Integral Transform and the Solution of Fractional Kinetic Equation Involving Some Special Functions

S.K.Sharma, A.S.Shekhawat "Integral Transform and the Solution of Fractional Kinetic Equation Involving Some Special Functions", *International Journal of Mathematics Trends and Technology (IJMTT). *V55(2):127-136 March 2018. ISSN:2231-5373. www.ijmttjournal.org. Published by Seventh Sense Research Group.

**Abstract**

The aim of the present paper is to establish the solution of advanced generalized fractional order kinetic equation and a main theorem based upon the multivariable I-function, Mittag –Leffler function, generalized M-series, generalized k4 – function, and generalized Mittag- Leffler function, Riemann–Liouville operator. The solution of the generalized fractional kinetic equation involving the multivariable I-function is obtained with help of the Laplace and Sumudu transform. Due to its simple formulation and consequent special and useful properties, the Sumudu and Laplace transform has already shown much promise. It is revealed herein and elsewhere that is can help to solve intricate problems in mathematical physics, especially in astrophysical problems. The results derived by using certain Corollaries used in this paper are interesting, computable and very general in nature.

**Reference**

[1] A.A. Kilbas, M.Saigo and R.K. Saxena, Generlized Mittag-Leffler function and generalized fractional calculus operators, Integral Transform and Special Functions, Vol.15, pp.31-49, 2004

[2] A.Faraz, T. Salim, A. Sadik and J.Ismail, Solution of generalization fractional kinetic Equation in terms of special functions, International Mathematical Forum, Hikari Ltd., Vol.9, No.33, pp.1647-1657, 2014.

[3] A.M. Mathai, R.K. Sexena and H.J. Houbold, On generalized fractional kinetic equation, Physica A 344, pp.653-664, 2004.

[4] A.Saichev and G. Zaslavsky, Fractional kinetic equations, solutions and applications, Chaos, Vol.7, pp.753-784. http://dx.doi.org/10.1063/1.166272, 1997.

[5] A.Faraz, T.Salim. S.Sadek and J.Ismail, Generalization K4 function and its application in solving kinetic equation of fractional order, J.Math. Comput. Sci., Vol.4, pp.1-10, 2014.

[6] A. Wiman, Uber de Fundamental Staz in der Theorie der Functionen E(X), Acta Mathematica, Vol. 29(1), pp.191-201, 1905.

[7] F.B.M. Belgacem, A.A. Karaballi, Sumdu transform fundamental properties investigations and applications, International J.Appl.Maths.Stoch. Anal., pp.1-23, 2005.

[8] F.B.M. Belgacem, A.A. Karaballi and S.L. Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Mathematical problems in Engineering, Vol. 3,pp.103-118, 2003.

[9] G. Zaslavsky, Fractional kinetic equation for Hamiltonian Chaos, Physica D, Vol.76, pp.110 - 122. http://dx.doi.org/10.1016/0167-2789 (94)90254-2, 1994.

[10] H.J. Haubold and A.M. Mathai, The fractional Kinetic equation and thermonuclearfunctions, Astrophys. Space Sci., Vol. 327, pp.53-63, 2000.

[11] I.Podlubny, Fractional differential equations, Vol.198 of mathematics in science and Engineering ,Academic Press ,San Diego, Calif. USA ,1999.

[12] K.S. Miller and B. Ross, An Introduction of Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993.

[13] M. Sharma, Fractional integration and fractional differentiation of the M-series, Fract. Calc. Appl. Anal., Vol.11(2), pp.187-192, 2008.

[14] M.A. Asiru, Sumudu transform and the solution of integral equations of convolution type, International Journal of Mathematical Education in science and technology, Vol.32, pp.906-910, 2001.

[15] R.Gorenflo, A.A. Kilbass and S.V. Rogosin, On the general Mittag-Leffler type function, Integral Transform and Special Functions,7(3-4) (1998), pp.215-224.

[16] R.K. Sexena, A.M.Mathai, and H. Haubold, Unified fractional kinetic equation and a fractional diffusion equation, Astrophysics and Space - Science, Vol.290, pp.299 – 310, 2002.

[17] R.K. Sexena, and S. Kalla, On the solution of certain fractional kinetic equations, Appl. Math. Comput., Vol.199, pp.504 – 511, 2008, http://dx.doi.org/10.1016/j.amc.2007.10.005.

[18] R.K. Sexena, A.M. Mathai, and H. Haubold, On fractional kinetic equations, Astrophysics and Space - Sci, Vol.282, pp.281 – 287, 2002.

[19] S.G. Samko, A.A Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.

[20] T.Salim and A.Faraz, A generalization of Mittag-Leffler type function and integral operator associated with fractional calculus, Jou. Frac. Cal. Appl. Vol.3, No., pp.1-13, 2012.

[21] R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Mathematical Journal,Vol.19,pp.7-15, 1971.

[22] V.B.L. Chaurasia and D. Kumar, On the solutions of Generalized fractionalKinetic equation, Adv. Stud. Theor. Phys.,Vol. 4(16), pp.773-780, 2010.

[23] M.Sharma and R.Jain, A note on a generalized M-series as a special function of fractional calculus and applied analysis, Vol.(12), pp. 449-452, 2009V.B.L. Chaurasia and J.Singh, Application of Sumudu Transform in Schrodinger Equation Occurring in Quantum Mechanics, Applied mathematical Sciences, Vol.4(57) ,pp. 2843-2850, 2010.

[24] Y.N. Prasad, Multivariable I-function, Vijanana Parishad Anusandhan Patrika, Vol. 29(4), 231-235, 1986.

[25] Y.N. Prasad, G.S. Yadav, Proc. Math. Soc. B.H.U., I, pp.127-136, 1985.

[26] A.Faraz, T.Salim. S.Sadek and J.Ismail, A generalized of M-series and Integral operator associated with fractional calculus, Asian Journal of Fuzzy and Applied Mathematics. Vol.2 (5), pp.142-155, 2014.

[27] C.F.Lorenzo, and T.T. Hartley, R-function relationship for application in the fractional calculus, NASA/ TM, 2000-210361, 2000.

[28] C.F.Lorenzo, and T.T. Hartley, Generalized functions for the fractional calculus. NASA/ TP, 1999-209424/ REVI, pp.17, 1999.

[29] K. Sharma, On Application of Fractional differ integral Operator to the K4-function, Bol. Soc.Paran. Math., Vol. 30(1), pp.91-97, 2012.

[30] T.T. Hartley and C.F.Lorenzo, A solution to the fundamental linear fractional order differential equations, NASA/TP-1998-208693, 1998.

**Keywords**

Special functions, Fractional Kinetic equation, Mittag-Leffler function, Riemann-Liouville operator, Laplace transform, Sumudu transform.