Differential equation

A differential equation (also differential equation , often DGL , DG , DGI. Or Dgl. Is abbreviated) a mathematical equation for a sought function of one or more variables, which also discharges occur this function. Many laws of nature can be formulated using differential equations. Differential equations are therefore an essential tool in mathematical modeling . A differential equation describes the behavior of these variables in relation to one another. Differential equations are an important subject of analysis in analysis , which investigates its solution theory. Not only because no explicit solution representation is possible for many differential equations, the approximate solution using numerical methods plays an essential role. A differential equation can be illustrated by a direction field .

Types of differential equations

There are different types of differential equations. They are roughly divided into the following sub-areas. All of the following types can essentially co-exist independently and simultaneously.

Ordinary differential equations

If the function you are looking for depends only on one variable, it is called an ordinary differential equation. There are only ordinary derivatives according to the one variable.

Examples:

{\ displaystyle y '(x) = - 2 \ cdot y (x) +5, \ qquad {\ ddot {z}} (t) +4 \ cdot z (t) = \ sin (3 \ cdot t)}

Writes the ordinary differential equation for the function you are looking for in the form ${\ displaystyle y (x)}$

{\ displaystyle F \ left (x, y (x), y '(x), \ ldots, y ^ {(n)} (x) \ right) = 0,}

so the ordinary differential equation is called implicit .

Is the differential equation solved for the highest derivative, i. i.e., it applies

{\ displaystyle y ^ {(n)} (x) = f \ left (x, y (x), y '(x), \ ldots, y ^ {(n-1)} (x) \ right), }

so one calls the ordinary differential equation explicit . In the applications, explicit ordinary differential equations are mathematically easier to process. The highest order of derivation that occurs is called the order of the differential equation . For example, an explicit ordinary first order differential equation has the form ${\ displaystyle n}$

{\ displaystyle y '(x) = f (x, y (x)) \ ,.}

There is a closed theory of solving explicit ordinary differential equations.

An ordinary differential equation is linear if it is linear in function and its derivatives:

{\ displaystyle y ^ {(n)} (x) + a_ {n-1} (x) \, y ^ {(n-1)} (x) + \ cdots + a_ {0} (x) \, y (x) = b (x)}

It is semilinear if it is linear in the derivatives and the function on the left, but the function can also depend on the function and its derivatives, except for the highest derivative: ${\ displaystyle b (x)}$ ${\ displaystyle y}$

{\ displaystyle y ^ {(n)} (x) + a_ {n-1} (x) \, y ^ {(n-1)} (x) + \ cdots + a_ {0} (x) \, y (x) = b (x, y (x), y '(x), \ cdots, y ^ {(n-1)} (x))}

Partial differential equation

If the function you are looking for depends on several variables and if partial derivatives occur in the equation for more than one variable, then one speaks of a partial differential equation. Partial differential equations are a large field and the theory is not mathematically closed, but is the subject of current research in several areas.

One example is the so-called heat conduction equation for a function ${\ displaystyle u (t, x)}$

{\ displaystyle {\ tfrac {\ partial} {\ partial t}} u (t, x) = a {\ tfrac {\ partial ^ {2}} {\ partial x ^ {2}}} u (t, x )}

A distinction is made between different types of partial differential equations. First there are linear partial differential equations . The function you are looking for and its derivatives are included linearly in the equation. The dependency with regard to the independent variables can definitely be non-linear. The theory of linear partial differential equations is the most advanced, but far from complete.

One speaks of a quasilinear equation if all derivatives of the highest order occur linearly, but this no longer applies to the function and derivatives of lower order. A quasi-linear equation is more difficult to deal with. A quasi-linear partial differential equation is semilinear if the coefficient function before the highest derivatives does not depend on lower derivatives and the unknown function. Most of the results are currently being achieved in the area of quasi-linear and semilinear equations.

Finally, if one cannot determine a linear dependency with regard to the highest derivatives, the equation is called a nonlinear partial differential equation or a completely nonlinear partial differential equation .

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

More types

With the type of stochastic differential equations, stochastic processes occur in the equation . Actually, stochastic differential equations are not differential equations in the above sense, but only certain differential relations that can be interpreted as differential equations.

The type of algebro differential equations is characterized by the fact that, in addition to the differential equation, there are also algebraic relations as secondary conditions .

There are also so-called retarded differential equations . In addition to a function and its derivatives , function values or derivatives from the past also occur at a point in time . ${\ displaystyle t}$

An integro-differential equation is an equation in which not only the function and its derivatives, but also integrations of the function appear. An important example is the Schrödinger equation in the momentum representation ( Fredholm 's integral equation ).

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

Systems of differential equations

One speaks of a system of differential equations when there is a vector-valued mapping and several equations ${\ displaystyle y = (y_ {1}, \ ldots, y_ {k})}$

{\ displaystyle F_ {l} \ left (x, y, Dy, \ ldots, D ^ {n} y \ right) = 0, \ qquad l = 1, \ ldots, k.}

must be fulfilled at the same time. If this implicit differential equation system cannot be converted locally into an explicit system everywhere, then it is an algebro differential equation .

Problems

The solution set of a differential equation is generally not uniquely determined by the equation itself, but requires additional initial or boundary values . So-called initial boundary value problems can also occur in the area of partial differential equations.

In the case of initial or initial boundary value problems, one of the variables is generally interpreted as time. With these problems, certain dates are prescribed at a certain point in time, namely the start point in time.

In the case of boundary value or initial boundary value problems, a solution to the differential equation is sought in a restricted or unrestricted area and we provide so-called boundary values as data, which are given on the boundary of the area. Depending on the type of boundary conditions, a distinction is made between other types of differential equations, such as Dirichlet problems or Neumann problems .

Solution methods

Due to the diversity of both the actual differential equations and the problem definitions, it is not possible to provide a generally applicable solution method. Only explicit ordinary differential equations can be solved with a closed theory. A differential equation is called integrable if it is possible to solve it analytically, i.e. to specify a solution function (the integral ). Many mathematical problems, in particular nonlinear and partial differential equations, cannot be integrated, including those that appear quite simple, such as the three-body problem , the double pendulum or most of the top types .

Lie theory

A structured general approach to solving differential equations is pursued through symmetry and continuous group theory. In 1870, Sophus Lie put the theory of differential equations on a generally applicable basis with the Lie theory . He showed that the older mathematical theories for solving differential equations can be summarized by the introduction of so-called Lie groups . A general approach to solving differential equations takes advantage of the symmetry property of differential equations. Continuous infinitesimal transformations are used that map solutions to (other) solutions of the differential equation. Continuous group theory, Lie algebras and differential geometry are used to grasp the deeper structure of the linear and nonlinear (partial) differential equations and to map the relationships that ultimately 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 numerical calculation of solutions are therefore completely or not at all solved, depending on the equation. Due to the importance of differential equations in practice, the application of numerical solution methods, especially with partial differential equations, is more advanced than their theoretical underpinning.

One of the Millennium Problems is the proof of the existence of a regular solution to the Navier-Stokes equations . These equations occur in fluid mechanics , for example .

Approximate methods

As a solution, differential equations have functions that fulfill conditions for their derivatives . An approximation usually takes place by dividing space and time into a finite number of parts using a computational grid ( discretization ). The derivatives are then no longer represented by a limit value, but are approximated by differences. In numerical mathematics , the resulting error is analyzed and estimated as well as possible.

Depending on the type of equation, different discretization approaches are chosen, in the case of partial differential equations, for example, finite difference methods , finite volume methods or finite element methods .

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

Appearance and applications

A multitude of phenomena in nature and technology can be described by differential equations and mathematical models based on them. Some typical examples are:

Many physical theories are based on differential equations: equations of motion or vibrations in Newtonian mechanics , the load behavior of components , electrodynamics is dominated by the Maxwell equations , quantum mechanics by the Schrödinger equation .
in astronomy the orbits of the celestial bodies and the turbulence in the interior of the sun,
in biology, for example, processes in growth , in currents or in muscles , or in evolutionary theory .
in chemistry the kinetics of reactions ,
in electrical engineering the behavior of networks with energy-storing elements,
the behavior of surfaces in differential geometry ,
in fluid mechanics the behavior of these very currents,
in economics the analysis of economic growth processes ( growth theory ).
in computer science, for example, image inpainting (calculating fonts or logos from images)

The field of differential equations has given mathematics decisive impulses. Many parts of current mathematics research the existence, uniqueness and stability theory of various types of differential equations.

Higher levels of abstraction

Differential equations or systems of differential equations require that a system can be described and quantified in algebraic form . Furthermore, that the descriptive functions can be differentiated at least in the areas of interest . In the scientific and technical environment, these requirements are often met, but in many cases they are not met. Then the structure of a system can only be described on a higher abstraction level. See in the order of increasing abstraction:

literature

GH Golub, JM Ortega: Scientific Computing and Differential Equations. An introduction to numerical mathematics . Heldermann Verlag, Lemgo 1995, ISBN 3-88538-106-0 .
G. Oberholz: Differential equations for technical professions. 4th edition. Verlag Anita Oberholz, Gelsenkirchen 1995, ISBN 3-9801902-4-2 .
PJ Olver: Equivalence, Invariants and Symmetry . Cambridge Press, 1995.
L. Papula: Mathematics for Engineers and Natural Scientists Volume 2 . Vieweg's reference books of technology, Wiesbaden 2001, ISBN 3-528-94237-1 .
H. Stephani: Differential Equations: Their Solution Using Symmetries. Edited by M. MacCallum. Cambridge University Press, 1989.
H. Benker: Differential equations with MATHCAD and MATLAB . Springer-Verlag, Berlin / Heidelberg / New York 2005.

Web links

Differential equations Matheplanet - Instructions for solving various differential equations with examples
Mathematics online course on differential equation at the University of Stuttgart
Dörte Haftendorn: Differential equations: numerics, examples, isoclines, ... University of Lüneburg
Arthur Mattuck: Differential Equations . Academic Earth - MIT

Individual evidence

↑ Guido Walz (ed.), Lexikon der Mathematik, Springer-Spektrum Verlag, 2017, article linear differential equation, semilinear differential equation
↑ Peterson, Ivars: Filling in Blanks . In: Society for Science & # 38 (Ed.): Science News . 161, No. 19, pp. 299-300. doi : 10.2307 / 4013521 . Retrieved May 11, 2008.

[1] Guido Walz (ed.), Lexikon der Mathematik, Springer-Spektrum Verlag, 2017, article linear differential equation, semilinear differential equation

[2] Peterson, Ivars: Filling in Blanks . In: Society for Science & # 38 (Ed.): Science News . 161, No. 19, pp. 299-300. doi : 10.2307 / 4013521 . Retrieved May 11, 2008.