Abstract
In this article, we introduce the regular and singular fractional Sturm–Liouville problems with the Hilfer and Hilfer–Prabhakar derivatives. We show that these problems with the corresponding boundary conditions have real eigenvalues and their eigenfunctions are orthogonal. Also, the finite fractional Sturm–Liouville transforms and their inversion formulas are established, and as an application, the formal solution of the fractional Laplace equation in prolate spheroidal coordinates is obtained using the finite Legendre transform.
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Abramowitz M, Stegun IA (1972) Handbook of mathematical functions with formulas, graphs and mathematical tables. Dover, New York
Askari H, Ansari A (2016) Fractional calculus of variations with a generalized fractional derivative. Fract Differ Calc 6(1):57–72
Ansari A (2015a) On finite fractional Sturm–Liouville transforms. Integral Transform Spec Funct 26(1):51–64
Ansari A (2015b) Some inverse fractional Legendre transforms of gamma function form. Kodai Math J 38:658–671
Bas E, Metin F (2013) Fractional singular Sturm–Liouville operator for Coulomb potential. Adv Differ Equ. https://doi.org/10.1186/1687-1847-1-13 2013-300
Eshaghi S, Ansari A (2015) Autoconvolution equations and generalized Mittag-Leffler functions. Int J Indus Math 7(4):335–341
Eshaghi S, Ansari A (2016) Lyapunov inequality for fractional differential equations with Prabhakar derivative. Math Inequal Appl 19(1):349–358
Hilfer R (2000) Applications of fractional calculus in physics. World Scientific Publishing Company, Singapore
Hilfer R, Luchko Y, Tomovski Z (2009) Operational method for solution of the fractional differential equations with the generalized Riemann–Liouville fractional derivatives. Fract Calc Appl Anal 12:299–318
Churchill CRV (1953) New operational mathematics the operational calculus of Legendre transforms. Technical Report No.1, Project 2137 Ordnance Corps, US Army, Contract No. DA-20-018-ORD-12916, August, 1953
Debnath L, Bhatta D (2007) Integral transforms and their applications, 2nd ed. Chapman & Hall/CRC, Taylor & Francis Group, New York
Erturk VS (2011) Computing eigenelements of Sturm–Liouville problems of fractional order via fractional differential transform method. Math Comput Appl 16:712–720
Garra R, Gorenflo R, Polito F, Tomovski Z (2014) Hilfer–Prabhakar derivatives and some applications. Appl Math Comput 242:576–589
Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam
Klimek M, Agrawal OP (2012) On a regular fractional Sturm–Liouville problem with derivatives of order in (0,1). In: Proceedings of the 13th international carpathian control conference, Vysoke Tatry (Podbanske), Slovakia, 28–31 May
Klimek M, Agrawal OP (2013) Fractional Sturm–Liouville problem. Comput Math Appl 66:795–812
Lebedev NN, Uflyand YS, Skalskaya IP (1965) Worked problems in applied mathematics. Dover, New York
D’Ovidio M (2012) From Sturm–Liouville problems to fractional and anomalous diffusions. Stoch Process Appl 122:3513–3544
Podlubny I (1999) Fractional differential equations. Academic Press, San Diego
Prabhakar TR (1971) A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math J 19:7–15
Rivero M, Trujillo JJ, Velasco MP (2013) A fractional approach to the Sturm–Liouville problem. Cent Eur J Phys 11(10):1246–1254
Sneddon IN (1979) The use of integral transforms. Mac Graw-Hill, New York
Srivastava HM, Tomovski Z (2009) Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl Math Comput 211(1):198–210
Tomovski Z, Hilfer R, Srivastava HM (2010) Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions. Integral Transf Spec Funct 21(11):797–814
Tranter CJ (1950) Legendre transforms. Q J Math Oxf J 1(2):1–8
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The authors wish to thank the referees for their constructive comments on earlier version of this paper.
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Eshaghi, S., Ansari, A. Finite Fractional Sturm–Liouville Transforms For Generalized Fractional Derivatives. Iran J Sci Technol Trans Sci 41, 931–937 (2017). https://doi.org/10.1007/s40995-017-0311-0
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DOI: https://doi.org/10.1007/s40995-017-0311-0