Fractional calculus
The Fractional Calculus refers to the extension of the derivation concept to non-integer orders. The term " fractional " is for historical reasons, the derivatives can be generally of real or even complex order.
- 2.1 Euler's approach
- 3.1 Iterative and fractional integrals
- 3.2 repatriation of fractional integrals on folds
- 3.3 Fractional Weylintegrale
- 3.4 Fractional Weylintegrale and folds
- 3.5 Examples
- 3.6 Integration of hypergeometric functions
- 3.7 Duality of and - operators
- 4.1 First generalization of fractional integration
- 4.2 Further generalization of the fractional integration
- 5.1 Tautochronenproblem
Caveat: important functions and integral transforms
( Incomplete ) gamma function
As a generalization of the function faculty gamma function is defined as follows:
For integer arguments arises. In case of incomplete gamma function is not to infinity but integrates only up to a certain value y:
Beta function
The Euler Beta function is defined as
Where they can be represented as a product of gamma functions
Hypergeometric function
As an extension of the geometric series, the hypergeometric function is defined as
Immediately understandable is the special case
Fourier transform
For one defines
As a Fourier transform, and
A reverse transformation.
Note that there are several possible definitions of the Fourier transform, which differ in transformation in which to write the minus sign in the exponential function, or where the factor of 2π appears.
Translational operator:
Stretching operator.
Folding:.
It follows from the convolution theorem: For is
The Fourier transform does so from the convolution of two functions, the multiplication of its Fourier transform.
Next is valid for
Laplace
Let be a locally integrable function, then the Laplace transform is defined as
The Laplacefaltung is defined similar to the Fourier convolution and provides a similar context:
History
Even the mathematician Gottfried Wilhelm Leibniz and Leonhard Euler dealt with the generalization of the derivative term. Leibniz describes in a letter to Guillaume François Antoine, Marquis de L'Hospital, the similarity between powers and the product rule of derivatives:
What is seemingly easy to
Can be generalized (with one in the case of α -n is negative ). However, with such a naive use of symbolism, problems occur. As an example, we choose a function f such that
Note the mathematical in itself is not correct " durchmultiplizieren " with dx. One thinks of such a function directly to the e-function, but was not yet known explicitly as such at the time. Where to now encounters a contradiction when you look at? To see this one simply sets:
Thus Leibniz can simple approach not be the appropriate solution to the problem.
Euler approach
Euler considered integer derivatives of power functions zm. In this economy:
He now tried to generalize this relationship by replacing the faculty by the found of him gamma function to non-integer powers:
This way also leads to contradictions. Again, consider the exponential function eλx which differentiates λneλx gives n times; thus generalized:
On the other hand, however, the exponential function is an infinite power series, namely.
Thus, one has to calculate two ways the α - derivative of ex:
This contradiction can be explained by the example α = -1 when back negative exponents of the differential operators are interpreted as integrals:
Illustrate the different lower limits that you " need to know " with this approach, from where to where you have to integrate to find the correct antiderivative. Thus, Euler 's approach, although better not suitable with the idea and execution forth to generalize the differential operator is correct to real powers.
Definition of fractional integral operators
Iterative and fractional integrals
One way to define fractional integrals consistent results from the formula
Which converts the double integration over two variables with the same lower bound into a single integral. This formula can be extended to any number of integrals.
If we introduce now even the integral operator as follows
Where F (x ) is an antiderivative of f ( x), then arbitrarily high powers of this operator can be reduced to a single integral thanks to the above formula of multiple integrals:
In contrast to the formulas at the beginning you can of integers n for real ( or complex) numbers generalize this integral operator with relative ease α by n by α and the faculty replaced by the gamma function and demands that:
This is called right-sided fractional Riemann - Liouville integral. Analogously, by
The left-side equivalent are defined.
Repatriation of fractional integrals on folds
If we define the distribution, the fractional integral can be attributed to a Laplacefaltung:
Because
Fractional Weylintegrale
If we let in the equations above, or go in magnitude to infinity, one obtains the so-called Weylintegrale and the corresponding partial integral operators
For the definition and quantity. This Bedinung is for example met with.
Fractional Weylintegrale and folds
Even fractional Weylintegrale can be traced back to folds. However, these are Fourier convolutions, as Weylintegrale have an infinite lower or upper limit.
Which may be converted by in
Therefore, the Fourier transformation yields
Thus we see that the fractional Riemann - Liouville integral operator is diagonalized by the corresponding Laplacefaltung, the fractional Weyl integral operator by the Fourier convolution.
Examples
:
Substituiere
In the special case α = 1, it
( with the substitution ):
It is thus seen that you " need to know " also in this integral operator, similar to Euler's approach, from where to where you have to integrate in order to obtain the actual derivative of a function, however, is packed with explicitly in the operator definition in it. Thus, the lower limit must be chosen such that in (see top, Def of Ia ) the F ( a) disappears and one F ( x ) (or the fractional equivalent thereto ) receives. So we have in this second example from - ∞ to x integrated, knowing that eax for a → - ∞ goes to 0. Therefore, we integrate this function just once, but this time with lower limit 0 ( and substitution ):
Substituting here now even az with t, then we have:
:
Substitution of z with y / x leads to
Compare this with. You can see that you simply a = 1, b = - β, c = α 1 and z = -x / c must set in order to obtain the above integral. So is
Integration of hypergeometric functions
Since you do with hypergeometric functions represent very many other functions, it is advisable to present here a formula for their integration.
Duality of the and - operators
The general rule
The two Riemann - Liouville and the two are so mutually Weylintegrale each dual. So you can move the fractional integral of a function to the easier to be integrated into integrals.
Definition of general frakt. Integral operators and
First generalization of fractional integration
Through the approach
To attempt to define a more general integral operator. The amount of strokes rather than simply parentheses already indicate that for this a kind of spherical symmetry is assumed. C ( α ) is to be determined so that the additivity of order ( IαIβ = Iα β ) is still applicable. You may already suspect that this operator is simply a linear combination of the Weylintegraloperatoren already known that one can also prove:
So is
Which can be generalized to higher dimensions:
Now the question is how C (d, α ) can be determined. If you want to make the choice as that is true, then results after detailed study of the Fourier transform of C (d, α ):
Taking advantage of the formulas and thus results in the one-dimensional case:
This is called fractional Riesz -Feller integral.
More generalization of fractional integration
The formula leads to the conclusion that by further such general integral operators
Can define what = makes the Riesz -Feller integral to the special case c = c- 1. For example, results for c = 1 and c = -1
These two operators are related by the Hilbert transform:
Feller for integrals of the form
Proved that the additivity of the order applies. These integrals can also be represented as a linear combination of the above form, you just have to choose.
Examples of fractional integral equations in physics
Tautochronenproblem
Problem in two dimensions: a mass point falls under the influence of gravity along a fixed but unknown path y = h ( x) of the height at the height y0 y1; the time that it is needed for specified with = time of the fall of solid y0 to y1 variable. The question now is: Can be made aware of the case alone determine times already h (x)?
We set v ( y) is equal to the magnitude of the instantaneous velocity is obtained for the period of the fall from P0 to P1: S ( y ) is equal to the distance traveled value as a function of height. Now, if y = h (x) is invertible, then x = h -1 ( y) = Φ (y) and the arc length of the arc length with differential
It follows from the energy conservation law. Use in the equation for T yields
Then, imagine that ( and considering that is ), then results