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:
[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

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

[1]

Jean Daniel Djida, Juan J. Nieto, Iván Area. Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 609-627. doi: 10.3934/dcdss.2020033
[2]

Mehmet Yavuz, Necati Özdemir. Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 995-1006. doi: 10.3934/dcdss.2020058
[3]

Ndolane Sene. Mittag-Leffler input stability of fractional differential equations and its applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 867-880. doi: 10.3934/dcdss.2020050
[4]

Antonio Coronel-Escamilla, José Francisco Gómez-Aguilar. A novel predictor-corrector scheme for solving variable-order fractional delay differential equations involving operators with Mittag-Leffler kernel. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 561-574. doi: 10.3934/dcdss.2020031
[5]

Behzad Ghanbari, Devendra Kumar, Jagdev Singh. An efficient numerical method for fractional model of allelopathic stimulatory phytoplankton species with Mittag-Leffler law. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020428
[6]

Russell Johnson, Luca Zampogni. On the inverse Sturm-Liouville problem. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 405-428. doi: 10.3934/dcds.2007.18.405
[7]

Muhammad Bilal Riaz, Syed Tauseef Saeed. Comprehensive analysis of integer-order, Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC) fractional time derivative for MHD Oldroyd-B fluid with slip effect and time dependent boundary condition. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020430
[8]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko. A partial inverse problem for the Sturm-Liouville operator on the lasso-graph. Inverse Problems & Imaging, 2019, 13 (1) : 69-79. doi: 10.3934/ipi.2019004
[9]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039
[10]

N. A. Chernyavskaya, L. A. Shuster. Spaces admissible for the Sturm-Liouville equation. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1023-1052. doi: 10.3934/cpaa.2018050
[11]

Guglielmo Feltrin. Multiple positive solutions of a sturm-liouville boundary value problem with conflicting nonlinearities. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1083-1102. doi: 10.3934/cpaa.2017052
[12]

Chuan-Fu Yang, Natalia Pavlovna Bondarenko, Xiao-Chuan Xu. An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition. Inverse Problems & Imaging, 2020, 14 (1) : 153-169. doi: 10.3934/ipi.2019068
[13]

Fahd Jarad, Thabet Abdeljawad. Variational principles in the frame of certain generalized fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 695-708. doi: 10.3934/dcdss.2020038
[14]

Ebenezer Bonyah, Samuel Kwesi Asiedu. Analysis of a Lymphatic filariasis-schistosomiasis coinfection with public health dynamics: Model obtained through Mittag-Leffler function. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 519-537. doi: 10.3934/dcdss.2020029
[15]

Peter Howard, Alim Sukhtayev. The Maslov and Morse indices for Sturm-Liouville systems on the half-line. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 983-1012. doi: 10.3934/dcds.2020068
[16]

Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067
[17]

Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236
[18]

Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control & Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016
[19]

Zbigniew Gomolka, Boguslaw Twarog, Jacek Bartman. Improvement of image processing by using homogeneous neural networks with fractional derivatives theorem. Conference Publications, 2011, 2011 (Special) : 505-514. doi: 10.3934/proc.2011.2011.505
[20]

Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 377-387. doi: 10.3934/dcdss.2020021

2019 Impact Factor: 1.233

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences