Applied Mathematical Sciences, Vol. 4, 2010, no. 21, 1021 - 1032
                                Applications of Fractional Calculus
                                                       Mehdi Dalir
                                                Department of Mathematics
                                                     Faculty of Sciences
                                       Islamic Azad University of Varamin(Pishva)
                                                    Varamin-Tehran-Iran
                                                      Majid Bashour
                                                Department of Mathematics
                                                     Faculty of Sciences
                                       Islamic Azad University of Varamin(Pishva)
                                                    Varamin-Tehran-Iran
                                                 majidbashour@yahoo.com
                                                          Abstract
                               Different de“nitions of fractional derivatives and fractional Integrals
                            (Differintegrals) are considered.  By means of them explicit formula
                            and graphs of some special functions are derived. Also we reviw some
                            applications of the theory of fractional calculus.
                         Mathematics Subject Classi“cation: 26A33
                         Keywords: fractional derivative, fractional Integral, differintegrals
                     1 Introduction
                     Fractional calculus is a “eld of mathematics study that qrows out of the tra-
                     ditional de“nitions of calculus integral and derivative operators in much the
                     samewayfractionalexponentsisanoutgrowthofexponentswithintegervalue.
                         The concept of fractional calculus( fractional derivatives and fractional in-
                     tegral) is not new. In 1695 LHospital asked the question as to the meaning
                     of dny/dxn if n =1/2; that is Ž what if n is fractional?Ž. Leibniz replied that
                        1/2                     √
                     Žd    x will be equal to x dx : xŽ.
                         It is generally known that integer-order derivatives and integrals have clear
                     physical and geometric interpretations. However, in case of fractional-order
                     integration and differentiation, which represent a rapidly qrowing “eld both in
                1022                                        M. Dalir and M. Bashour
                theoryandinapplicationstorealworldproblems, itisnotso. Sincetheappear-
                ance of the idea of differentiation and integration of arbitrary (not necessary
                integer) order there was not any acceptable geometric and physical interpre-
                tation of these operations for more than 300 year. In [11], it is shown that
                geometric interpretation of fractional integration is ŽShadows on the wallsŽ
                and its Physical interpretation is ŽShadows of the pastŽ.
                   In the last years has found use in studies of viscoelastic materials, as well as
                in many“eldsofscienceandengineeringincluding ”uid”ow, rheology, diffusive
                transport, electerical networks, electromagnetic theory and probability.
                   In this paper we consider different de“nitions of fractional derivatives and
                integrals (differintegrals). For some elementary functions, explicit formula of
                fractional drevative and integral are presented. Also we present some applica-
                tions of fractional calculus in science and engineering.
                2 Different De“nitions
                In this section we consider different de“nitions of fractional calculus.
                   1. L. Euler(1730):
                     Euler generalized the formula
                                   n m
                                   d x                         mŠn
                                     n =m(mŠ1)···(mŠn+1)x
                                   dx
                     by using of the following property of Gamma function,
                              (m+1)=m(mŠ1)···(mŠn+1)(mŠn+1)
                     to obtain
                                      n m
                                     d x      (m+1) mŠn
                                        n =            x    .
                                      dx    (mŠn+1)
                     Gammafunction is de“ned as follows.
                                          ∞ Št zŠ1
                                   (z)=     e t   dt,   Re(z) > 0
                                          0
                   2. J. B. J. Fourier (1820 - 1822):
                     By means of integral representation
                       Applications of fractional calculus                                               1023
                                              f(x)= 1  ∞ f(z)dz ∞ cos(pxŠpz)dp
                                                       2π Š∞             Š∞
                             he wrote
                                         dnf(x) = 1  ∞ f(z)dz ∞ cos(pxŠpz+nπ)dp,
                                             n
                                           dx       2π Š∞              Š∞                    2
                          3. N. H. Abel (1823- 1826):
                             Abel considered the integral representation  x s′(η)dη = ψ(x) for ar-
                                                                                   (xŠη)α
                                                                                0
                             bitrary α and then wrote
                                                       s(x)=        1     dŠαψ(x).
                                                                               Šα
                                                                (1Šα) dx
                          4. J. Lioville (1832 - 1855):
                                I. In his “rst de“nition, according to exponential representation of a
                                                                                                     m ax
                                                         ∞       a x                                  d e
                                   function f(x)=            cne n , he generalized the formula          n  =
                                                         n=0                                           dx
                                    m ax
                                   a e    as
                                                             ν          ∞
                                                            d f(x)      ν ax
                                                                     =      cna e n
                                                                 ν             n
                                                              dx        n=0
                               II. Second type of his de“nition was Fractional Integral
                                                 µ         µ          1       ∞              µŠ1
                                                    Φ(x)dx =           µ           Φ(x+α)α        dα
                                                                 (Š1) (µ) 0
                                                 µ         µ      1    ∞              µŠ1
                                                    Φ(x)dx = (µ) 0 Φ(xŠα)α                 dα
                                   By substituting of τ = x +α and τ = xŠα in the above formulas
                                   respectively, he obtained
                                                 µ         µ         1       ∞          µŠ1
                                                    Φ(x)dx =          µ           (τ Šx)      Φ(τ)dτ
                                                                 (Š1) (µ) x
                                                 µ         µ      1    x          µŠ1
                                                    Φ(x)dx = (µ) Š∞(xŠτ)              Φ(τ)dτ.
                       1024                                                          M. Dalir and M. Bashour
                              III. Third de“nition, includes Fractional derivative,
                                     µ                  µ                                                     
                                    d F(x)         (Š1)          µ                 µ(µŠ1)
                                        µ     =       µ    F(x) F(x+h)++                     F(x+2h)Š···
                                      dx             h           1                   1·2
                                    dµF(x)          1       µ                 µ(µŠ1)                       
                                        µ     = µ F(x) F(xŠh)++                          F(xŠ2h)Š··· .
                                      dx           h         1                    1·2
                          5. G. F. B. Riemann (1847 - 1876):
                              His de“nition of Fractional Integral is
                                                Šν            1   x         νŠ1
                                              D f(x)=(ν) c (xŠt)                f(t)dt +ψ(t)
                          6. N. Ya. Sonin (1869), A. V. Letnikov (1872), H. Laurent (1884),
                              N. Nekrasove (1888), K. Nishimoto (1987-):
                              They considered to the Cauchy Integral formula
                                                      f(n)(z)= n!           f(t)    dt
                                                                  2πi    (t Š z)n+1
                                                                        c
                              and substituted n by ν to obtain
                                                 Dνf(z)=(ν+1) x+               f(t)    dt.
                                                                2πi           (t Šz)ν+1
                                                                         c
                          7. Riemann-Liouvill de“nition:
                              The popular de“nition of fractional calculus is this which shows joining
                              of two previous de“nitions.
                                                                       
                                                               1        d n t        f(τ)dτ
                                              Dαf(t)=
                                             a  t         (nŠα) dt               (t Š τ)αŠn+1
                                                                               a
                                                                (nŠ1≤α