AIMS Mathematics, 2019, 4(3): 593-612. doi: 10.3934/math.2019.3.593

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Theory of discrete fractional Sturm–Liouville equations and visual results

Department of Mathematics, Firat University, 23119, Elazig, Turkey

In this article, we study discrete fractional Sturm-Liouville (DFSL) operators within Riemann-Liouville and Grünwald-Letnikov fractional operators with both delta and nabla operators. Self-adjointness of the DFSL operator is analyzed and fundamental spectral properties are proved. Besides, we get sum representation of solutions for DFSL problem by means of Laplace transform for nabla fractional difference equations and find the analytical solutions of the problem. Moreover, the results for DFSL problem, discrete Sturm-Liouville (DSL) problem with the second order, and fractional Sturm-Liouville (FSL) problem are compared with the second order classical Sturm-Liouville (CSL) problem. We display the results comparatively by tables and figures.
  Article Metrics


1.J. B. Diaz, T. J. Osler, Differences of fractional order, Math. Comput., 28 (1974), 185-202.    

2.K. S. Miller, B. Ross, Fractional difference calculus. In: Proceedings of the International Symposium on Univalent Functions, fractional calculus and their applications, pp. 139-152, 1989.

3.H. L. Gray, N. F. Zhang, On a new definition of the fractional difference, Math. Comput., 50 (1988), 513-529.    

4.C. Goodrich, A. C. Peterson, Discrete fractional calculus, Berlin: Springer, 2015.

5.D. Baleanu, S. Rezapour, S. Salehi, On some self-adjoint fractional finite difference equations, J. Comput. Anal. Appl., 19 (2015), 59-67.

6.K. Ahrendt, T. Rolling, L. Dewolf, et al. Initial and boundary value problems for the caputo fractional self-adjoint difference equations, EPAM, 2 (2016), 105-141.

7.F. M. Atı cı, P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theo., 3 (2009), 1-12.

8.N. Acar, F. M. Atı cı, Exponential functions of discrete fractional calculus, Appl. Anal. Discr. Math., 7 (2013), 343-353.    

9.F. Atici, P. Eloe, Initial value problems in discrete fractional calculus, P. Am. Math. Soc., 137 (2008), 981-989.    

10.F. M. Atici, P. W. Eloe, Linear forward fractional difference equations, Communications in Applied Analysis, 19 (2015), 31-42.

11.F. M. Atı cı, M. Atı cı, M. Belcher, et al. A new approach for modeling with discrete fractional equations, Fund. Inform., 151 (2017), 313-324.    

12.T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.    

13.T. Abdeljawad, D. Baleanu, Fractional Differences and Integration by Parts, J. Comput. Anal. Appl., 13 (2011), 574-582.

14.T. Abdeljawad, On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013), 1-12.

15.D. Mozyrska, M. Wyrwas, The Z-Transform Method and Delta Type Fractional Difference Operators, Discrete Dyn. Nat. Soc., 2015 (2015), 1-12.

16.D. Mozyrska, M. Wyrwas, Solutions of fractional linear difference systems with Caputo-type operator via transform method. In: ICFDA'14 International Conference on Fractional Differentiation and Its Applications 2014, pp. 1-6.

17.D. Mozyrska, M. Wyrwas, Fractional linear equations with discrete operators of positive order. In: K. Latawiec, M. Łukaniszyn, R. Stanisławski (eds) Advances in Modelling and Control of Non-integer-Order Systems, Lecture Notes in Electrical Engineering, Vol. 320, Springer, Cham, 2015.

18.G. A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Model., 51 (2010), 562-571.    

19.G. A. Anastassiou, Right nabla discrete fractional calculus, International Journal of Difference Equations, 6 (2011), 91-104.

20.J. Hein, Z. McCarthy, N. Gaswick, et al. Laplace transforms for the nabla-difference operator, Panamerican Mathematical Journal, 21 (2011), 79-97.

21.J. F. Cheng, Y. M. Chu, Fractional difference equations with real variable, Abstr. Appl. Anal., 2012 (2012), 1-24.

22.R. Yilmazer, F. Tchier, D. Baleanu, Particular solutions of the confluent hypergeometric differential equation by using the nabla fractional calculus operator, Entropy, 18 (2016), 49.

23.G. C. Wu, D. Baleanu, S. D. Zeng, et al. Discrete fractional diffusion equation, Nonlinear Dynam., 80 (2015), 281-286.    

24.D. Baleanu, O. Defterli, O. P. Agrawal, A central difference numerical scheme for fractional optimal control problems, J. Vib. Control, 15 (2009), 583-597.    

25.G. C. Wu, D. Baleanu, S. D. Zeng, et al. Mittag-Leffler function for discrete fractional modelling, Journal of King Saud University-Science, 28 (2016), 99-102.    

26.R. Almeida, A. Malinowska, M. L. Morgado, et al. Variational methods for the solution of fractional discrete/continuous Sturm–Liouville problems, J. Mech. Mater. Struct., 12 (2017), 3-21.    

27.B. M. Levitan, I. S. Sargsian, Introduction to spectral theory: selfadjoint ordinary differential operators, Translations of mathematical monographs, American Mathematical Society, 1975.

28.M. Dehghan, A. B. Mingarelli, Fractional Sturm-Liouville eigenvalue problems, I, arXiv preprint arXiv:1712.09891, 2017.

29.M. Dehghan, A. B. Mingarelli, Fractional Sturm-Liouville eigenvalue problems, II, arXiv preprint arXiv:1712.09894, 2017.

30.E. Bas, F. Metin, Fractional singular Sturm-Liouville operator for Coulomb potential, Adv. Differ. Equ-Ny, 2013 (2013), 300.

31.E. Bas, The Inverse Nodal problem for the fractional diffusion equation, Acta Sci-Technol, 37 (2015), 251-257.    

32.M. Klimek, O. P. Agrawal, Fractional Sturm–Liouville problem, Comput. Math. Appl., 66 (2013), 795-812.    

33.M. Klimek, O. P. Agrawal, On a regular fractional Sturm-Liouville problem with derivatives of order in (0, 1). In: Proceedings of the 13th International Carpathian Control Conference (ICCC), pp. 284-289, 2012.

34.M. Klimek, T. Odzijewicz, A. B. Malinowska, Variational methods for the fractional Sturm–Liouville problem, J. Math. Anal. Appl., 416 (2014), 402-426.    

35.A. B. Malinowska, T. Odzijewicz, D. F. Torres, Advanced methods in the fractional calculus of variations, Cham: Springer, 2015.

36.M. Al-Refai, T. Abdeljawad, Fundamental results of conformable Sturm-Liouville eigenvalue problems, Complexity, 2017 (2017), 1-7.

37.T. Abdeljawad, R. Mert, A. Peterson, Sturm Liouville Equations in the frame of fractional operators with exponential kernels and their discrete versions, Quaest. Math., (2018), 1-19.

38.S. Qureshi, A. Yusuf, A. A. Shaikh, et al. Fractional modeling of blood ethanol concentration system with real data application, Chaos, 29 (2019), 013143.

39.A. Yusuf, S. Qureshi, A. I. Aliyu, et al. Two-strain epidemic model involving fractional derivative with Mittag-Leffler kernel, Chaos, 28 (2018), 123121.

40.S. Qureshi, A. Yusuf, Modeling chickenpox disease with fractional derivatives: From caputo to atangana-baleanu, Chaos Soliton. Fract., 122 (2019), 111-118.    

41.E. Bas, R. Ozarslan, Real world applications of fractional models by Atangana–Baleanu fractional derivative, Chaos Soliton. Fract., 116 (2018), 121-125.    

42.E. Bas, R. Ozarslan, D. Baleanu, et al. Comparative simulations for solutions of fractional Sturm–Liouville problems with non-singular operators, Adv. Differ. Equ-Ny, 2018 (2018), 350.

43.E. Bas, B. Acay, R. Ozarslan, Fractional models with singular and non-singular kernels for energy efficient buildings, Chaos, 29 (2019), 023110.

44.R. Ozarslan, A. Ercan, E. Bas, Novel Fractional Models Compatible with Real World Problems, Fractal and Fractional, 3 (2019), 15.

45.R. Mert, T. Abdeljawad, A. Peterson, Sturm Liouville Equations in the frame of fractional operators with Mittag-Leffler kernels and their discrete versions, arXiv preprint arXiv:1803.05013, 2018.

46.M. Bohner, A. C. Peterson, Advances in dynamic equations on time scales, Springer Science & Business Media, 2003.

47.T. Abdeljawad, F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), 1-13.

48.L. Bourdin, J. Cresson, I. Greff, et al. Variational integrator for fractional Euler–Lagrange equations, Appl. Numer. Math., 71 (2013), 14-23.    

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved