Differential equation

A differential equation (also differential equation, often abbreviated by DGL or DG) is a mathematical equation for an unknown function of one or more variables, in which occur also derivatives of this function. Many natural laws can be formulated by means of differential equations. Differential equations are therefore an essential tool of mathematical modeling. Here, a differential equation describes the changing behavior of each of these variables. Differential equations are an important object of study of the analysis, which examines their solution theory. Not only because for many differential equations no explicit solution representation is possible to play the approximate solution using numerical methods a significant role. A differential equation can be illustrated by a field direction.

• 3.1 Lie theory
• 3.2 Existence and Uniqueness
• 3.3 Approximate methods

Types of differential equations

There are different types of differential equations. Roughly speaking, they are divided into the following sub- areas. All of the following types can occur independently and simultaneously substantially adjacent.

Ordinary Differential Equations

→ Main article ordinary differential equation

Depends on the unknown function only of one variable from, this is called an ordinary differential equation. There are only ordinary derivatives before after a variable.

Examples:

Writes to the ordinary differential equation for the unknown function in the form

Is the name of the ordinary differential equation implicitly.

Is the differential equation solved for the maximum discharge, i.e. it is

It is called explicitly the ordinary differential equation. In the applications of explicit ordinary differential equations are mathematically easier to work with. The highest occurring derivative of order is called the order of the differential equation. For example, an explicit ordinary differential equation first order the shape

There is a completed solution theory of explicit ordinary differential equations.

Partial differential equation

→ Main article Partial Differential Equation

Depends on the unknown function of several variables and enter into the equation in partial derivatives by more than one variable, it is called a partial differential equation. Partial differential equations are a large field and the theory is mathematically not completed, but the subject of current research in several areas.

An example is the so-called heat conduction equation for a function

There are different types of partial differential equations. First, there are linear partial differential equations. Here, the unknown function and its derivatives go straight into the equation. The dependence with respect to the independent variables can be quite non-linear. The theory of linear partial differential equations is the most advanced, however, is far from being completed.

One speaks of a semilinear equation, if all the derivatives of the highest order occur linearly, but this is no longer true for the function and derivatives of lower order. A semilinear equation is more difficult to treat.

From a quasi-linear partial differential equation is called, if the coefficient functions depend additionally on before the highest derivatives of lower derivatives and the unknown function. Especially in the field of quasi-linear equations, most results are obtained at the time.

Finally, one can also detect not a linear dependency with respect to the same derivative, is called the equation is a non-linear partial differential equation or a fully non- linear partial differential equation.

Of particular interest in the field of partial differential equations, the second order equations. In these special cases, there are other classification options.

Other types

When type of stochastic differential equations occur in the equation on so-called stochastic processes. Actually stochastic differential equations are not as defined above differential equations, but rather only certain differential ratios, which can be interpreted as a differential equation.

The type of differential-algebraic equations is characterized in that in addition to the differential equation algebraic relations are still given as constraints.

Next, there is so-called retarded differential equations. Here come next to a function and its derivatives at a time even function values ​​and derivatives from the past.

Under an integro - differential equation is understood an equation in which not only the function and its derivatives, but also the functional integrations emerge. An important example is the Schrödinger equation in the momentum representation ( Fredholm integral equation ).

Depending on the application and methodology, there are other types of differential equations.

Systems of differential equations

We speak of a system of differential equations, if a vector- illustration, and more equations

Have to be met simultaneously. Can this implicit system of differential equations are not locally everywhere in an explicit system to convert, it is a differential-algebraic equation.

Problems

The solution set of a differential equation is in general not uniquely determined by the equation itself, but additionally needs another initial or boundary values ​​. In the field of partial differential equations, so-called initial boundary value problems may occur.

Basically, one of the variables is interpreted as a time at the beginning or initial boundary value problems. These issues are some of the data at a certain point, namely the initial time prescribed.

The boundary-value or initial-boundary value problems a solution of the differential equation is sought in a restricted or unrestricted area and we as a so-called edge data values ​​which are just given on the edge of the area. Depending on the type of boundary conditions are distinguished further types of differential equations, such as Dirichlet or Neumann problems - problems.

Solution methods

Due to the diversities in both the actual differential equations as well as the problems it is not possible to give a general solution methodology. Only explicit ordinary differential equations can be solved with a closed theory. A differential equation is called integrable if it is possible, to solve analytically, so a solution function (integral ) indicated. Very many mathematical problems, in particular non-linear and partial differential equations are not integrable, including quite seem simple, like that of the three-body problem, the double pendulum or the most centrifugal types.

Lie theory

A structured general approach for solving differential equations is pursued through the symmetry and the continuous group theory. 1870 presented Sophus Lie in his work the theory of differential equations with the Lie theory on a general basis. He showed that the older mathematical theories for solving differential equations by the introduction of so-called Lie groups can be combined. A general approach for solving differential equations makes use of the symmetry property of the differential equations. This continuous infinitesimal transformations are applied that represent solutions to ( other ) solutions of the differential equation. Continuous group theory, Lie algebras and differential geometry are used to capture the deeper structure of the linear and nonlinear ( partial) differential equations, and map the relationships that eventually lead to the exact analytical solutions of a differential equation. Symmetry methods are used to solve differential equations exactly.

Existence and uniqueness

The questions of existence, uniqueness, representation and the numerical calculation of solutions are thus not resolved, depending on the equation completely up. Because of the importance of differential equations in practice, in this case the application of the numerical solution methods for partial differential equations is particularly more advanced than their theoretical underpinnings.

One of the Millennium problems is the proof of existence of a regular solution for so-called Navier -Stokes equations. These equations occur for example in fluid mechanics.

Approximate methods

Differential equations have a solution functions satisfy the conditions on their derivatives. An approximation is usually done by time and space are decomposed by a computational grid in a finite number of parts ( discretization ). The derivatives are then no longer represented by a limit, but approximated by differences. In the numerical analysis period, the resulting error is analyzed and estimated as well as possible.

Depending on the type of the equation different discretization are chosen for partial differential equations such as finite difference method, finite volume method or finite element method.

The discretized differential equation contains no more leads, but only purely algebraic expressions. This results in either a direct solution procedure or a linear or non-linear system of equations, which can then be solved using numerical methods.

Occurrence and applications

A variety of phenomena in nature and technology can be described by differential equations and that build mathematical models. Some typical examples are:

• Many physical theories are differential equations to reason: equations of motion or vibrations in the Newtonian mechanics, the stress behavior of components, electro-dynamics is dominated by the Maxwell equations, the quantum mechanics of the Schrödinger equation.
• In astronomy, the orbits of the heavenly bodies and the turbulence in the interior of the sun,
• In biology about processes in growth, in currents, or in muscles, or in the theory of evolution.
• In the chemical kinetics of reactions
• In electrical engineering the behavior of networks with energy-storing elements,
• In differential geometry, the behavior of surfaces,
• In fluid mechanics, the behavior of these flows precisely,
• In economics, the analysis of economic growth processes (growth theory).
• In computer science, for example, the Image inpainting ( the removal of Computing writing or logos from pictures )

The field of differential equations, the mathematics given decisive impetus. Many parts of the current mathematics research at the existence, uniqueness and stability theory of different types of differential equations.

Higher levels of abstraction

Differential equations or systems of differential equations assume that a system can be described in algebraic form and quantified. Further that the described functions are differentiated, at least in the regions of interest. In the scientific-technical environment, these conditions are indeed often given, in many cases, but they are not met. Then, the structure of a system can only be described at a higher abstraction level. See in order of increasing abstraction:

• System Theory ( disambiguation)
• Ontology ( computer science )
• Ontology (philosophy)
• Formal concept analysis (mathematics)
• Order relation (mathematics)