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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 variable] s and their [partial derivative] s 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.

** Introduction **

A partial differential equation (PDE) for the function u(x_1,...x_n) is of the form

: F(x_1, \cdots x_n,u,\frac{\partial}{\partial x_1}u, \cdots \frac{\partial}{\partial x_n}u,\frac{\partial^2}{\partial x_1 \partial x_1}u, \frac{\partial^2}{\partial x_1 \partial x_2}u, \cdots ) = 0 \,

If * F * is a linear function of * u * and its derivatives, then the PDE is linear. Common examples of linear PDEs include the [heat equation] , the [wave equation] and [Laplace's equation] .

A relatively simple PDE is

: \frac{\partial}{\partial x}u(x,y)=0\, .

This relation [implies] that the function * u * ( * x * , * y * ) is independent of * x * . Hence the general solution of this equation is

: u(x,y) = f(y),\,

where * f * is an arbitrary function of * y * . The analogous [ordinary differential equation] is

: \frac{du(x)}{dx}=0\,

which has the solution

: u(x) = c,\,

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