Mathematical Physics Vol 1

8.2 Basic Definitions of Fractional Order Differintegrals

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Derivative (8.27) is strongly connected to the Riemann-Liouville fractional derivative and is today frequently used in applications. It is interesting to note that Rabotnov [43,44] introduced the same differential operator into the Russian viscoelastic literature a year before Caputo’s paper was published. Namely, the fractional exponential function ∋ α ( β , t ) introduced by Rabotnov as (8.28) is known also as the Rabotnov function and as a special case of the Mittag-Leffler function widely used in fractional calculus. The corresponding kernel is:

∞ ∑ n = 0

β n ( t − τ ) n ( 1 + α ) Γ [( n + 1 )( 1 + α )] .

∋ α ( β , t − τ )=( t − τ ) α

(8.28)

In the late 20 th century, fractional calculus began to undergo a large increase in popularity and research output. By the second half of the twentieth century, the field of fractional calculus had grown to such extent that in 1974 the first conference “ The First Conference on Fractional Calculus and its Applications ” concerned solely with the theory and applications of fractional calculus was held in New Haven. In the same year, the first book on fractional calculus by Oldham and Spanier [3] was published after a joint collaboration started in 1968. A number of additional books have appeared since then, for example McBride (1979) [45], Nishimoto (1991) [46], Miller and Ross (1993), [4], Samko et al. (1993),[47], Kiryakova (1994) [48], Rubin (1996) [49], Carpinteri and Mainardi (1997),[50], Davison and C. Essex (1998), [51],Podlubny (1999) [52], R. Hilfer (2000) [53], Kilbas et.al (2006),[5], Das (2007)[54], J. Sabatier et. al (2007) [55], and others. In 1998 the first issue of the mathematical journal “Fractional calculus & applied analysis” was printed. This journal is solely concerned with topics on the theory of fractional calculus and its applications. Finally, in 2004 the first conference “Fractional differentiation and its applications ” was held in Bordeaux, and it is organized every second year since 2004,[56]. Some conferences dedicated, entirely or partly (with special sessions), to FC during the last decades such as: International Carpathian Control Conference (ICCC 2000,2001. . . ), conference on Non-integer Order Calculus and its Applications is organized every year from 2009(R α RNR), Fractional Calculus Day (FCDay first 2009, 2013, 2015,. . . ), International Conference on Ana lytic Methods of Analysis and Differential Equations is organized every second year (AMADE 1999, 2001, 2003. . . ), Problems and Applications of Operator (OTHA 2011, 2012,. . . ). 8.2 Basic Definitions of Fractional Order Differintegrals There are many different forms of fractional operators in use today. “Fractional” here does not mean only fractions but it stands for arbitrary quantity including integers, fractions, general complex numbers. Riemann-Liouville, Grunwald-Letnikov, Caputo, Weyl and Erdely-Kober derivatives and integrals are the ones mentioned in the previous historical survey. In addition, most of these operators can be defined in two distinct forms, as the left and as right fractional operators. The three most frequently used definitions for the general fractional differintegral are: the Grunwald-Letnikov (GL), the Caputo(C) and the Riemann-Liouville (RL) definitions,[3 5],[52]. A short account of these most frequently used operators is given next. For expression of the Riemann-Liouville definition, we will consider the Riemann-Liouville n -fold integral for n ∈ N , n > 0 defined as (this expression is usually referred to as the Cauchy repeated integration formula ) t Z a t n Z a t n − 1 Z a ··· t 3 Z a t 2 Z a | {z } n − fold f ( t 1 ) d t 1 d t 2 ··· d t n − 1 d t n = 1 Γ ( n ) t Z a ( t − τ ) n − 1 f ( τ ) d τ . (8.29)