Mathematical Physics Vol 1

8.1 Brief History of Fractional Calculus

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and by choosing c = − ∞ , formula (8.19) is equivalent to Liouville’s first definition (8.10). These two facts explain why equation (8.19) is called Riemann-Liouville fractional integral . While the notation of fractional integration and differentiation only differ in the sign of the parameter α in (8.19), the change from fractional integration to differentiation cannot be achieved directly by inserting negative α at the right-hand side of (8.19). The problem originates from the integral at the right side of (8.19) which is divergent for negative integration orders. However, by analytic continuation it can be shown that  , (8.20) holds, which is known today as the definition of the Riemann-Liouville fractional derivative . In (8.20) n =[ α ] is the smallest integer greater than α with 0 < β = n − α < 1. For either c = 0 or c = ∞ the integral in (8.20) is the Beta-integral for a wide class of functions and thus easily evaluated. The Riemann-Liouville model can be used to describe processes with power-law behaviour, due to the power-function kernel in the definition of the integral transform, but there are many other types of behaviours that occur in nature and that cannot be described by simple power functions. Nearly simultaneously, Grunwald and Letnikov provided the basis for another definition of fractional derivative [27] which is also frequently used today. Disturbed by the restrictions of the Liouville‘s approach Grunwald (1867) adopted the definition of a derivative as the limit of a difference quotient as its starting point. He arrived at definite-integral formulas for ordinary derivatives, showed that Riemann’s definite integral had to be interpreted as having a finite lower limit, and also that the Liouville’s definition, in which no distinguishable lower limit appeared, correspond to a lower limit − ∞ . Formally, (8.21) which is today called the Grunwald-Letnikov fractional derivative . In definition (8.21), α k is the generalized binomial coefficient, wherein the factorials are replaced by Euler’s Gamma function. Letnikov [24] also showed that definition (8.21) coincides, under certain relatively mild conditions, with the definitions given by Riemann and Liouville. Today, the Grunwald-Letnikov definition is mainly used for derivation of various numerical methods, which use formula (8.21) with finite sum to approximate fractional derivatives. Also, in 1888-1891 Nekrasov [28,29] gave applications of fractional integro-differentiation in the form (8.18) to the integration of high order differential euations. Together with the advances in fractional calculus at the end of the nineteenth century the work of O. Heaviside [30] has to be mentioned. The operational calculus of Heaviside, developed to solve certain problems of electromagnetic theory, was an important next step in the application of generalized derivatives. The connection to fractional calculus has been established by the fact that Heaviside used arbitrary powers of p ,mostly √ p , to obtain solutions of various engineering problems. Weyl [31] and Hardy, [32,33], also examined some rather special, but natural, properties of differintegrals of functions belonging to Lebesgue and Lipschitz classes in 1917. Moreover, Weyl showed that the following fractional integrals could be written for 0 < α < 1 assuming that the integrals in (8.22) are convergent over an infinite interval D α c , x f ( x )= D n − β c , x f ( x )= D n c , x f ( x ) D − β c , x f ( x )= d n dx n   1 Γ ( β ) x Z c ( x − t ) β − 1 f ( t ) dt  GL D α x f ( x )= lim h → 0 ∆ α h f ( x ) h α = lim h → 0 ∞ ∑ k = 0 ( − 1 ) k α k f ( x − kh ) h α α > 0

∞ Z x

x Z − ∞

1 Γ ( α )

1 Γ ( α )

α − 1 ϕ

α − 1 ϕ

I α + ϕ ( x )=

α −

ϕ ( x )=

( x − t )

( t ) dt , I

( t − x )

( t ) dt ,

(8.22)