Mathematical Physics Vol 1

Chapter 8. Introduction to the Fractional Calculus

444

known as the left Grunwald-Letnikov (GL) derivative . This derivative can be seen as a limit of the fractional order backward difference. The right sided derivative is defined accordingly

( ∆ α

f ( x ))

α x f ( x )= lim h → 0

− h

GL D

h α

( − 1 ) | j |

j

(8.47)

α

f ( x )= ∑

∆ α

f ( x + jh ) , h > 0 ,

− h

0 ≤| j | < ∞

Definitions (8.28) and (8.47) are valid for both α > 0 (fractional derivative) and for α < 0 (fractional integral) and, commonly, these two notions are grouped into one single operator called GL differintegral . The GL derivative and RL derivative are equivalent if the functions they act upon are sufficiently smooth. The generalized binomial coefficients, calculation for α ∈ R and k ∈ N 0 , is the following α j = α ! j ! ( α − j ) ! = α ( α − 1 ) ··· ( α − j + 1 ) j ! = Γ ( α + 1 ) Γ ( j + 1 ) Γ ( α − j + 1 ) , α 0 = 1 . (8.48) Let us consider n =( x − a ) / h where a is a real constant. This constant can be interpreted as the lower terminal (an analogue of the lower integration limit, necessary even for the derivative operator due to its non-local properties). The GL differentigral can be expressed as a limit

( − 1 ) j

j

[ x − a

h ]

α

1 h α

∑ j = 0

α a , t f ( x )= lim h → 0

GL D

f ( x − jh )

(8.49)

where [ x ] means the integer part of x , a and t are the bounds of the operation for GL D α a , t f ( x ) . For the numerical calculation of fractional-order derivatives we can use the following relation (8.50) derived from the GL definition (8.49). The relation to the explicit numerical approximation of the α -th derivative at the points kh , ( k = 1 , 2 ,... ) has the following form, [52]

N ( x ) ∑ j = 0

D ± α

x f ( x ) ≈ h ∓ α

b ± α

j f ( x − jh )

(8.50)

x − L

where L is the “memory length”, h is the step size of the calculation, N ( x )= min h x h i , L h

(8.51)

( ± α ) j

[ x ] is the integer part of x and b

is the binomial coefficient given by

= 1 −

j

1 ± α

b ( ± α ) 0

( ± α ) j

b µα

= 1 , b

(8.52)

j − 1

This approach is based on the fact that (for a wide class of functions and assuming all initial conditions are zero) the three most commonly used definitions - GL, RL, and Caputo’s - are equivalent,[57]. 8.3 Basic Properties of Fractional Order Differintegrals As stated previously, for a wide class of functions, Grunwald-Letnikov definition of the fractional derivative operator coincides with the Riemann-Liouville definition. Thus, in the present section only Riemann-Liouville and Caputo derivatives will be considered. Also, left-side operators are used primarily. Thus, all of the properties presented next will be accounted for this kind of