Abstract
This paper is intended to investigate a fractional differential Bessel’s equation of order 2α with \( \alpha \in]0,1] \) involving the Riemann–Liouville derivative. We seek a possible solution in terms of power series by using operational approach for the Laplace and Mellin transform. A recurrence relation for coefficients is obtained. The existence and uniqueness of solutions is discussed via Banach fixed point theorem.
Mathematics Subject Classification (2010). Primary 35R11; Secondary 34B30, 42A38, 47H10.
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.
References
J. Butzer, Mellin transform theory and the role of its differential and integral operators. Transform Methods and Special Functions, Varna’96 (Conf. Proc.), Bulgarian Acad. Sci., (1998), 63–83.
S.D. Eidelman, S.D. Ivasyshen and A.N. Kochubei, Analytic methods in the theory of differential and pseudo-differential equations of parabolic type, Operator Theory: Advances and Applications 152, Birkhäuser, Basel, 2004.
A.A. Kilbas, M. Rivero, L. Rodríguez-Germá and J.J. Trujillo, Analytic solutions of some linear fractional differential equations with variable coefficients Appl. Math. Comput. 187 (2007), 239–249.
A. Kilbas, H. Srivastava and J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.
N.N. Lebedev, Special functions and their applications, Translated from Russian by R.A. Silverman, N.J.: Prentice-Hall Inc., Englewood Cliffs, 1965.
F. Mainardi andM. Tomirotti, Seismic pulse propagation with constant Q and stable probability distributions. Annali di Geofisica 40 (1997), 1311–1328.
F. Mainardi, Fractional diffusive waves in viscoelastic solids, IUTAM Symposium – Nonlinear Waves in Solids (Conf. Proc.), J.L. Wagnern and F.R. Norwood (editors), (1995), 93–97.
O.I. Marichev, Handbook of integral transforms of higher transcendental functions: theory and algorithmic tables, translated from Russian by L.W. Longdon, Ellis Horwood Limited, Chichester, 1983.
I. Podlubny, Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solutions and some applications, Mathematics in Science and Engineering 198, Academic Press, San Diego, 1999.
A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series, Volume 3: More special functions, translated from Russian by G.G. Gould, Gordon and Breach Publisher, New York, 1990.
S.G. Samko, A.A. Kilbas, and O.I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach, New York, 1993, 277–284.
E.C. Titchmarsh, Introduction to the theory of Fourier integrals, Clarendon Press, Oxford, 1948.
S. Yakubovich and Y.F. Luchko, The hypergeometric approach to integral transforms and convolutions, Mathematics and Applications 287, Kluwer Academic Publishers, Dordrecht, 1994.
S. Yakubovich,M.M. Rodrigues and N. Vieira, Fractional two-parameter Schrödinger equation. Preprint CMUP, No. 8, 2011, 12 pp.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this paper
Cite this paper
Rodrigues, M.M., Vieira, N., Yakubovich, S. (2013). Operational Calculus for Bessel’s Fractional Equation. In: Almeida, A., Castro, L., Speck, FO. (eds) Advances in Harmonic Analysis and Operator Theory. Operator Theory: Advances and Applications, vol 229. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0516-2_20
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0516-2_20
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0515-5
Online ISBN: 978-3-0348-0516-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)