Partial Differential Equation

INTRODUCTION

A very simple partial differential equation is
:

rac{\part u}{\part x}=0\,

where ''u'' is an unknown function of ''x'' and ''y''. This relation implies that the values ''u''(''x'',''y'') are actually 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

:

rac{du}{dx}=0,\,

which has the solution

:

u(x) = c,\,

where ''c'' is any constant value (independent of ''x''). These two examples illustrate that general solutions of ordinary differential equations involve arbitrary constants, but solutions of partial differential equations involve arbitrary functions. 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. For instance, the upper half-plane consists of points (''x'',''y'') where ''x'' is any real number and

y \ge 0

. The Cauchy problem for a partial differential equation of second order in the upper half-plane consists in specifying the value of the function and its derivative with respect to ''y'' where ''y'' = 0.

Although the issue of the existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard–Lindelöf Theorem , that is far from the case for partial differential equations. There is a general theorem (the Cauchy-Kovalevskaya Theorem ) that states that the Cauchy problem for any partial differential equation that is analytic in the unknown function and its derivatives has a unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: see Lewy (1957). Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties.

An example of pathological behavior is the sequence of Cauchy problems (depending upon ''n'') for the Laplace equation

:

rac{\part^2 u}{\part x^2} + rac{\part^2 u}{\part y^2}=0,

with initial conditions

:

u(x,0) = 0, \,

rac{\part u}{\part y}(x,0) = rac{\sin n x}{n^2},\,

where ''n'' is an integer. The derivative of ''u'' with respect to ''y'' approaches 0 uniformly in ''x'' as ''n'' increases, but the solution is

:

u(x,y) = rac{(\sinh ny)(\sin nx)}{n^2}.\,

This solution approaches infinity if ''nx'' is not an integer multiple of
π for any non-zero value of ''y''. The Cauchy problem for the Laplace equation is called ''ill-posed'' or ''not well posed'', since the solution does not depend continuously upon the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications. Only special types of partial differential equation problems are well posed, and well posed problems are generally closely related to physical applications.

NOTATION AND EXAMPLES

In PDEs, it is common to denote partial derivatives using subscripts. That is:

:

u_x = {\part u \over \part x}

u_{xy} = {\part^2 u \over \part y\, \part x} = {\part  \over \part y } \left({\part u  \over \part x}
ight).

Especially in (mathematical) physics, one often prefers use of the gradient operator Nabla Operator

abla=(\part_x,\part_y,\part_z)

for spatial derivatives and a dot (

\dot u

) for time derivatives, e.g. to write the Wave Equation (see below) as

:

\ddot u=c^2
abla^2u.\,

Heat equation in one space dimension

The equation for conduction of heat in one dimension has the form

:

u_t = \alpha u_{xx} \,

where ''u''(''t'',''x'') is temperature, and α is a positive constant that describes the rate of diffusion. The Cauchy problem for this equation consists in specifying

u(0,x)= f(x)

,
where ''f(x)'' is an arbitrary function.

General solutions of the heat equation can be found by the method of Separation Of Variables . Some examples appear in Heat Equation .
They are examples of Fourier Series for periodic ''f'' and Fourier Integral s for non-periodic ''f''. Using the Fourier integral, a
general solution of the heat equation has the form

:

u(t,x) = rac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} F(\xi) e^{-\alpha \xi^2 t} e^{i \xi x} d\xi, \,

where ''F'' is an arbitrary function. In order to satisfy the initial condition, ''F'' (the Fourier transform of ''f'') is given by the Fourier Integral

:

F(\xi) = rac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-i \xi x}\, dx. \,

If ''f'' represents a very small but intense source of heat, then the preceding integral can be approximated by the Delta Distribution , multiplied by the strength of the source. For a source whose strength is normalized to 1, the result is

:

F(\xi) = rac{1}{\sqrt{2\pi}}, \,

and the resulting solution of the heat equation is

:

u(t,x) = rac{1}{2\pi} \int_{-\infty}^{\infty}e^{-\alpha \xi^2 t} e^{i \xi x} d\xi. \,

This is a Gaussian Integral . It may be evaluated to obtain

:

u(t,x) = rac{1}{2\sqrt{\pi \alpha t}} \exp\left(-rac{x^2}{4 \alpha t} 
ight). \,

This result corresponds to a normal probability density for ''x'' with mean 0 and variance 2α''t''. The heat equation and similar Diffusion Equation s are useful tools to study random phenomena.

Wave equation in one space dimension

The Wave Equation is an equation for an unknown function ''u''(''t'', ''x'') of the form

:

u_{tt} \, = c^2 u_{xx}. \,

Here ''u'' might describe the displacement of a stretched string from equilibrium, or the difference in air pressure in a tube, or the magnitude of an electromagnetic field in a tube, and ''c'' is a number that corresponds to the velocity of the wave. The Cauchy problem for this equation consists in prescribing the initial displacement and velocity of the string or other medium:

:

u(0,x) = f(x), \,

u_t(0,x) = g(x), \,

where ''f'' and ''g'' are arbitrary functions. The solution of this problem is given in Wave Equation :

:

+ rac{1}{2c}\int_{x-ct}^{x+ct} g(y)\, dy. \,

This formula implies that the solution at (''t'',''x'') depends only upon the data on the segment of the initial line that is cut out by the characteristic curves

:

x - ct = \hbox{constant,} \quad x + ct = \hbox{constant}, \,

that are drawn backwards from that point. These curves correspond to signals that propagate with velocity ''c'' forward and backward.
Conversely, the influence of the data at any given point on the initial line propagates with the finite velocity ''c'': there is no effect outside a triangle through that point whose sides are characteristic curves. This behavior
is very different from the solution for the heat equation, where the effect of a point source appears (with small amplitude) instantaneously at every point in space. The solution given above is also valid if ''t'' is negative, and the explicit formula shows that the solution depends smoothly upon the data: both the forward and backward Cauchy problems for the wave equation are well-posed.

Spherical waves

Spherical waves are waves whose amplitude depends only upon the radial distance ''r'' from a central point. For such waves, the three-dimensional wave equation takes the form

:

u_{tt} = c^2 \left + rac{2}{r} u_r 
ight . \,

This is equivalent to

:

(ru)_{tt} = c^2 \left[(ru)_{rr} 
ight],\,

and hence the quantity ''ru'' satisfies the one-dimensional wave equation. Therefore a general solution for spherical waves has the form

:

u(t,r) = rac{1}{r} \left + G(r+ct) 
ight,\,

where ''F'' and ''G'' are completely arbitrary functions. Radiation from an antenna corresponds to the case where ''G'' is identically zero. Thus the wave form transmitted from an antenna has no distortion in time: the only distorting factor is 1/''r''. This feature of undistorted propagation of waves is not present if there are two spatial dimensions.

Laplace equation in two dimensions

The Laplace Equation for an unknown function of two variables φ has the form

:

arphi_{xx} + arphi_{yy} = 0.\,

:

Connection with analytic functions

Solutions of the Laplace equation are intimately connected with analytic functions of a complex variable: the real and imaginary parts of any analytic function are conjugate harmonic functions: they both satisfy the Laplace equation, and their gradients are othogonal. If ''f''=''u''+''iv'', then the Cauchy-Riemann Equations state that

:

u_x = v_y, \quad v_x = -u_y,\,

and it follows that
:

u_{xx} + u_{yy} = 0, \quad v_{xx} + v_{yy}=0. \,

Conversely, given any harmonic function, it is the real part of an analytic function, at least locally. Details are given in Laplace Equation .

A typical boundary value problem

A typical problem for Laplace's equation is to find a solution that satisfies arbitrary values on the boundary of a domain. For example, we may seek a harmonic function that takes on the values ''u''(θ) on a circle of radius one. The solution was given by Poisson :

:

arphi(r,	heta) = rac{1}{2\pi} \int_0^{2\pi} rac{1-r^2}{1 +r^2 -2r\cos (	heta -	heta')} u(	heta')d	heta'.\,

Petrovsky (1967, p. 248) shows how this formula can be obtained by summing a Fourier series for φ. If ''r''<1, the derivatives of φ may be computed by differentiating under the integral sign, and one can verify that φ is analytic, even if ''u'' is continuous but not necessarily differentiable. This behavior is typical for solutions of , and more general Hyperbolic Partial Differential Equation s, which typically have no more derivatives than the data.

Euler-Tricomi equation

The Euler-Tricomi Equation is used in the investigation of Transonic flow. It is

:

u_{xx} \, =xu_{yy}

Advection equation

The Advection Equation describes the transport of a conserved scalar

\psi

in a velocity field

{\bold  u}=(u,v,w)

. It is:

:

\psi_t+(u\psi)_x+(v\psi)_y+(w\psi)_z \, =0.

If the velocity field is Solenoidal (that is,

abla\cdot{\bold u}=0

), then the equation may be simplified to

:

\psi_t+u\psi_x+v\psi_y+w\psi_z \, =0.

The one dimensional steady flow advection equation

\psi_t+u.\psi_x=0

(where

u

is constant) is commonly referred to as the Pigpen Problem . If

u

is not constant and equal to

\psi

the equation is referred to as Burgers' Equation .

Ginzburg-Landau equation

The Ginzburg-Landau Equation is used in modelling Superconductivity . It is
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	CATEGORIES ABOUT PARTIAL DIFFERENTIAL EQUATION

	multivariable calculus

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Information About

Partial Differential Equation