is again a function, and, if f(x, y) denotes the original function of the variables x and y, the partial derivative with respect to x—i.e., when only x is allowed to vary—is typically written as f_{x}(x, y) or ∂f/∂x. The operation of finding a partial derivative can be applied to a function that is itself a partial derivative of another function to get what is called a secondorder partial derivative. For example, taking the partial derivative of f_{x}(x, y) with respect to y produces a new function f_{xy}(x, y), or ∂^{2}f/∂y∂x. The order and degree of partial differential equations are defined the same as for ordinary differential equations.In general, partial differential equations are difficult to solve, but techniques have been developed for simpler classes of equations called linear, and for classes known loosely as “almost” linear, in which all derivatives of an order higher than one occur to the first power and their coefficients involve only the independent variables.Many physically important partial differential equations are secondorder and linear. For example:● u_{xx} + u_{yy} = 0 (twodimensional Laplace equation (Laplace's equation))● u_{xx} = u_{t} (onedimensional heat equation)● u_{xx} − u_{yy} = 0 (onedimensional wave equation)The behaviour of such an equation depends heavily on the coefficients a, b, and c of au_{xx} + bu_{xy} + cu_{yy}. They are called elliptic, parabolic, or hyperbolic equations according as b^{2} − 4ac < 0, b^{2} − 4ac = 0, or b^{2} − 4ac > 0, respectively. Thus, the Laplace equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic.* * *
Universalium. 2010.
Look at other dictionaries:
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
partial differential equation — noun a differential equation involving a functions of more than one variable • Hypernyms: ↑differential equation … Useful english dictionary
partial differential equation — noun Date: 1845 a differential equation containing at least one partial derivative … New Collegiate Dictionary
partial differential equation — noun a differential equation that involves the partial derivatives of a function of several variables … Wiktionary
partial differential equation — noun Mathematics an equation containing one or more partial derivatives … English new terms dictionary
Hyperbolic partial differential equation — In mathematics, a hyperbolic partial differential equation is usually a second order partial differential equation (PDE) of the form :A u {xx} + 2 B u {xy} + C u {yy} + D u x + E u y + F = 0 with: det egin{pmatrix} A B B C end{pmatrix} = A C B^2 … Wikipedia
First order partial differential equation — In mathematics, a first order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables. The equation takes the form: F(x 1,ldots,x n,u,u {x 1},ldots u {x n}) =0 … Wikipedia
List of partial differential equation topics — This is a list of partial differential equation topics, by Wikipedia page. Contents 1 General topics 2 Specific partial differential equations 3 Numerical methods for PDEs 4 … 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
Dispersive partial differential equation — In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities.… … Wikipedia