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A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators

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  • In this work, we use integration by parts formulas derived for fractional operators with Mittag-Leffler kernels to formulate and investigate fractional Sturm-Liouville Problems ($ FSLPs $). We analyze the self-adjointness, eigenvalue and eigenfunction properties of the associated Fractional Sturm-Liouville Operators ($ FSLOs $). The discrete analogue of the obtained results is formulated and analyzed by following nabla analysis.

    Mathematics Subject Classification: 26A33, 65Q10.

    Citation:

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  • [1] T. Abdeljawad, On Delta and Nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013), Art. ID 406910, 12 pp. doi: 10.1155/2013/406910.
    [2] T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ., 2013 (2013), 16 pp. doi: 10.1186/1687-1847-2013-36.
    [3] T. Abdeljawad, F. Jarad and D. Baleanu, A semigroup-like property for discrete Mittag-Leffler functions, Adv. Differ. Equ., 2012 (2012), 7 pp. doi: 10.1186/1687-1847-2012-72.
    [4] T. Abdeljawad and F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), Art. ID 406757, 13 pp. doi: 10.1155/2012/406757.
    [5] T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.
    [6] T. Abdeljawad and D. Baleanu, Fractional differences and integration by parts, J. Comput. Anal. Appl., 13 (2011), 574-582. 
    [7] T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098-1107.  doi: 10.22436/jnsa.010.03.20.
    [8] T. Abdeljawad and D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11-27.  doi: 10.1016/S0034-4877(17)30059-9.
    [9] T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), 11 pp. doi: 10.1186/s13660-017-1400-5.
    [10] T. Abdeljawad, Fractional operators with exponential kernels and a Lyapunov type inequality, Adv. Differ. Equ., 2017 (2017), 11 pp. doi: 10.1186/s13662-017-1285-0.
    [11] T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, J. Comput. Appl. Math., 339 (2018), 218-230.  doi: 10.1016/j.cam.2017.10.021.
    [12] T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ., 2016 (2016), 18 pp. doi: 10.1186/s13662-016-0949-5.
    [13] T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., 2017 (2017), 9 pp. doi: 10.1186/s13662-017-1126-1.
    [14] T. Abdeljawad and D. Baleanu, Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel, Chaos Soliton. Fract., 102 (2017), 106-110.  doi: 10.1016/j.chaos.2017.04.006.
    [15] T. Abdeljawad, Q. M. Al-Mdallal and M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dyn. Nat. Soc., 2017 (2017), Art. ID 4149320, 8 pp. doi: 10.1155/2017/4149320.
    [16] T. Abdeljawad and F. Madjidi, A Lyaponuv inequality for fractional difference operators with discrete Mittag-Leffler kernel of order 2 ≤ ν < 5/2, Eur. Phys. J., 226 (2017), 3355-3368. 
    [17] T. Abdeljawad, Different type kernel h-fractional differences and their fractional h-sums, Chaos, Solitons and Fractals, 116 (2018), 146-156.  doi: 10.1016/j.chaos.2018.09.022.
    [18] T. Abdeljawad, R. Mert and A. Peterson, Sturm-Liouville equations in the frame of fractional operators with exponential kernels and their discrete versions, Quaestiones Mathematicae, (2018). doi: 10.2989/16073606.2018.1514540.
    [19] T. Abdeljawad, R. Mert and A. Peterson, Sturm-Liouville equations in the frame of fractional operators with exponential kernels and their discrete versions, Quaestiones Mathematicae, (2018). doi: 10.2989/16073606.2018.1514540.
    [20] B. S. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos Soliton. Fract., 89 (2016), 547-551. 
    [21] Q. M. Al-Mdallal, On the numerical solution of fractional Sturm-Liouville problems, International Journal of Computer Mathematics, 87 (2010), 2837-2845.  doi: 10.1080/00207160802562549.
    [22] Q. M. Al-Mdallal, An efficient method for solving fractional Sturm-Liouville problems, Chaos, Solitons and Fractals, 40 (2009), 183-189.  doi: 10.1016/j.chaos.2007.07.041.
    [23] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763-769. 
    [24] A. Atangana and J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, Advances in Mechanical Engineering, 7 (2015). doi: 10.1177/1687814015613758.
    [25] A. Atangana and S. Jain, Models of fluid flowing in non-conventional media: New numerical analysis, Discrete and Continuous Dynamical Systems-S, (2019), 757–763. doi: 10.3934/dcdss.2020026.
    [26] A. Atangana and D. Baleanu, Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer, J. Eng. Mech., 143 (2017). doi: 10.1061/(ASCE)EM.1943-7889.0001091.
    [27] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Soliton. Fract., 89 (2016), 447-454.  doi: 10.1016/j.chaos.2016.02.012.
    [28] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.
    [29] A. Atangana and J. F. Gómez-Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos, Solitons and Fractals, 114 (2018), 516-535.  doi: 10.1016/j.chaos.2018.07.033.
    [30] F. M. Atici and P. W. Eloe, A Transform method in discrete fractional calculus, International Journal of Difference Equations, 2 (2007), 165-176. 
    [31] F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, P. Amer. Math. Soc., 137 (2009), 981-989.  doi: 10.1090/S0002-9939-08-09626-3.
    [32] F. M. Atici and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., Spec. Ed. I, 2009 (2009), 12 pp. doi: 10.14232/ejqtde.2009.4.3.
    [33] E. Bas and R. Ozarslana, Sturm-Liouville problem via Coulomb type in difference equations, Filomat, 31 (2017), 989-998.  doi: 10.2298/FIL1704989B.
    [34] W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, Inc., New York-London-Sydney, 1965.
    [35] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 73-85. 
    [36] M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11.  doi: 10.18576/pfda/020101.
    [37] C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, Cham, 2015. doi: 10.1007/978-3-319-25562-0.
    [38] H. L. Gray and N. F. Zhang, On a new definition of the fractional difference, Math. Comp., 50 (1988), 513-529.  doi: 10.1090/S0025-5718-1988-0929549-2.
    [39] F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos, Solitons and Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.
    [40] B. Karaagac, Analysis of the cable equation with non-local and non-singular kernel fractional derivative, Eur. Phys. J. Plus, 133 (2018). doi: 10.1140/epjp/i2018-11916-1.
    [41] A. A. Kilbas, M. H. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
    [42] A. A. KilbasM. Saigo and R. K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms Spec. Funct., 15 (2004), 31-49.  doi: 10.1080/10652460310001600717.
    [43] M. Klimek and O. P. Agrawal, Fractional Sturm-Liouville problem, Comput. Math. Appl., 66 (2013), 795-812.  doi: 10.1016/j.camwa.2012.12.011.
    [44] J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87-92. 
    [45] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006.
    [46] K. S. Miller and B. Ross, Fractional difference calculus, Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Ser. Math. Appl., Horwood, Chichester, (1989), 139–152.
    [47] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.
    [48] M. RiveroJ. J. Trujillo and M. P. Velasco, A fractional approach to the Sturm-Liouville problem, Cent. Eur. J. Phys., 11 (2013), 1246-1254.  doi: 10.2478/s11534-013-0216-2.
    [49] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.
    [50] I. SuwanT. Abdeljawad and F. Jarad, Monotonicity analysis for nabla h-discrete fractional Atangana-Baleanu differences, Chaos, Solitons and Fractals, 117 (2018), 50-59.  doi: 10.1016/j.chaos.2018.10.010.
    [51] 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 (2017), Art. ID cna-00334, 217–232. doi: 10.5899/2017/cna-00334.
    [52] M. Zayernouri and G. E. Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation, Journal of Computational Physics, 252 (2013), 495-517.  doi: 10.1016/j.jcp.2013.06.031.
    [53] A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005.
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