Abstract
An alternative method for solving the fractional kinetic equations solved earlier by Haubold and Mathai (Astrophysics and Space Science 273:53–63, 2000) and Saxena et al. (Astrophysics and Space Science 282:281–287, 2002; Physica A 344:653–664, 2004a; Astrophysics and Space Science 290:299–310, 2004b) is recently given by Saxena and Kalla (Applied Mathematics and Computation 199:504–511, 2007). This method can also be applied in solving more general fractional kinetic equations than the ones solved by the aforesaid authors. In view of the usefulness and importance of the kinetic equation in certain physical problems governing reaction-diffusion in complex systems and anomalous diffusion, the authors present an alternative simple method for deriving the solution of the generalized forms of the fractional kinetic equations solved by the aforesaid authors and Nonnenmacher and Metzler (Fractals 3:557–566, 1995). The method depends on the use of the Riemann-Liouville fractional calculus operators. It has been shown by the application of Riemann-Liouville fractional integral operator and its interesting properties, that the solution of the given fractional kinetic equation can be obtained in a straight-forward manner. This method does not make use of the Laplace transform.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Al-Saqabi, B.N., Tuan, V.K.: Solution of a fractional differ integral equation. Integral Transforms and Special Functions 4, 321–326 (1996)
Dzherbashyan, M.M.: Integral Transforms and Representation of Functions in Complex Domain (in Russian). Nauka, Moscow (1966)
Dzherbashyan, M.M.: Harmonic Analysis and Boundary Value Problems in the Complex Domain. Birkhaeuser-Verlag, Basel and London (1993)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Tables of Integral Transforms. Vol. 2, McGraw-Hill, New York, Toronto, and London (1954)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions. Vol. 3, McGraw-Hill, New York, Toronto, and London (1955)
Gloeckle, W.G., Nonnenmacher, T.F.: Fractional integral operators and Fox function in the theory of viscoelasticity. Macromolecules 24, 6426–6434 (1991)
Haubold, H.J., Mathai, A.M.: The fractional kinetic equation and thermonuclear functions. Astrophysics and Space Science 273, 53–63 (2000)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Publishing Co., New York (2000)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier, Amsterdam (2006)
Mathai, A.M., Saxena, R.K.: The H-function with Applications in Statistics and Other Disciplines. John Wiley and Sons Inc., New York, London and Sydney (1978)
Miller, K.S., Ross, B.: An Introduction to Fractional Calculus and Fractional Differential Equations. Wiley and Sons, New York (1993)
Mittag-Leffler, G.M.: Sur la nouvelle function. C.R. Acad. Sci., Paris, 137, 554–558 (1903)
Nonnenmacher, T.F., Metzler, R.: On the Riemann-Liouville fractional calculus and some recent applications. Fractals 3, 557–566 (1995)
Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Applications of Differentiation and Integration of Arbitrary Order. Academic Press, New York (1974)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Saichev, A., Zaslavsky, M.: Fractional kinetic equations: solutions and applications, Chaos 7, 753–764 (1997)
Saxena, R.K., Kalla, S.L.: 2007, On the solution of certain fractional kinetic equations. In Applied Mathematics and Computation 199, 504–511 (2007)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: On fractional kinetic equations. Astrophysics and Space Science 282, 281–287 (2002)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: On generalized fractional kinetic equations. Physica A 344, 653–664 (2004a)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: Unified fractional kinetic equation and a fractional diffusion equation. Astrophysics and Space Science 290, 299–310 (2004b)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: Fractional reaction-diffusion equations. Astrophysics and Space Science 305, 289–296 (2006a)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: Solution of generalized fractional reaction-diffusion equations. Astrophysics and Space Science 305, 305–313 (2006b)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: Reaction-diffusion systems and nonlinear waves. Astrophysics and Space Science 305, 297–303 (2006c)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: Solution of fractional reaction-diffusion equations in terms of Mittag-Leffler functions. International Journal of Scientific Research 15, 1–17 (2006d)
Wiman, A.: Ueber den Fundamentalsatz in der Theorie der Funktionen. Acta Mathematica 29, 191–201 (1905)
Zaslavsky, G.M.: Fractional kinetic equation for Hamiltonian chaos. Physica D 78, 110–122 (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Saxena, R.K., Mathai, A.M., Haubold, H.J. (2010). An Alternative Method for Solving a Certain Class of Fractional Kinetic Equations. In: Haubold, H., Mathai, A. (eds) Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science. Astrophysics and Space Science Proceedings. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03325-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-03325-4_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03388-9
Online ISBN: 978-3-642-03325-4
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)