Mathematical Physics Vol 1

Chapter 8. Introduction to the Fractional Calculus

436

positive integer

d n dx n

m ! ( m − n ) !

( x m )=

x m − n , m ≥ n

D n

x y =

(8.1)

Replacing the factorial symbol by Gamma function (8.3), he developed the formula for the fractional derivative of a power function

Γ ( β + 1 ) Γ ( β − α + 1 )

D α

β

x β − α

x x

(8.2)

=

where α and β are fractional numbers and where the gamma function Γ ( z ) (see Section 5.6.1, on page 244) is defined for z > 0 as:

∞ Z 0

e − x x z − 1 dx

Γ ( z )=

(8.3)

In particular, Lacroix calculated

x 1 / 2 = 2 r

Γ ( 2 ) Γ ( 3 / 2 )

x π

D 1 / 2

x x =

(8.4)

Surprisingly, the previous definition gives a nonzero value for the fractional derivative of a constant function ( β = 0 ) , since

1 Γ ( 1 − α )

D α

α x x

x − α̸

0 =

x 1 = D

= 0

(8.5)

Using linearity of fractional derivatives, the method of Lacroix is applicable to any analytic function by term-vise differentiation of its power series expansion. Unfortunately, this class of functions is too narrow in order for the method to be considered general. It is interesting to note that simultaneously with these initial theoretical developments, first practical applications of fractional calculus can also be found. In a sense, the first of these was the discovery by Abel in 1823,[7-9]. Abel considered the solution of the integral equation related to the tautochrone problem 2 . He found that the solution could be accomplished via an integral transform, which could be written as a semi-derivative. More precisely, the integral transform considered by Abel was

x Z 0

1 / 2 f

( x − t ) −

K =

( t ) dt , K = const .

(8.6)

He wrote the right hand side of (8.6) by means of a fractional derivative of order 1 / 2,

( f ( x )) !

d − 1 / 2 dx − 1 / 2

√ π

(8.7)

Abel’s solution had attracted the attention of Joseph Liouville, who made the first major study of fractional calculus,[11-14]. The most critical advances in the subject came around 1832 when he began to study fractional calculus in earnest and then managed to apply his results to problems 2 The tautochrone problem consists of the determination of a curve in the (x, y) plane such that the time required for a particle to slide down the curve to its lowest point under the influence of gravity is independent of its initial position x 0 , y 0 on the curve.