Certain fractional integral operators and the generalized multi-index Mittag-Leffler functions

详细信息    查看全文

• 作者：PRAVEEN AGARWAL ; SERGEI V ROGOSIN ; JUAN J TRUJILLO
• 关键词：Marichev–Saigo–Maeda fractional integral operators ; generalized multiindex Mittag ; Leffler functions ; Appell functions ; generalized Wright function. ; 33E12 ; 26A33 ; 33C65.
• 刊名：Proceedings Mathematical Sciences
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：125
• 期：3
• 页码：291-306
• 全文大小：211 KB
• 参考文献：[1]Agarwal P, Chnad M and Jain S, Certain integrals involving generalized Mittag-Leffler functions, Proc. Nat. Acad. Sci. India Sect. A (under printing)
  [2]Al-Bassam M A and Luchko Y F, On generalized fractional calculus and it application to the solution of integro-differential equations, J. Fract. Calc. 7 (1995) 69-8MathSciNet MATH
  [3]Baleanu D, Diethelm K, Scalas E and Trujillo J J, Fractional Calculus: Models and Numerical Methods (2012) (N. Jersey, London, Singapore: World Scientific Publishers)
  [4]Caponetto R, Dongola G, Fortuna L and Petrá? I, Fractional Order Systems: Modeling and Control Applications (2010) (Singapore: World Scientific Publ. Co Inc)
  [5]Caputo M and Mainardi F, Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II) 1 (1971) 161-98CrossRef
  [6]Choi J, and Agarwal P, Certain integral transform and fractional integral formulas for the generalized Gauss hypergeometric functions, Abstr. Appl. Anal. 2014 (2014) 735946,7 pages; available online at http://?dx.?doi.?org/-0.-155/-014/-35946
  [7]Diethelm K, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type (2010) (Berlin: Springer) Springer Lecture Notes in Mathematics No 2004MATH
  [8]Dzrbashjan M M, On the integral transforms generated by the generalized Mittag-Leffler function, Izv. AN Arm. SSR 13 (3) (1960) 21-3
  [9]Gorenflo R and Mainardi F, Fractional calculus: integral and differential equations of fractional order, in: A Carpinteri and F Mainardi (eds) Fractals and Fractional Calculus in Continuum Mechanics (1997) (Wien: Springer Verlag)
  [10]Haubold H J, Mathai A M, and Saxena R K, Mittag-Leffler functions and their applications, J. Appl. Math. 2011 (2011) 298628, 51 pages; available online at http://?dx.?doi.?org/-0.-155/-011/-98628
  [11]Hilfer R (ed) Applications of Fractional Calculus in Physics (2000) (New Jersey, London, Hong Kong, Word Scientific Publishing Co.)
  [12]Hille E and Tamarkin J D, On the theory of linear integral equations, Ann. Math. 31 (1930) 479-28MathSciNet CrossRef MATH
  [13]Kilbas A A, Fractional calculus of the generalized Wright function, Fract. Calc. Appl. Anal. 8 (2) (2005) 113-26MathSciNet MATH
  [14]Kilbas A A, Koroleva A A and Rogosin S V, Multi-parametric Mittag-Leffler functions and their extension, Fract. Calc. Appl. Anal. 16 (2) (2013) 378-04MathSciNet CrossRef
  [15]Kilbas A A and Saigo M, H-Transform. Theory and Applications (2004) (Boca Raton-London- New York-Washington, D.C.: Chapman and Hall/CRC)CrossRef
  [16]Kilbas A A, Saigo M and Saxena R K, Solution of Volterra integro-differential equations with generalized Mittag-Leffler function in the kernels, J. Integral Equations Appl. 14 (4) (2002) 377-86MathSciNet CrossRef MATH
  [17]Kilbas A A, Srivastava H M, Trujillo J J, Theory and Applications of Fractional Differential Equations North-Holland Mathematics Studies 204 (2006) (Elsevier, Amsterdam, etc)
  [18]Kiryakova V, Generalized Fractional Calculus and Applications (1994) (Harlow: Longman)MATH
  [19]Kiryakova V, Multiindex Mittag-Leffler functions, related Gelfond-Leontiev operators and Laplace type integral transforms, Fract. Calc. Appl. Anal. 2 (4) (1999) 445-62MathSciNet MATH
  [20]Kiryakova V, Multiple (multi-index) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math. 118 (2000) 241-59MathSciNet CrossRef MATH
  [21]Kiryakova V, On two Saigo’s fractional integral operators in the class of univalent functions, Fract. Calc. Appl. Anal. 9 (2) (2006) 160-76MathSciNet
  [22]Kiryakova V, Some special functions related to fractional calculus and fractional (non-integer) order control systems and equations, Facta Universitatis (Sci. J. of University of Nis), Series: Automatic Control and Robotics 7 (1) (2008) 79-8MathSciNet
  [23]Kiryakova V S, The special functions of fractional calculus as generalized fractional calculus operators of some basic functions, Comp. Math. Appl. 59 (3) (2010) 1128-1141MathSciNet CrossRef MATH
  [24]Kiryakova V S, The multi-index Mittag-Leffler function as an important class of special functions of fractional calculus, Comp. Math. Appl. 59 (5) (2010) 1885-895MathSciNet CrossRef MATH
  [25]Mainardi F, Fractional Calculus and Waves in Linear Viscoelasticity (2010) (London: Imperial College Press)CrossRef MATH
  [26]Marichev O I, Volterra equation of Mellin convolution type with a Horn function in the kernel, Izv. AN BSSR Ser. Fiz.-Mat. Nauk. No. 1 (1974) 128-29
  [27]Mathai A M and Haubold H J, Special Functions for Applied Scientists (2008) (New York: Springer)CrossRef MATH
  [28]Mathai A M, Saxena R K and Haubold H J, The H-function: Theory and applications (2010) (Dordrecht: Springer)CrossRef
  [29]McBride A C, Fractional Calculus and Integral Transforms of Generalized Functions, Research Notes in Math. 31 (1979) (London: Pitman)
  [30]M
• 作者单位：PRAVEEN AGARWAL (1)
  SERGEI V ROGOSIN (1)
  JUAN J TRUJILLO (3)
  
  1. Department of Mathematics, Anand International College of Engineering, Jaipur, 303 012, India
  3. Department de Analisis Matemático, Universidad de La Laguna, C/Astr. Fco. Sánchez s/n, 38271, La Laguna Tenerife, Spain
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer India
• ISSN：0973-7685

文摘

In this paper, we obtain formulas of fractional integration (of Marichev-Saigo–Maeda type) of the generalized multi-index Mittag-Leffler functions E _γ,κ [(α _j ,β _j ) _m ; z] generalizing 2m-parametric Mittag-Leffler functions studied by Saxena and Nishimoto (J. Fract. Calc. 37 (2010] 43-2). Some interesting special cases of our main results are considered too. Keywords Marichev–Saigo–Maeda fractional integral operators generalized multiindex Mittag-Leffler functions Appell functions generalized Wright function.