Partial derivative equation

NB: Being translated according to the English article ---- In Mathematical, more precisely in differential Calculus, a equation with the derivative partial or equation partial Différentielle (EDP) is an equation whose solutions are the unknown functions checking certain partial conditions relating to their derivative.

A EDP often has very many solutions, the conditions being less strict than in the case of a differential equation ordinary (with only one Variable); the problems often include Boundary conditions which restricts the Ensemble solutions . Whereas the whole of solutions of a differential equation ordinary is parameterized by one or more Paramètre S corresponding to the additional conditions, in the case of EDP the boundary conditions are rather appeared as function; intuitively that means that the whole of the solutions is much larger, which is true in the near total of the problems.

The EDP are omnipresent in sciences, since they as well appear in dynamic structures, Mécanique of the fluids as in the theories of the Gravitation or the electromagnetism (Maxwell's equations). They are paramount in fields such as simulation Aéronautique, the Synthèse of image S, or the weather forecasting. Lastly, the most important equations of the General relativity and the quantum Mécanique are also EDP .

One of the ten problems to a million dollars proposed by the Clay foundation consists in showing the existence and continuity compared to the initial data of a system of EDP called equations of Navier-Stokes. These equations are useful enormously in the Mécanique of the fluids.

Introduction

A very simple differential equation is:

\ frac {\ partial U} {\ partial X} =0 \,

where U is an unknown function of X and there . This relation implies that the values U ( X , there ) are independent of X . The solutions of this equation are:

$u (X, there) = F (there), \,$

where F is a function of there . The ordinary equation,

$\ frac {of the} {dx} =0 \,$

has as a solution:

$u (X) = C, \,$

with C a constant value (independent of X ). These two examples illustrate that in general, the solutions of an ordinary differential equation brings into play an arbitrary constant, while the partial derivative equations bring into play arbitrary functions. A solution of the partial derivative equations is generally not single.

Notations and examples

For EDP , by preoccupation with a simplification, it is of use to write U the unknown function and D _X U (French notation) or U _X (Anglo-Saxon notation, more widespread) its partial derivative compared to X, is with the usual notations of differential calculus:

$u_x = {\ share U \ over \ share X}$

and for the derivative partial seconds:

$u_ {xy} = {\ part^2 U \ over \ share X \, \ share there}$

Equation of Laplace

The equation of Laplace is a very important basic EDP:

${\ part^2 U \ over \ share x^2} + {\ part^2 U \ over \ share y^2} + {\ part^2 U \ over \ share z^2} = 0$

where U (X, there, Z) indicates the unknown function.

Equation of propagation (or equation of the vibrating cords)

This EDP describes the phenomena of propagation of the sound waves and the electromagnetic waves (of which the light). The function of unknown wave is noted U (X, there, Z, T), T representing time:

${\ part^2 U \ over \ share x^2} + {\ part^2 U \ over \ share y^2} + {\ part^2 U \ over \ share z^2} = {1 \ over c^2} {\ part^2 U \ over \ share t^2}$

The number C represents the celerity or propagation velocity of the wave U.

Equation of Fourier

${\ part^2 U \ over \ share x^2} + {\ part^2 U \ over \ share y^2} + {\ part^2 U \ over \ share z^2} = {1 \ over \ alpha} {\ share U \ over \ share T}$

This EDP is also called equation of heat. The function U represents the temperature. The derivative of order 1 compared to time translates the irreversibility of the phenomenon. The number $\ alpha$ is called thermal diffusivity of the medium.

Equation of Schrödinger

Methods of numerical resolution

The numerical methods most usually used for the resolution of the partial derivative equations are:

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