Historical Remarks on the Fractional Calculus

The term fractional calculus is by no means new. It is a generalization of the ordinary differentiation by non-integer derivatives. The subject is as old as the calculus of differentiation and goes back to times when Leibniz, Gauß, and Newton invented this kind of calculation. In a letter to L`Hospital in 1695 Leibniz raised the following question:

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Gottfried Wilhelm von Leibniz; Born: 1 July 1646 in Leipzig,  Died: 14 Nov 1716 in Hannover  Leibniz developed the present day notation for the differential and integral calculus. He never thought of the derivative as a limit.

Can the meaning of derivatives with integral order (d^ny(x))/dx^n be generalized to derivatives with non-integral orders; so that in general n ∈ . This question goes back to a query of Bernoulli who was interested in the noninteger differentiation of a product. The story goes that L`Hospital was somewhat curious about that question of Leibniz and replied by another question. What if n = 1/2? Leibniz in a letter dated September 30, 1695 replied: Il y a de l'apparence qu'on tirera un jour des consequences bien utiles de ces paradoxes, car il n'y a gueres de paradoxes sans utilité. The translation reads: It will lead to a paradox, from which one day useful consequences will be drawn. The question raised by Leibniz for a fractional derivative was an ongoing topic in the last 300 years. Several mathematicians contributed to this subject over the years. People like Liouville, Riemann, and Weyl made major contributions to the theory of fractional calculus.

In fact a fractional derivative is useful for some types of functions. For example let us consider the nth derivative of a power x^m. We know that the general expression for the nth derivative is given by

(d^nx^m)/dx^n = m !/(m - n) ! x^(m - n)

We also know that a factorial is connected with Euler's Γ-function by the relation n != Γ(n + 1). Replacing the factorials in (1) by the Γ-function we can write

(d^nx^m)/dx^n = Γ(m + 1)/Γ(m - n + 1) x^(m - n).

This representation is equivalent with (1), however contains the potential of a generalization. We know that the Γ-function is defined for continuous arguments over the complex domain. If we now change the integer value of n to a number q∈ , we are able to generalize the meaning of an integer differentiation to a non-integer form. We can even define a complex differentiation. Replacing n by q in (2) resulting in general to

(d^qx^m)/dx^q = Γ(m + 1)/Γ(m - q + 1) x^(m - q)

The relation (3) has a well defined meaning, however is restricted to powers x^m. However, if we try to fractionally differentiate such simple functions with Mathematica  we end up with the following result

∂_ {x, 1/2} x^2

1 D :: dvar : Multiple derivative in {x, -} does not have the form {x, n}. 2

∂_ {x, 1/2} x^2

This shows us that Mathematica is not capable to deal with fractional differentiation orders.The developer of Mathematica  however designed the system in such a way that the user can extend the definition of derivatives. This extension will be our subject in the following. Telling Mathematica that fractional derivatives of powers are useful mathematical constructs is realized by the following lines

Unprotect[D] ;

D[x_^m_., {x_, q_}] := Gamma[m + 1]/Gamma[m - q + 1] x^(m - q)/;Head[q] == Real || Head[q] == Rational || Head[q] == Complex

Protect[D] ;

The definition of the fractional derivative of powers is based on equation (3) and restricts the order of differentiation either to the rational, the real or the complex numbers. An example for a rational number reads

∂_ {x, 1/2} x

(2 x^(1/2))/π^(1/2)

If we set the order of differentiation q to a real number we find

RowBox[{RowBox[{∂, _, RowBox[{{, RowBox[{x, ,, 2.1}], }}]}], , x^2}]

RowBox[{1.87156, /, RowBox[{x, ^, 0.1}]}]

Even if we use complex numbers in the order of differentiation we get a result

RowBox[{RowBox[{∂, _, RowBox[{{, RowBox[{x, ,, RowBox[{11.5, +, I}]}], }}]}], , x^4}]

RowBox[{RowBox[{(, RowBox[{RowBox[{57152.1, }], -, RowBox[{143371., , I}]}], )}], , RowBox[{x, ^, RowBox[{(, RowBox[{RowBox[{-, 7.5}], -, I}], )}]}]}]

This kind of formulas were discussed by Lacroix in 1819 based on the work by Euler [1738]. In retrospect these formulas are the first analytical answer of Leibniz's question on fractional derivatives. The answer lied 100 years dormant and needed the work of Euler to get a preliminary answer. The story on  fractional calculus continued with contributions from Fourier, Abel, Liouville, Riemann and Weyl. For a historical survey the reader may consult the books of Oldham and Spanier or  Miller and Ross. The historical developments culminated in two main calculi which are based on the work of Riemann and Liouville at the one side and on the work of Weyl at the other hand. Both formulations are connected and Weyl's calculus forms a subset of the Riemann-Liouville (RL) calculus.

Created by Mathematica  (September 15, 2003)