American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020171

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

Raziye Mert 1, , Thabet Abdeljawad 2,3,4,, and  Allan Peterson 5,

1. 

Mechatronic Engineering Department, University of Turkish Aeronautical Association, 06790, Ankara, Turkey
2. 

Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia
3. 

Department of Medical Research, China Medical University, Taichung 40402, Taiwan
4. 

Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan
5. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE, 68588-0130, USA

* Corresponding author: T. Abdeljawad (tabdeljawad@psu.edu.sa)

Received  March 2019 Revised  April 2019 Published  December 2019

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.

Keywords: Fractional Sturm-Liouville problem, ABR and ABC fractional derivatives, ABR and ABC fractional differences, Mittag-Leffler kernel.
Mathematics Subject Classification: 26A33, 65Q10.
Citation: Raziye Mert, Thabet Abdeljawad, Allan Peterson. A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020171
References:
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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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T. Abdeljawad and D. Baleanu, Fractional differences and integration by parts, J. Comput. Anal. Appl., 13 (2011), 574-582.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[20]

B. S. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos Soliton. Fract., 89 (2016), 547-551.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

F. M. Atici and P. W. Eloe, A Transform method in discrete fractional calculus, International Journal of Difference Equations, 2 (2007), 165-176.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

E. Bas and R. Ozarslana, Sturm-Liouville problem via Coulomb type in difference equations, Filomat, 31 (2017), 989-998.  doi: 10.2298/FIL1704989B.  Google Scholar

[34]

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, Inc., New York-London-Sydney, 1965.  Google Scholar

[35]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 73-85.   Google Scholar

[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.  Google Scholar

[37]

C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, Cham, 2015. doi: 10.1007/978-3-319-25562-0.  Google Scholar

[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.  Google Scholar

[39]

F. Jarad, T. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

A. A. Kilbas, M. 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.  Google Scholar

[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.  Google Scholar

[44]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87-92.   Google Scholar

[45]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

M. Rivero, J. 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.  Google Scholar

[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.  Google Scholar

[50]

I. Suwan, T. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[53]

A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl., 62 (2011), 1602-1611.  doi: 10.1016/j.camwa.2011.03.036.  Google Scholar

[6]

T. Abdeljawad and D. Baleanu, Fractional differences and integration by parts, J. Comput. Anal. Appl., 13 (2011), 574-582.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

B. S. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos Soliton. Fract., 89 (2016), 547-551.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

F. M. Atici and P. W. Eloe, A Transform method in discrete fractional calculus, International Journal of Difference Equations, 2 (2007), 165-176.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

E. Bas and R. Ozarslana, Sturm-Liouville problem via Coulomb type in difference equations, Filomat, 31 (2017), 989-998.  doi: 10.2298/FIL1704989B.  Google Scholar

[34]

W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, Inc., New York-London-Sydney, 1965.  Google Scholar

[35]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernal, Progr. Fract. Differ. Appl., 1 (2015), 73-85.   Google Scholar

[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.  Google Scholar

[37]

C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, Cham, 2015. doi: 10.1007/978-3-319-25562-0.  Google Scholar

[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.  Google Scholar

[39]

F. Jarad, T. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

A. A. Kilbas, M. 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.  Google Scholar

[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.  Google Scholar

[44]

J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 87-92.   Google Scholar

[45]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

M. Rivero, J. 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.  Google Scholar

[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.  Google Scholar

[50]

I. Suwan, T. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[53]

A. Zettl, Sturm-Liouville Theory, Mathematical Surveys and Monographs, 121. American Mathematical Society, Providence, RI, 2005.  Google Scholar

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