Operational Calculus for Bessel’s Fractional Equation
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.
KeywordsFractional differential equations Riemann Liouville derivative Mellin transform Laplace transform Bessel equation.
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- 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.Google Scholar
- 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.Google Scholar
- A. Kilbas, H. Srivastava and J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204, Elsevier, Amsterdam, 2006.Google Scholar
- N.N. Lebedev, Special functions and their applications, Translated from Russian by R.A. Silverman, N.J.: Prentice-Hall Inc., Englewood Cliffs, 1965.Google Scholar
- F. Mainardi andM. Tomirotti, Seismic pulse propagation with constant Q and stable probability distributions. Annali di Geofisica 40 (1997), 1311–1328.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- 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.Google Scholar
- E.C. Titchmarsh, Introduction to the theory of Fourier integrals, Clarendon Press, Oxford, 1948.Google Scholar
- S. Yakubovich and Y.F. Luchko, The hypergeometric approach to integral transforms and convolutions, Mathematics and Applications 287, Kluwer Academic Publishers, Dordrecht, 1994.Google Scholar
- S. Yakubovich,M.M. Rodrigues and N. Vieira, Fractional two-parameter Schrödinger equation. Preprint CMUP, No. 8, 2011, 12 pp.Google Scholar