Partial differential equation

a partial differential equation (abbreviation PDGL or PDE for close. partial differential equation) is a differential equation, which contains partial derivatives. They serve the mathematical modelling of many physical procedures. The solution theory of partial differential equations is investigated for linear equations extensively, with nonlinear equations contains the mathematical theory still many gaps. For the practical computation of solutions usually numeric procedures are consulted.

1 definition
2 example
3 organization
4 theory
5 numeric procedures of partial differential equations
of 6 examples

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definition

somewhat more exactly said is a PDE an equation (or a set of equations) for one or more unknown functions, which fulfills the following criteria:

the unknown function hangs from at least two variables (if it only of variable depends, calls one it ordinary differential equation, or briefly only differential equation)
into the PDE comes partial derivatives to at least 2 variables forwards
in the equation comes only the function, as well as their partial derivatives, in each case at the same point evaluated forwards.

The implicit form of a partial differential equation for a function <math> u< /math>, those of two variables <math> x< /math> and <math> y< /math> depends,reads

< math> F \ left (x, y, u (x, y), \ frac {\ partial u (x, y)}{\ partial x}, \ frac {\ partial u (x, y)}{\ partial y},

\ ldots \ frac {\ partial^2 u (x, y)}{\ partial x \ partial y}, \ ldots \ right) = 0, </math>

whereby F is any function.

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example

many physical processes depend both on the place and on the time. The change concerning both is not important always: the movement of a mass point is described only by derivatives after the time ( speed and acceleration). One calls so a kind of equation ordinary differential equation. Often is not sufficient however: The waves, which from a water drop, which results falls on a water surface, depend both on the time derivative (speed of the wave) thus also on the space derivative (profile of the wave). Since derivatives emerge after several variables, thus a partial differential equation is necessary for the description of the procedure.

The simplest possible example of a partial differential equation is the following: A function u (x, t) may depend on 2 variables (z. B. of place x and time t). The partial derivative <math> \ frac {\ partial u (x, t)}{\ partial t}< /math> indicates, how strongly the function in the time changes, similarly gives <math> \ frac {\ partial u (x, t)}{\ partial x} </math> the change of the function values in the local variable on. If equal these two changes are, the following differential equation results

< math> \ frac {\ partial u (x, t)}{\ partial x} = \ frac {\ partial u (x, t)}{\ partial t} </math>

A solution of this equation would be <math> u (x, t) = f (x+t)< /math> with any function <math> f< /math>.

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organization

one can divide PDG according to different criteria. One calls the degree of the highest derivative, to which in the equation seems, the order. For example only partial first derivatives arise in an equation of first order.

Further can be divided according to linearity. If the unknown function, as well as all arising derivatives occur linear, one speaks of a linear partial differential equation. If all derivatives of highest order do not arise linear, but the function and derivatives of low order, one speaks of a semilinearen equation. Here it means linear that the coefficients functions before the unknown function and/or. Their derivatives only on the variable ones depend. Hang the coefficients functions before the highest derivative additionally from lower derivatives and the unknown function speak one of a quasi-linear partial differential equation. Otherwise one speaks of a nonlinear PDG. A nonlinear partial differential equation one knows through out-differentiates into a quasi-linear form to always transfer, in which the highest derivatives emerge linear.

The simplest case is naturally the case of the linear equations. But contrary to ordinary differential equations even here formula solutions are possible only in exceptional cases.

With linear PDG one continues to differentiate between hyperbolic (e.g. the wave equation), parabolic (e.g. the thermal conduction equation) and elliptical (e.g. the Poisson equation) differential equations. Descriptive regarded the types differ by the kind of the propagation from disturbances in the solution. This classification is not clear however any longer, it gives thus partial differential equations, which have a mixed character. As example of the organization into elliptical, parabolically and hyperbolically a partial differential equation of the order 2 in 2 variables consulted:

<math> A (x, y) \ frac {\ partial^2 u (x, y)}{\ partial x^2} + b (x, y) \ frac {\ partial^2 u (x, y)}{\ partial x \ partial y} +C (x, y) \ frac {\ partial^2 u (x, y)}{\ partial y^2} + D (x, y) \ frac {\ partial u (x, y)}{\ partial y} = 0 </math>

During the organization in each case the coefficients of the highest derivatives become (here 2. Order) in the equation regards:

If <math> A (x, y) C (x, y) - b (x, y) ^2/4 > 0 </math> the equation is in (x, y) elliptically

if <math> A (x, y) C (x, y) - b (x, y) ^2/4 = 0 </math> the equation is in (x, y) parabolically

if <math> A (x, y) C (x, y) - b (x, y) ^2/4 < 0 </math> the equation is hyperbolic into (x, y)

this distinction can one also to attribute whether the matrix (A (x, y) b (x, y) /2; b (x, y) /2c (x, y)) positively definitely (⇒elliptisch), positively semidefinit, but not definitely (singularly) (⇒parabolisch), or indefinit (with exactly a negative eigenvalue) (⇒hyperbolisch) is. There A, b and C of x and y depend depend the type of the differential equation on the place. For more than three variable ones is this organization no longer completely, i.e. one can e.g. in <math> \ mathbb {R} ^4< /math> easily examples do not design those elliptically, parabolically or hyperbolic are.

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edge and initial value problems

a partial differential equation by SE have generally several solutions. Around a clear solution to gotten it needs certain additional conditions, i.e. edge - and/or initial conditions. This situation is similar as with the ordinary differential equations, where one needs initial conditions in one point. With PDGen the default of a function value at one point is not sufficient, one must function values (and/or derivatives) on a diversity give.

Not each additional condition leads the whole to a reasonable solution, depends on the kind of the equation. Typical examples are

Dirichlet boundary conditions (for elliptical problems)
Neumann boundary conditions (for elliptical problems)
at the beginning of and boundary conditions (for parabolic problems)
Cauchy problems (for hyperbolic problems)

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elliptical partial differential equations

these arise typically in connection with to time-independent (stationary) problems. A further characteristic is that elliptical equations often describe a state of minimum energy, thus from variation problems comes. The prime example is the Laplace equation, and/or the Poisson equation. These equations describe for instance (stationary) the temperature distribution in a body, or also the electrostatic charge pattern in a body. In addition (Newton's ones) the gravitation potential is a solution of the Poisson equation.

With elliptical equations the most frequently arising boundary conditions are either Dirichlet boundary conditions or Neumann boundary conditions. The first means that the values of the looked for function on the edge are given, while second is a default of the normal derivative of the looked for function. By the example of the temperature distribution the difference is to be made clear: If one is an object into ice water, then the temperature at the edge is 0 degrees. Thus the temperature distribution is inside the solution of a Dirichlet boundary value problem. Another case arises, if one isolates the body. Here not the temperature is well-known, but by isolation the heat flow at the edge is 0. Since the river can be brought with the normal derivative in connection, this leads on a Neumann problem. Something similar applies in electrostatics: If one knows the tension at the edge is put on, one comes to a Dirichlet problem, knows one however the amperage at the edge comes one to a Neumann problem.

A nonlinear equation, which is elliptical, is the equation for minimum minimum (minimum surface equation), these describes a soap skin, which forms, if one dips a wire rack into soap solution.

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parabolic partial differential equations

this type of equations describes similar phenomena as elliptical equations, but in the intermittent case. The by far most important example of a parabolic equation is the thermal conduction equation, which describes the cooling and a heating of a body. Diffusion processes will likewise describe by this equation. Parabolic equations lead to a left-hand margin value problem. For example the temperature or the temperature river must be given during the thermal conduction equation at (spatial) the edge of the area for all times either. This corresponds to the case of Dirichlet or Neumann conditions in the elliptical case. Additionally still the temperature distribution at the beginning (at the time 0) must be given. Altogether thus parabolic equations need condition at the spatial edge and to the start time. A further (nonlinear) representative of parabolic equations is Korteweg de Vries Equation, who describes water waves in bank proximity.

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hyperbolic partial differential equations

the typical hyperbolic equation is the wave equation. Generally by this kind by equations waves and their propagation are described. In addition equations of first order are always hyperbolic. In contrast to parabolic and elliptical equation solutions are not absorbed to at all by hyperbolic equations few. That leads on the one hand to the fact that the solution theory becomes more difficult, since on less differentiability can be counted. On the other hand waves can spread only by this missing absorption over far distances.

At the beginning of and boundary values belonging to to this type lead on Cauchy problems: That means that as additionally in the parabolic case for spatial boundary conditions initial values are needed. With hyperbolic equations of second order one needs however two initial values: The function value and the temporal derivative of the same at the beginning. By the example of a clamped string this is to be clarified: The deflection of the string fulfills the wave equation. If the string at the ends is clamped, this leads on the spatial boundary conditions, in this case is deflection at the edge 0 (because clamped), thus is the function value at the edge well-known and it arises Dirichlet boundary conditions. (In the case of freely swinging objects, like in woodwind instruments comes one that-correspond on Neumann conditions). Additionally now still two initial conditions must be given: Deflection (corresponds to the function value) at the beginning is angezupft, and the speed with that the string at the beginning (corresponds to the temporal derivative). With this conditions deflection can be solved at all later times clearly.

Hyperbolic equations with in pairs different eigenvalues are called strictly hyperbolic. Here the solution theory is well-known also for nonlinear systems, is the equation not-strictly hyperbolically, as for example the multidimensional Euler equations or the equations of the magnetohydrodynamics, is this no more the case.

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