Partial Differential Equation: Difference between revisions
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A [[Partial Differential Equation]] is a [[differential equation]] that involve | A [[Partial Differential Equation]] is a [[differential equation]] that involve functions and their partial derivatives. | ||
* <B>AKA:</B> [[PDE]]. | * <B>AKA:</B> [[PDE]]. | ||
* <B>Context</U>:</B> | * <B>Context</U>:</B> | ||
** | |||
** It can be associated to a [[Partial Derivative]]. | ** It can be associated to a [[Partial Derivative]]. | ||
** It can be solved by [[PDE Algorithm]]. | ** It can be solved by [[PDE Algorithm]]. | ||
* <B>Example(s):</B> | * <B>Example(s):</B> | ||
** one dimensional wave equation (<math>\frac{\partial^2 u}{\partial t^2}=c^2 \frac{\partial^2 u}{\partial x^2}</math>). | |||
** a [[Schrödinger Equation]]. | ** a [[Schrödinger Equation]]. | ||
* <B>Counter-Example(s):</B> | * <B>Counter-Example(s):</B> |
Revision as of 13:24, 7 January 2016
A Partial Differential Equation is a differential equation that involve functions and their partial derivatives.
- AKA: PDE.
- Context:
- It can be associated to a Partial Derivative.
- It can be solved by PDE Algorithm.
- Example(s):
- one dimensional wave equation ([math]\displaystyle{ \frac{\partial^2 u}{\partial t^2}=c^2 \frac{\partial^2 u}{\partial x^2} }[/math]).
- a Schrödinger Equation.
- Counter-Example(s):
- See: Differential Equation, Finite Element Algorithm, System of Equations, Fluid Flow, Continuous Distribution, Electrodynamics, Linear Map, Heat Equation, Wave Equation, Laplace's Equation.
References
2014
- (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/partial_differential_equation#Introduction Retrieved:2014-12-3.
- Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. The position of a rigid body is specified by six numbers, but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid flow, electrodynamics, and heat transfer.
A partial differential equation (PDE) for the function [math]\displaystyle{ u(x_1, \cdots, x_n) }[/math] is an equation of the form
: [math]\displaystyle{ F \left (x_1, \ldots, x_n, u, \frac{\partial u}{\partial x_1}, \ldots, \frac{\partial u}{\partial x_n}, \frac{\partial^2 u}{\partial x_1 \partial x_1}, \ldots, \frac{\partial^2 u}{\partial x_1 \partial x_n}, \ldots \right) = 0. }[/math]
If F is a linear function of u and its derivatives, then the PDE is called linear. Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein–Gordon equation, and Poisson's equation.
A relatively simple PDE is
: [math]\displaystyle{ \frac{\partial u}{\partial x}(x,y) = 0.~ }[/math]
This relation implies that the function u(x,y) is independent of x. However, the equation gives no information on the function's dependence on the variable y. Hence the general solution of this equation is
: [math]\displaystyle{ u(x,y) = f(y), }[/math]
where f is an arbitrary function of y. The analogous ordinary differential equation is
: [math]\displaystyle{ \frac{\mathrm{d} u}{\mathrm{d} x}(x) = 0, }[/math]
which has the solution
: [math]\displaystyle{ u(x) = c, }[/math]
where c is any constant value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions. A solution of a PDE is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function f(y) can be determined if u is specified on the line x = 0.
- Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. The position of a rigid body is specified by six numbers, but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid flow, electrodynamics, and heat transfer.
2009
- http://en.wikipedia.org/wiki/Partial_differential_equation
- In mathematics, partial differential equations (PDE) are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and its (or their) partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic.
1950
- (Hopf, 1950) ⇒ Eberhard Hopf. (1950). "The partial differential equation ut + uux = xx." In: Communications on Pure and Applied Mathematics, 3(3).
1944
- (Bateman, 1944) ⇒ Harry Bateman. (1944). "Partial Differential Equations." Read Books. ISBN:1406743747