Nirmal JANGİD, Sunil JOSHİ, Sunil Dutt PROHİT, Dineshlal SUTHAR

Fractional derivatives and expansion formulae of incomplete $H$ and $\overline{H}$-functions

In this paper, we investigate the fractional derivatives and expansion formulae of incomplete $H$ and $\overline{H}$-functions for one variable. Further, we also obtain results for repeated fractional order derivatives and some special cases are also discussed. Various other analogues results are also established. The results obtained here are very much helpful for the further research and useful in the study of applied problems of sciences, engineering and technology.

Keywords:

Fractional calculus operators, Incomplete Gamma functions, Incomplete $H$-functions, Mellin-Barnes type contour,

PDF

___

[1] A.K. Arora, C.L. Koul, Applications of fractional calculus, Indian J. Pure Appl. Math. 18 (1987) 931-937.
[2] R.G. Buschman, H.M. Srivastava, The H-function associated with a certain class of Feynman integrals, J. Phys. A 23 (1990) 4707-4710.
[3] B.B. Jaimini, N. Shrivastava, H.M. Srivastava, The integral analogue of the Leibniz rule for fractional calculus and its applications involving functions of several variables, Comput. Math. Appl. 41 (2001) 149-155.
[4] C.M. Joshi, N.L. Joshi, Fractional derivatives and expansion formulas involving H-functions of one and more variables, J. Math. Anal. Appl. 207 (1997) 1-11.
[5] K. Jothimani, N. Valliammal, C. Ravichandran, Existence Result for a Neutral Fractional Integro-Differential Equation with State Dependent Delay, J. Appl. Nonlinear Dyn. 7 (2018) 371-381.
[6] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematical Studies, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York 204 (2006).
[7] A. Kumar, H.V.S. Chauhan, C. Ravichandran, K.S. Nisar, Existence of soltions of non-autonomous fractional differential equations with integral impulse condition, Adv. Differ. Equ. 434 (2020).
[8] A.M. Mathai, R.K. Saxena, The H-function with applications in statistics and other disciplines, Wiley Eastern Limited, New Delhi; John Wiley and Sons, New York (1978).
[9] A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-functions: Theory a applications, Springer, New York (2010).
[10] S. Meena, S. Bhatter, K. Jangid, S.D. Purohit, Certain expansion formulae of incomplete H-functions associated with Leibniz rule, TWMS J. App. & Eng. Math (2020), Accepted.
[11] S. Min, Some algebra of Leibniz rule for fractional calculus, Int. J. Innov. Sci. Math. 4 (2016) 204-208.
[12] K.B. Oldham, J. Spanier, The fractional calculus, Academic Press, New York/London (1974).
[13] S.D. Purohit, Summation formulae for basic hypergeometric functions via q-fractional calculus, Le Matematiche 64 (2009) 67-75.
[14] C. Ravichandran, K. Logeswari, F. Jarad, New results on existence in the framework of Atangana-Baleanu derivative for fractional integro-di?erential equations, Chaos Solitons Fractals 125 (2019) 194-200.
[15] C. Ravichandran, K. Logeswari S. K. Panda, K. S. Nisar, On new approach of fractional derivative by Mittag-Leffer kernel to neutral integro-differential systems with impulsive conditions, Chaos Solitons Fractals 139 (2020) 110012.
[16] B. Ross, F.H. Northover, A use for a derivative of complex order in the fractional calculus, Indian J. Pure Appl. Math. 9 (1978) 400-406.
[17] B. Ross, Fractional calculus and its applications, Lecture Notes in Math, Springer-Verlag, New York 457 (1975).
[18] H.M. Srivastava, K.C. Gupta, S.P. Goyal, The H-functions of one and two variables with applications, South Asian Publishers, New Delhi and Madras (1982).
[19] H.M. Srivastava, M.A. Chaudhry, R.P. Agarwal, The incomplete pochhammer symbols and their applications to hyperge- ometric and related functions, Integral Transforms Spec. Funct. 23 (2012) 659-683.
[20] H.M. Srivastava, R.K. Saxena, R.K. Parmar, Some families of the incomplete H-functions and the incomplete H-functions and associated integral transforms and operators of fractional calculus with applications, Russ. J. Math. Phys. 25 (2018) 116-138. [21] R. Subashini, K. Jothimani, K.S. Nisar, C. Ravichandran, New results on nonlocal functional integro-differential equations via Hilfer fractional derivative, Alex. Eng. J. 59 (2020) 2891-2899.
[22] N. Valliammal, C. Ravichandran, K.S. Nisar, Solutions to fractional neutral delay differential nonlocal systems, Chaos Solitons Fractals 138 (2020) 109912.
[23] R.K. Yadav, S.D. Purohit, V.K. Vyas, On transformations involving generalized basic hypergeometric functions of two variables, Rev. Tec. Ing. Univ. Zulia. 33 (2010) 176-182.