Fractional calculus in the Mellin setting and Hadamard-type fractional integrals

By Paul L. Butzer, Anatoly A. Kilbas, Juan Trujillo. 2002

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Abstract The purpose of this paper and some to follow is to present a new approach to fractional integration and differentiation on the half-axis R + =(0,∞) in terms of Mellin analysis. The natural operator of fractional integration in this setting is not the classical Liouville fractional integral Iα0+f but J α 0+,c f (x):= 1 Γ(α) ∫ 0 x u x c log x u α−1 f(u) du u (x>0) for α>0, c∈ R . The Mellin transform of this operator is simply (c−s) −α M [f](s) , for s=c+it, c,t∈ R . The Mellin transform of the associated fractional differentiation operator D α 0+,c f is similar: (c−s) α M [f](s) . The operator D α 0+,c f may even be represented as a series in terms of xkf(k)(x), k∈ N 0 , the coefficients being certain generalized Stirling functions Sc(α,k) of second kind. It turns out that the new fractional integral J α 0+,c f and three further related ones are not the classical fractional integrals of Hadamard (J. Mat. Pure Appl. Ser. 4, 8 (1892) 101–186) but far reaching generalizations and modifications of these. These four new integral operators are first studied in detail in this paper. More specifically, conditions will be given for these four operators to be bounded in the space Xcp of Lebesgue measurable functions f on (0,∞), for c∈(−∞,∞), such that ∫ ∞ 0 |u c f(u)| p du/u for 1⩽p ess sup u>0 [u c |f(u)|] for p=∞, in particular in the space Lp(0,∞) for 1⩽p⩽∞. Connections of these operators with the Liouville fractional integration operators are discussed. The Mellin convolution product in the above spaces plays an important role.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/d0ea41e5056c8f807418034dfd55751785b25941

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Citations in the corpus

Mellin transform analysis and integration by parts for Hadamard-type fractional integrals (2002, methodology): “13), and hence if Re(μ) >−c, thenJ α 0+,μ ∈ [Xr −c,Xp ′ −c] by Theorem 7(a) in [1], and ∥∥J α 0+,μg∥∥Xp′ −c K1‖g‖Xr −c , (4.”
Compositions of Hadamard-type fractional integration operators and the semigroup property (2002, methodology): “Similarly, if Re(μ) > −c and f (x) ∈ X c , then by Theorem 4(b) in [1] the operators J α −,μJ β −,μ and J α+β −,μ are bounded in X p c , and relation (2.”
Exponential Polynomials, Stirling Numbers, and Evaluation of Some Gamma Integrals (2009, background): “tor . Proofs and details are left to the reader. , (3.18) 15 , (3.19) and , (3.20) analogous to (1.2). (3.5), and (3.12) correspondingly For a comprehensive study of the Mellin derivative we refer to [11]. More Stirling numbers. The polynomials , form a basis in the linear space of all polynomials. ”
An Existence Theorem for Fractional Hybrid Differential Inclusions of Hadamard Type with Dirichlet Boundary Conditions (2014, background): “A detailed description of Hadamard fractional derivative and integral can be found in [2, 21, 22, 23, 24, 25].”
A NEW APPROACH TO GENERALIZED FRACTIONAL DERIVATIVES (2011, background): “, investigate properties of Hadamard fractional integral and deriva tive [1, 2, 3, 9, 11, 12, 14, 17].”
On Hadamard fractional integro-differential boundary value problems (2015, background): “Details and properties of Hadamard fractional derivative and integral can be found in [2,15–19].”
Initial-value problems for hybrid Hadamard fractional differential equations (2014, background): “A detailed description of Hadamard fractional derivative and integral can be found in [1, 5, 6, 7, 15, 16, 17].”
Weighted Fractional Calculus: A General Class of Operators (2022, background): “If w(x) = xβ is a power function and φ(x) = log(x) is the natural logarithm function, then the operators of w-weighted fractional calculus with respect to φ become precisely those of Hadamard-type fractional calculus, defined [46,47] as follows:”
New fractional integral unifying six existing fractional integrals (2016, background): “Recent results about the Hadamard and Hadamard-type integrals such as Hadamard-type fractional calculus [21], composition and semigroup properties [7], Mellin transforms [6], integration by parts formulae [8], Gtransform representations [22], impulsive differential equations with Hadamard fractional”