Abstract
In view of the usefulness and a great importance of the kinetic equation in certain astrophysical problems the authors develop a new and further generalized form of the fractional kinetic equation involving the G-function, a generalized function for the fractional calculus. This new generalization can be used for the computation of the change of chemical composition in stars like the Sun. The Mellin-Barnes contour integral representation of the G-function is also established. The manifold generality of the G-function is discussed in terms of the solution of the above fractional kinetic equation. A compact and easily computable solution is established. Special cases, involving the generalized Mittag-leffler function and the R-function, are considered. The obtained results imply more precisely the known results.
Similar content being viewed by others
References
Agarwal, R.P.: A Propos d’une note de M. Pierre Humbert. C. R. Acad. Sci. Paris 296, 2031–2032 (1953)
Buschman, R.G., Srivastava, H.M.: The \(\overline{H}\) -function associated with a certain class of Feynman integrals. J. Phys. A Math. Gen. 23, 4707–4710 (1990)
Clayton, D.D.: Principles of Stellar Evolution and Nucleosynthesis, 2nd edn. The University of Chicago and London (1983)
Dotsenko, M.R.: On some applications of Wright’s hypergeometric function. C. R. Acad. Bulg. Sci. 44(6), 13–16 (1991)
Dzherbashyan, M.M.: Integral Transforms and Representation of Functions in Complex Domain. Nauka, Moscow (1966). (In Russian)
Dzherbashyan, M.M.: Harmonic Analysis and Boundary Value Problems in the Complex Domain. Birkhauser, Basel (1993)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of Integral Transforms, vol. 2. McGraw-Hill, New York (1954)
Fox, C.: The asymototic expansion of generalized hypergeometric functions. Proc. Lond. Math. Soc. Ser. 2 27, 389–400 (1928)
Hartley, T.T., Lorenzo, C.F.: A solution to the fundamental linear fractional order differential equation. NASA/TP-1998-208963 (1998)
Haubold, H.J., Mathai, A.M.: The fractional kinetic equation and thermonuclear functions. Astrophys. Space Sci. 273, 53–63 (2000)
Inayat-Hussain, A.A.: New properties of hypergeometric series derivable from Feynman integrals, II. A generalization of the H-function. J. Phys. A Math. Gen. 20, 4119–4128 (1987)
Kourganoff, V.: Introduction to the Physics of Stellars Interiors. D. Reidel Publishing Company, Dordrecht (1973)
Lorenzo, C.F., Hartley, T.T.: Generalized function for the fractional calculus. NASA/TP-1999-209424 (1999)
Mathai, A.M., Saxena, R.K.: The H-Function with Applications in Statistics and other Disciplines. Halsted Press, Wiley, New York (1978)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Mittag-Leffler, G.M.: Sur la nouvelle fonction E α (x). C. R. Acad. Sci. Paris Ser. 2 137, 554–558 (1903)
Mittag-Leffler, G.M.: Sur la representation analytique d’une fonction branche uniforme dune fonction. Acta Math. 29, 101–181 (1905)
Oldham, K.B., Spanier, J.: The Fractional Calculus. Theory and Applications of Differentiation and Integration of Arbitrary Order. Academic Press, New York (1974)
Perdang, J.: Lecturer notes in Stellar stability, Parts I and II. Instituto di Astronomia, Padova (1976)
Prabhakar, T.R.: A singular integral equation with the generalized Mittag-Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York (1993)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: On fractional kinetic equations. Astrophys. Space Sci. 282, 281–287 (2002)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: On generalized fractional kinetic equations. Physica A 344, 657–664 (2004a)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: Unified fractional kinetic equation and a fractional diffusion equation. Astrophys. Space Sci. 290, 299–310 (2004b)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: Solution of certain fractional kinetic equation and a fractional diffusion equation (2007). arXiv:0704.1916v1
Srivastava, H.M., Karlsson, P.W.: Multiple Gaussian Hypergeometric Series. Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York (1985)
Srivastava, H.M., Gupta, K.C., Goyal, S.P.: The H-function of One and Two Variables with Applications. South Asian Publ., New Delhi and Madras (1982)
Wiman, A.: Uber den Fundamentalsatz in der Theorie der Functionen E α (x). Acta. Math. 29, 191–201 (1905)
Wright, E.M.: The asymptotic expansion of the generalized hypergeometric functions. J. Lond. Math. Soc. 10, 286–293 (1935)
Wright, E.M.: The asymptotic expansion of the generalized hypergeometric functions. Proc. Lond. Math. Soc. 46(2), 389–408 (1940)
Wylie, C.R.: Advance Engineering Mathematics, 4th edn. McGraw-Hill, New York (1975)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chaurasia, V.B.L., Pandey, S.C. On the new computable solution of the generalized fractional kinetic equations involving the generalized function for the fractional calculus and related functions. Astrophys Space Sci 317, 213–219 (2008). https://doi.org/10.1007/s10509-008-9880-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10509-008-9880-x
Keywords
- Generalized function for the fractional calculus and fractional kinetic equations: general
- Fractional kinetic equations and generalized functions: individual (the G-function and its relationships with other special functions, the Mellin-Barnes contour integral representation of the G-function, generalized fractional kinetic equations)