Eigenvalues of fractional Sturm-Liouville problems by successive method

Document Type : Research Paper


1 Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.

2 Department of Applied Mathematics, Mathematical Science Faculty, University of Tabriz, Tabriz, Iran.

3 Department of Mathematics, Sahand University of Technology, Tabriz, Iran.


In this paper, we consider a fractional Sturm-Liouville equation of the form, − cDα 0+ ◦ Dα 0+ y(t) + q(t)y(t) = λy(t), 0 < α < 1, t ∈ [0, 1], with Dirichlet boundary conditions I 1−α 0+ y(t)|t=0 = 0, and I 1−α 0+ y(t)|t=1 = 0, where, the sign ◦ is composition of two operators and q ∈ L2 (0, 1), is a real valued potential function. We use a recursive method based on Picard’s successive method to find the solution of this problem. We prove the method is convergent and show that the eigenvalues are obtained from the zeros of the Mittag-Leffler function and its derivatives.


  • [1]          S. Abbasbandy and A. Shirzadi, Homotopy analysis method for multiple solutions of the frac- tional Sturm-Liouville problems, Numerical Algorithms, 54(4) (2010), 521–532.
  • [2]          Q. M. Al-Mdallal, An efficient method for solving fractional Sturm–Liouville problems, Chaos, Solitons & Fractals, 40(1) (2009),183–189.
  • [3]          Q. M. Al-Mdallal, On the numerical solution of fractional Sturm–Liouville problems, Interna- tional Journal of Computer Mathematics, 87(12) (2010), 2837–2845.
  • [4]          Q. M. Al-Mdallal, On fractional-legendre spectral Galerkin method for fractional Sturm– Liouville problems, Chaos, Solitons & Fractals, 116 (2-18), 261–267.
  • [5]          Q. M. Al-Mdallal, M. Al-Refai, M. Syam, and M. K. Al-Srihin, Theoretical and computational perspectives on the eigenvalues of fourth-order fractional Sturm–Liouville problem, International Journal of Computer Mathematics, 95(8) (2018), 1548–1564.
  • [6]          T. S. Aleroev, The Sturm-Liouville problem for a second-order differential equation with frac- tional derivatives in the lower terms, Differentsial’nye Uravneniya, 18(2)( 1982), 341–343.
  • [7]          A. Ansari, On finite fractional Sturm–Liouville transforms, Integral Transforms and Special Functions, 26(1) (2015), 51–64.
  • [8]          A. Ansari, Some inverse fractional Legendre transforms of gamma function form, Kodai Math- ematical Journal, 38(3) (2015),658–671.
  • [9]          P. J. Collins, Differential and integral equations. , Oxford University Press, 2006.
  • [10]        F. Dastmalchi Saei, S. Abbasi, and Z. Mirzayi, Inverse laplace transform method for multiple solutions of the fractional sturm-liouville problems, Computational Methods for Differential Equations, 2(1) (2014), 56–61.
  • [11]        M. Dehghan and A. Mingarelli, Fractional Sturm-Liouville eigenvalue problems ii, arXiv preprint arXiv:1712.09894, 2017.
  • [12]        M. Dehghan and A. B. Mingarelli, Fractional Sturm–Liouville eigenvalue problems i, Re- vista  de  la  Real  Academia  de  Ciencias  Exactas,  F´ısicas  y  Naturales.  Serie  A.  Matem´aticas, 114(2)(2020),1–15.
  • [13]        M. H. Derakhshan and A. Ansari, Fractional Sturm–Liouville problems for weber fractional derivatives, International Journal of Computer Mathematics, 96(2) (2019),217–237.
  • [14]        A. Erd´elyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions, New York, 1, 1955.
  • [15]        S. Eshaghi and A. Ansari, Finite fractional Sturm–Liouville transforms for generalized frac- tional derivatives, Iranian Journal of Science and Technology, Transactions A: Science, 41(4) (2017), 931–937.
  • [16]        R. Gorenflo and A. Kilbas, Mittag-Leffler functions, related topics and applications.
  • [17]        M. A. Hajji, Q. M. Al-Mdallal, and F. M. Allan, An efficient algorithm for solving higher-order fractional Sturm–Liouville eigenvalue problems, Journal of Computational Physics, 272 (2014), 550–558.
  • [18]        A. Kilbas, Theory and applications of fractional differential equations.
  • [19]        M. Klimek and O. P. Agrawal, Fractional Sturm–Liouville problem, Computers & Mathematics with Applications, 66(5) (2013), 795–812.
  • [20]        M. Klimek, T. Odzijewicz, and A. B. Malinowska, Variational methods for the fractional Sturm– Liouville problem, Journal of Mathematical Analysis and Applications, 416(1) (2014), 402–426.
  • [21]        F. Mainardi, Fractional calculus: In Fractals and fractional calculus in continuum mechanics, Springer, 1997, 291–348.
  • [22]        F. Mainardi, Fractional calculus and waves in linear viscoelasticity: an introduction to mathe- matical models, World Scientific, 2010.
  • [23]        K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John-Wily and Sons Inc. New York, 1993.
  • [24]        G. M. Mittag-Leffler, Sur la nouvelle fonction eα(x), CR Acad. Sci. Paris, 137(2) (1903), 554– 558.
  • [25]        A. Neamaty, R. Darzi, A. Dabbaghian, and J. Golipoor, Introducing an iterative method for solving a special FDE , International Mathematical Forum, 4 (2009), 1449–1456.
  • [26]        I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, frac- tional differential equations, to methods of their solution and some of their applications, Else- vier, 1998.
  • [27]        M. I. Syam, Q. M. Al-Mdallal, and M. Al-Refai, A numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems, Communications in Numerical Analysis, (2017), 217–232.
  • [28]        M. Zayernouri and G. E. Karniadakis, Fractional Sturm–Liouville eigen-problems: theory and numerical approximation , Journal of Computational Physics, 252 (2013), 495–517.