Computation of eigenvalues of fractional Sturm–Liouville problems

Document Type : Research Article


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

2 University of Tabriz, Tabriz, Iran.

3 Sahand university of Technology, Tabriz, Iran.


We consider the eigenvalues of the fractional-order Sturm--Liouville equation of the form
-{}^{c}D_{0^+}^{\alpha}\circ D_{0^+}^{\alpha} y(t)+q(t)y(t)=\lambda y(t),\quad 0<\alpha\leq 1,\quad t\in[0,1],
with Dirichlet boundary conditions
$$I_{0^+}^{1-\alpha}y(t)\vert_{t=0}=0\quad\mbox{and}\quad I_{0^+}^{1-\alpha}y(t)\vert_{t=1}=0,$$
where $q\in L^2(0,1)$ is a real-valued potential function. The method is used based on a Picard's iterative procedure. We show that the eigenvalues are obtained from the zeros of the Mittag-Leffler function and its derivatives.


1. Abbasbandy, S. and Shirzadi, A. Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems, Numer. Algorithms, 54(4) (2010) 521–532.
2. Al-Mdallal, Q.M. An efficient method for solving fractional Sturm–Liouville problems, Chaos, Solitons Fractals, 40(1) (2009) 183–189.
3. Al-Mdallal, Q.M. On the numerical solution of fractional Sturm–Liouville problems, Int. J. Comput.  Math., 87(12) (2010) 2837–2845.
4. Ansari, A. On finite fractional Sturm–Liouville transforms, Integral Transform Spec. Funct. 26(1) (2015) 51–64.
5. Dastmalchi Saei, F., Abbasi, S. and Mirzay, Z. Inverse Laplace transform method for multiple solutions of the fractional Sturm-Liouville problems, Computational Methods for Differential Equations, 2(1) (2014) 56–61.
6. Dehghan, M. and Mingarelli, A.B. Fractional Sturm-Liouville eigenvalue problems, I, RACSAM 114, 46 (2020).
7. Dehghan, M. and Mingarelli, A.B. Fractional Sturm-Liouville eigenvalue problems, II, arXiv preprint arXiv:1712.09894, 2017.
8. Duan, J.-S. Time-and space-fractional partial differential equations, J. Math. Phys., 46(1) (2005) 013504.
9. Duan, J.-S., Rach, R., Baleanu, D. and Wazwaz, A.-M. A review of the Adomian decomposition method and its applications to fractional differential equations, Communications in Fractional Calculus, 3(2) (2012) 73–99.
10. Duan, J.-S.,Wang, Z., Liu, Y.-L. and Qiu, X. Eigenvalue problems for fractional ordinary differential equations, Chaos Solitons Fractals 46(2013), 46–53.
11. Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. Higher transcendental functions,Vol. III. Based, in part, on notes left by Harry Bateman. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
12. Eshaghi, S. and Ansari, A. Finite fractional Sturm–Liouville transforms for generalized fractional derivatives, Iran J. Sci. Technol. Trans. Sci. 41(4) (2017) 931–937.
13. Gorenflo, R., Kilbas, A.A., Mainardi, F., and Rogosin, S.V. a Mittag Leffler functions, related topics and applications, volume 2. Springer, 2014.
14. Gorenflo, R. and Mainardi, F. Fractional oscillations and Mittag-Leffler functions, Citeseer, 1996.
15. Joseph, K., Ralf, M. and Cheng, L.S. Fractional dynamics: Recent advances, World Scientific, 2011.
16. Kilbas, A.A. and Srivastava, H.M. Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
17. Mainardi, F. Fractional calculus and waves in linear viscoelasticity: Anintroduction to mathematical models, World Scientific, 2010.
18. Mainardi, F. Fractional calculus: some basic problems in continuum and statistical mechanics, arXiv:1201.0863, 2012.
19. Miller, K.S. and Ross, B. An introduction to the fractional calculus and fractional differential equations, John-Wiley and sons. Inc. New York, 1993.
20. Mittag-Leffler, G.M. Sur la nouvelle fonction (x), CR Acad. Sci. Paris, 137(2) (1903) 554–558.
21. Oldham, K.B. and Spanier, J. The fractional calculus. Theory and applications of differentiation and integration to arbitrary order. With an annotated chronological bibliography by Bertram Ross. Mathematics inScience and Engineering, Vol. 111. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974.
22. Podlubny, I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, 1999. vol. 198 of Mathematics in Science and Engineering.
23. Rossikhin, Y.A. and Shitikova, M.V. Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results, ASME. Appl. Mech. Rev. , 63(1) (2010) 010801.
24. Syam, M.I., Al-Mdallal, Q.M. and Al-Refai, M. A numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems, Commun. Num. Anal., (2017) 217–232.
25. Zayernouri, M. and Karniadakis, G.E. Fractional Sturm–Liouville eigen problems: Theory and numerical approximation, J. Comput. Phys., 252(2013) 495–517.