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Fractional Calculus of P-transforms

Kumar, Dilip, Kilbas, Anatoly (2010)

Fractional Calculus and Applied Analysis

MSC 2010: 44A20, 33C60, 44A10, 26A33, 33C20, 85A99The fractional calculus of the P-transform or pathway transform which is a generalization of many well known integral transforms is studied. The Mellin and Laplace transforms of a P-transform are obtained. The composition formulae for the various fractional operators such as Saigo operator, Kober operator and Riemann-Liouville fractional integral and differential operators with P-transform are proved. Application of the P-transform in reaction rate...

Fractional Calculus of the Generalized Wright Function

Kilbas, Anatoly (2005)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A33, 33C20.The paper is devoted to the study of the fractional calculus of the generalized Wright function pΨq(z) defined for z ∈ C, complex ai, bj ∈ C and real αi, βj ∈ R (i = 1, 2, · · · p; j = 1, 2, · · · , q) by the series pΨq (z) It is proved that the Riemann-Liouville fractional integrals and derivative of the Wright function are also the Wright functions but of greater order. Special cases are considered.* The present investigation was partially supported...

Fractional Derivatives and Fractional Powers as Tools in Understanding Wentzell Boundary Value Problems for Pseudo-Differential Operators Generating Markov Processes

Jacob, N., Knopova, V. (2005)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A33, 31C25, 35S99, 47D07.Wentzell boundary value problem for pseudo-differential operators generating Markov processes but not satisfying the transmission condition are not well understood. Studying fractional derivatives and fractional powers of such operators gives some insights in this problem. Since an L^p – theory for such operators will provide a helpful tool we investigate the L^p –domains of certain model operators.* This work is partially supported...

Fractional Derivatives in Spaces of Generalized Functions

Stojanović, Mirjana (2011)

Fractional Calculus and Applied Analysis

MSC 2010: 26A33, 46Fxx, 58C05 Dedicated to 80-th birthday of Prof. Rudolf GorenfloWe generalize the two forms of the fractional derivatives (in Riemann-Liouville and Caputo sense) to spaces of generalized functions using appropriate techniques such as the multiplication of absolutely continuous function by the Heaviside function, and the analytical continuation. As an application, we give the two forms of the fractional derivatives of discontinuous functions in spaces of distributions.

Fractional Extensions of Jacobi Polynomials and Gauss Hypergeometric Function

Gogovcheva, Elena, Boyadjiev, Lyubomir (2005)

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: 26A33, 33C45This paper refers to a fractional order generalization of the classical Jacobi polynomials. Rodrigues’ type representation formula of fractional order is considered. By means of the Riemann–Liouville operator of fractional calculus fractional Jacobi functions are defined, some of their properties are given and compared with the corresponding properties of the classical Jacobi polynomials. These functions appear as a special case of a fractional Gauss...

Fractional Integration and Fractional Differentiation of the M-Series

Sharma, Manoj (2008)

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A33, 33C60, 44A15In this paper a new special function called as M-series is introduced. This series is a particular case of the H-function of Inayat-Hussain. The M-series is interesting because the pFq -hypergeometric function and the Mittag-Leffler function follow as its particular cases, and these functions have recently found essential applications in solving problems in physics, biology, engineering and applied sciences. Let us note that the Mittag-Leffler ...

Fractional Integration of the Product of Bessel Functions of the First Kind

Kilbas, Anatoly, Sebastian, Nicy (2010)

Fractional Calculus and Applied Analysis

Dedicated to 75th birthday of Prof. A.M. Mathai, Mathematical Subject Classification 2010:26A33, 33C10, 33C20, 33C50, 33C60, 26A09Two integral transforms involving the Gauss-hypergeometric function in the kernels are considered. They generalize the classical Riemann-Liouville and Erdélyi-Kober fractional integral operators. Formulas for compositions of such generalized fractional integrals with the product of Bessel functions of the first kind are proved. Special cases for the product of cosine...

Fractional integro-differentiation in harmonic mixed norm spaces on a half-space

Karen L. Avetisyan (2001)

Commentationes Mathematicae Universitatis Carolinae

In this paper some embedding theorems related to fractional integration and differentiation in harmonic mixed norm spaces h ( p , q , α ) on the half-space are established. We prove that mixed norm is equivalent to a “fractional derivative norm” and that harmonic conjugation is bounded in h ( p , q , α ) for the range 0 < p ≤ ∞ , 0 < q ≤ ∞ . As an application of the above, we give a characterization of h ( p , q , α ) by means of an integral representation with the use of Besov spaces.

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