Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access September 25, 2015

Some fractional integral formulas for the Mittag-Leffler type function with four parameters

  • Praveen Agarwal EMAIL logo and Juan J. Nieto
From the journal Open Mathematics

Abstract

In this paper we present some results from the theory of fractional integration operators (of Marichev- Saigo-Maeda type) involving the Mittag-Leffler type function with four parameters ζ , γ, Eμ, ν[z] which has been recently introduced by Garg et al. Some interesting special cases are given to fractional integration operators involving some Special functions.

References

[1] Agarwal P, Trujillo J J, Rogosin S V, Certain fractional integral operators and the generalized multiindex Mittag-Leffler functions, Proc. Indian Acad. Sci. Math. Sci. (In press)Search in Google Scholar

[2] Agarwal P, Chnad M and Jain S, Certain integrals involving generalized Mittag-Leffler functions, Proc. Nat. Acad. Sci. India Sect. A, (2015); doi:10.1007/s40010-015-0209-110.1007/s40010-015-0209-1Search in Google Scholar

[3] Agarwal P, Jain S, Chnad M, Dwivedi S K, Kumar S, Bessel functions Associated with Saigo-Maeda fractional derivative operators, J. Fract. Calc. 5(2) (2014) 102-112Search in Google Scholar

[4] Al-Bassam M A, Luchko Y F, On generalized fractional calculus and it application to the solution of integro-differential equations. J. Fract. Calc. 7 (1995) 69-88Search in Google Scholar

[5] Baleanu D, Diethelm K, Scalas E, Trujillo J J, Fractional Calculus: Models and Numerical Methods (2012) (N. Jersey, London, Singapore: World Scientific Publishers)10.1142/8180Search in Google Scholar

[6] Capelas de Oliveira E, Mainardi F, VAZ J Jr, Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics, Eur. Phys. J. Special Topics 193 (2011) 161-171;http://dx.doi.org/10.1140/epjst/e2011-01388-0 10.1140/epjst/e2011-01388-0Search in Google Scholar

[7] Caponetto R, Dongola G, Fortuna L, and Petráš I, Fractional Order Systems: Modeling and Control Applications (2010) (Singapore: World Scientific Pub Co Inc)10.1142/7709Search in Google Scholar

[8] Caputo M, Mainardi F, Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento (Ser. II). 1 (1971) 161-198Search in Google Scholar

[9] Choi J, Agarwal P, A note on fractional integral operator associate with multiindex Mittag-Leffler functions, Filomat (In press)Search in Google Scholar

[10] Choi J, 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/10.1155/2014/735946 10.1155/2014/735946Search in Google Scholar

[11] 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 200410.1007/978-3-642-14574-2_8Search in Google Scholar

[12] Dzrbashjan M M, On the integral transforms generated by the generalized Mittag-Leffler function, Izv. AN Arm. SSR 13(3) (1960) 21-63Search in Google Scholar

[13] Garg M, A. Sharma and P. Manohar, A Generalized Mittag-Leffler Type Function with Four Parameters, Thai.J. Math., (In press)Search in Google Scholar

[14] Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V., Mittag-Leffler Functions, Related Topics and Applications (Springer 2014) 454 pages. http://www.springer.com/us/book/9783662439296 10.1007/978-3-662-43930-2Search in Google Scholar

[15] Gorenflo R, Mainardi F, Fractional calculus: integral and differential equations of fractional order, in: A. Carpinteri and F. Mainardi (Editors) Fractals and Fractional Calculus in Continuum Mechanics 223-276 (1997) (Springer Verlag, Wien)10.1007/978-3-7091-2664-6_5Search in Google Scholar

[16] 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/10.1155/2011/298628 10.1155/2011/298628Search in Google Scholar

[17] Hilfer R (Edt), Applications of Fractional Calculus in Physics (2000) (New Jersey, London, Hong Kong:Word Scientific Publishing Co.)Search in Google Scholar

[18] Kilbas A A, Koroleva A A, Rogosin S V, Multi-parametric Mittag-Leffler functions and their extension, Fract. Calc. Appl. Anal. 16(2) (2013) 378-40410.2478/s13540-013-0024-9Search in Google Scholar

[19] 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) 377-38610.1216/jiea/1181074929Search in Google Scholar

[20] Kilbas A A, Srivastava H M, Trujillo J J, Theory and Applications of Fractional Differential Equations (2006)North-Holland Mathematics Studies. 204 (Elsevier, Amsterdam, etc)Search in Google Scholar

[21] Kiryakova V,Generalized Fractional Calculus and Applications (1994) (Harlow, Longman)Search in Google Scholar

[22] Kiryakova V, Multiindex Mittag-Leffler functions, related Gelfond-Leontiev operators and Laplace type integral transforms, Fract. Calc. Appl. Anal. 2(4) (1999) 445-462Search in Google Scholar

[23] Kiryakova V, Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus, J. Comput. Appl. Math. 118 (2000) 241-25910.1016/S0377-0427(00)00292-2Search in Google Scholar

[24] Kiryakova V, On two Saigo’s fractional integral operators in the class of univalent functions, Fract. Calc. Appl. Anal. 9(2) (2006) 160-176Search in Google Scholar

[25] Kiryakova V S, The special functions of fractional calculus as generalized fractional calculus operators od some basic functions, Comp. Math. Appl. 59(3) (2010) 1128-114110.1016/j.camwa.2009.05.014Search in Google Scholar

[26] 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) 1885-189510.1016/j.camwa.2009.08.025Search in Google Scholar

[27] Mainardi F, Fractional Calculus and Waves in Linear Viscoelasticity (2010) (London: Imperial College Press)10.1142/p614Search in Google Scholar

[28] 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) 128-129Search in Google Scholar

[29] Mathai A M, Saxena R K, The H-function with Applications in Statistics and Other Disciplines, Halsted Press [John Wiley & Sons], New York, London, Sydney, 1978Search in Google Scholar

[30] Mathai A M, Saxena R K, Haubold H J, The H-function. Theory and Applications (2010) (Dordrecht: Springer)10.1007/978-1-4419-0916-9Search in Google Scholar

[31] McBride A C, Fractional Calculus and Integral Transforms of Gen- eralized Functions (1979) (Research Notes in Math. 31) (Pitman, London)Search in Google Scholar

[32] Miller K S, Ross B, An Introduction to the Fractional Calculus and Fractional Differential Equations (1993) (New York: John Wiley and Sons)Search in Google Scholar

[33] Mittag-Leffler G M, Sur la nouvelle fonction E .x/, C. R. Acad. Sci. Paris 137 (1903) 554-558Search in Google Scholar

[34] NIST Handbook of Mathematical Functions. Edited by Frank W.J. Olver (editor-in-chief), D.W. Lozier, R.F. Boisvert, and C.W. Clark. Gaithersburg, Maryland, National Institute of Standards and Technology, and New York, Cambridge University Press, 951 + xv pages and a CD, (2010)Search in Google Scholar

[35] Oldham K B, Spanier J, The Fractional Calculus (1974) (New York:Academic Press)Search in Google Scholar

[36] Podlubny I, Fractional Differential Equations (1999) (New York: Academic Press)Search in Google Scholar

[37] Prabhakar T R, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971) 7-15Search in Google Scholar

[38] Prakasa Rao B L S, Statistical inference for fractional diffusion processes (2010) (Chichester: John Wiley & Sons Ltd.)Search in Google Scholar

[39] Rabotnov Yu N, Elements of Hereditary Solid Mechanics (1980) (Moscow:MIR)Search in Google Scholar

[40] Rogosin S.V., "The Role of the Mittag-Leffler Function in Fractional Modeling" Mathematics 2015, 3, 368-381; doi:10.3390/math302036810.3390/math3020368Search in Google Scholar

[41] Saigo M, On generalized fractional calculus operators. In: Recent Advances in Applied Mathematics (Proc. Internat. Workshop held at Kuwait Univ.). Kuwait Univ., Kuwait, (1996) 441-450Search in Google Scholar

[42] Saigo M, Maeda N, More generalization of fractional calculus, In: Transform Methods and Special Functions, Varna 1996 (Proc. 2nd Intern. Workshop, P. Rusev, I. Dimovski, V. Kiryakova Eds.), IMI-BAS, Sofia, (1998) 386-400Search in Google Scholar

[43] Samko S G, Kilbas A A, Marichev O I, Fractional Integrals and Derivatives: Theory and Applications (1993) (New York and London: Gordon and Breach Science Publishers)Search in Google Scholar

[44] Saxena R K and Nishimoto K, N-Fractional Calculus of Generalized Mittag- Leffler functions, J. Fract. Calc. 37(2010) 43-52Search in Google Scholar

[45] Saxena R K, Saigo M, Generalized fractional calculus of the H-function associated with the Appell function, J. Fract. Calc. 19 (2001) 89-104 Search in Google Scholar

[46] Shukla A K and Prajapati J C, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl. 336 (2007) 797-81110.1016/j.jmaa.2007.03.018Search in Google Scholar

[47] Srivastava H M and Agarwal P, Certain fractional integral operators and the generalized incomplete hypergeometric functions, Appl. Appl. Math. 8(2) (2013) 333-345Search in Google Scholar

[48] Srivastava H M, Choi J, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 200110.1007/978-94-015-9672-5Search in Google Scholar

[49] Srivastava H M, Choi J, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 201210.1016/B978-0-12-385218-2.00002-5Search in Google Scholar

[50] Srivastava H M, Gupta K C, Goyal S P, The H-functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, Madras, 1982Search in Google Scholar

[51] Srivastava H M and Saigo M, Multiplication of fractional calculus operators and boundary value problems involving the eulerdarboux equation, J. Math. Anal. Appl. 121 (1987) 325-36910.1016/0022-247X(87)90251-4Search in Google Scholar

[52] Srivastava H M, Tomovski LZ, Fractional claculus with an integral operator containing generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211(1) (2009) 198-21010.1016/j.amc.2009.01.055Search in Google Scholar

[53] Tarasov V E, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media (2010) (Beijing: Springer, Heidelberg; Higher Education Press)Search in Google Scholar

[54] Tenreiro Machado J A, Kiryakova V, Mainardi F, A poster about the old history of fractional calculus, Fract. Calc. Appl. Anal. 13 (4)(2010) 447-454Search in Google Scholar

[55] Tenreiro Machado J A, Kiryakova V, Mainardi F, A poster about the recent history of fractional calculus, Fract. Calc. Appl. Anal. 13(3)(2010) 329-334Search in Google Scholar

[56] Tenreiro Machado J A, Kiryakova V, Mainardi F, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simulat. 16(3) (2011) 1140-1153; available online at http://dx.doi.org/10.1016/j.cnsns.2010.05.027 10.1016/j.cnsns.2010.05.027Search in Google Scholar

[57] Uchaikin V V, Fractional derivatives for physicists and engineers. Vol. I. Background and theory(2013) (Beijing: Springer, Berlin - Higher Education Press)10.1007/978-3-642-33911-0Search in Google Scholar

[58] Uchaikin V V, Fractional derivatives for physicists and engineers, Vol. II Applications(2013) (Beijing: Springer, Berlin - Higher Education Press)Search in Google Scholar

[59] Wiman A, Über den Fundamentalsatz in der Theorie der Funktionen E .x/, Acta Math. 29(1905) 191-20110.1007/BF02403202Search in Google Scholar

[60] Zaslavsky G M, Hamiltonian Chaos and Fractional Dynamics (2005) (Oxford: Oxford University Press)Search in Google Scholar

Received: 2015-6-16
Accepted: 2015-8-16
Published Online: 2015-9-25

© 2015

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 4.3.2024 from https://www.degruyter.com/document/doi/10.1515/math-2015-0051/html
Scroll to top button