Differential operator
In mathematics, a differential operator is a function that assigns a function to a function and contains the derivative according to one or more variables. In particular, differential operators degrade the regularity of the function to which they are applied.
Probably the most important differential operator is the ordinary derivative; H. the mapping (spoken: "d to dx"), which assigns its derivative to a differentiable function :
Differential operators can be linked with one another. By omitting the function on which they act, pure operator equations are obtained.
There are different definitions of a differential operator, all of which are special cases or generalizations of one another. Since the most general formulation is accordingly difficult to understand, different definitions with different general validity are given here. Ordinary differential operators consist of the concatenation of whole derivatives, while partial derivatives also appear in partial differential operators.
Unless otherwise stated, this article is a restricted and open set . In addition, with denotes the set of- times continuously differentiable functions and with the set of continuous functions. The restriction that mapping between real subsets is not necessary, but is usually assumed in this article. If other definition and image areas are necessary or useful, this is explicitly stated below.
This article is also largely restricted to differential operators that operate on the spaces of continuously differentiable functions just mentioned. There are weakening of the definitions. For example, the study of differential operators led to the definition of the weak derivative and thus to the Sobolev spaces , which are a generalization of the spaces of continuously differentiable functions. This led to the idea of investigating linear differential operators with the help of functional analysis in operator theory. However, these aspects will not be discussed further in this article for the time being. A generalization of a differential operator is the pseudo-differential operator .
First order linear differential operator
definition
Be an open subset . A first order linear differential operator is a map
by
can be represented, where is a continuous function .
Examples
- The most important example of a first order differential operator is the common derivative
- in -direction is a first order partial differential operator.
- Other differential operators of this kind can be obtained by multiplying by a continuous function. Let it be a continuous function, then it is through
- defined operator again a first order differential operator.
- Three further examples are the operators gradient (grad), divergence (div) and rotation (red) from vector analysis . They are denoted by the Nabla symbol , which in the three-dimensional case represents the shape in Cartesian coordinates
- Has.
- and
- are two other examples of differential operators. The special thing about these operators is that you can use them to examine functions for holomorphism , because then the function is holomorphic.
Ordinary differential operator
Ordinary differential operators occur in particular in connection with ordinary differential equations .
definition
Analogous to the definition of the first order differential operator, an ordinary order differential operator is a mapping
by
given is. Here is another continuous function for everyone . In the case for all , this operator is called an ordinary, linear differential operator.
example
- The -th order derivative
- is the simplest case of an ordinary differential operator. It is the special case resulting from for and .
Linear partial differential operator
definition
Be an open subset. A linear partial differential operator of order is a linear operator
the through
can be represented. Where is a continuous function for all multi-indices .
Examples
- The Laplace operator in Cartesian coordinates is
- This is an elementary example of a partial differential operator. It is also the most important example of an elliptic differential operator . Elliptic differential operators are a special class of partial differential operators.
- The operator corresponding to the heat conduction or diffusion equation is
- This is an example of a parabolic differential operator.
- where corresponds to speed is another important partial differential operator. This is a hyperbolic operator and is used in the wave equation .
Partial differential operator
definition
A (non-linear) partial differential operator of order is also a mapping again
This is given by
Here are for everyone and continuous functions.
Linear differential operators
In the above definitions it was mentioned briefly when an ordinary or a partial differential operator is called linear. For the sake of completeness, the abstract definition of a linear differential operator will now be mentioned. This is analogous to the definition of linear mapping . Unless otherwise stated, all of the above examples are linear differential operators.
definition
Let be an (arbitrary) differential operator. This is called linear if
holds for all functions and all constants .
The most prominent example of this is the differential operator
which assigns its derivative to a function .
The solution space of a linear differential equation forms a vector space . After Fourier transformation , they can often be traced back to algebraic equations and concepts of linear algebra. Nonlinear differential operators are much more difficult to deal with.
Algebra of the differential operators
The set of all linear differential operators of the order that operate on is denoted by . The amount
is used together with the series connection of linear differential operators as multiplication
to one - graduate algebra . In general, however, multiplication is not commutative. An exception are, for example, differential operators with constant coefficients, in which the commutativity follows from the interchangeability of the partial derivatives .
One can also formally form power series with the differential operators and use z. B. Exponential Functions . The Baker-Campbell-Hausdorff formulas apply to computing with such exponential expressions of linear operators .
Differential operator on a manifold
Since one only has the local coordinate systems in the form of maps and no globally valid coordinate systems available on manifolds, one has to define coordinate-independent on these differential operators. Such differential operators on manifolds are also called geometric differential operators.
Coordinate-invariant definition
Let be a smooth manifold and be vector bundles . A differential operator of the order between the sections of and is a linear mapping
with the following characteristics:
- The operator is local, that is, it holds
- For there is an open environment of , bundle maps and as well as a differential operator so that the diagram commutes . With is pullback of a smooth vector field in the space designated.
Examples
Examples of geometric differential operators are shown below.
- The set of differential forms forms a smooth vector bundle over a smooth manifold. The Cartan derivative and its adjoint operator are differential operators on this vector bundle.
- The Laplace-Beltrami operator, as well as other generalized Laplace operators, are differential operators.
- The tensor bundle is a vector bundle. For each fixed vector field , the mapping is defined by , where the covariant derivative is, a differential operator.
- The Lie derivative is a differential operator on the differential forms.
Symbol of a differential operator
If the partial derivatives are formally replaced by variables and only the terms of the highest - i.e. second - order are considered, the second order differential operators given in the examples correspond to a quadratic form in the . In the elliptical case all coefficients of the form have the same sign , in the hyperbolic case the sign changes, in the parabolic case the highest order term is missing for one of the coefficients . The corresponding partial differential equations each show very different behavior. The names come from the analogues of conic section equations .
This can also be extended to other cases using the concept of the main symbol of the differential operator. Only terms of the highest order are retained, derivatives are replaced by new variables, and a polynomial is obtained in these new variables with which the differential operator can be characterized. For example, it is of the elliptical type if: the main symbol is not equal to zero, if at least one is not equal to zero. However, there are already “mixed” cases for differential operators of the 2nd order that cannot be assigned to any of the three classes.
The following definitions record this again with mathematical precision.
symbol
Be it
a general differential operator of order . The coefficient function can be matrix-valued. The polynomial
in is the symbol of . However, since the most important information can be found in the term of the highest order, as already indicated in the introduction, the following definition of the main symbol is usually used.
Main symbol
Let again be the differential operator of order defined above . The homogeneous polynomial
in is the main symbol of . Often the main symbol is simply called a symbol if confusion with the definition given above is impossible.
Examples
- The symbol and the main symbol of the Laplace operator are
Main symbol of a differential operator between vector bundles
Differential operators on manifolds can also be assigned a symbol and a main symbol. In the definition, it must of course be taken into account that the main symbol and the symbol under Change of cards are invariant defined. Since changing cards for symbols is very complicated, you usually limit yourself to the definition of the main symbol.
Let be a (coordinate-invariant) differential operator that operates between intersections of vector bundles. Be , and . Choose and with , and . Then the expression
regardless of the choice of and . The function
is then called the main symbol of .
Pseudo differential operators
The order of a differential operator is always an integer and positive. This is generalized in the theory of pseudo-differential operators. Linear differential operators of the order with smooth and bounded coefficients can be understood as pseudo-differential operators of the same order. If such a differential operator is used, then one can apply the Fourier transformation and then the inverse Fourier transformation . That is, it applies
This is a special case of a pseudo differential operator
This also shows that certain differential operators can be represented as integral operators and thus differential operators and integral operators are not completely opposite.
literature
- Otto Forster : Analysis 2. Differential calculus in R ^{n} . Ordinary differential equations. Vieweg-Verlag, 7th edition, 2006, ISBN 3-528-47231-6 .
- Konrad Königsberger : Analysis 2. Springer-Verlag, Berlin / Heidelberg 2000, ISBN 3-540-43580-8 .
- Dirk Werner : Functional Analysis. Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 .
- Lawrence Evans : Partial Differential Equations. American Mathematical Society, ISBN 0-8218-0772-2 .
- Liviu I. Nicolaescu: Lectures on the geometry of manifolds. World Scientific Pub Co (for differential operators between vector bundles).