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Interpolation of exponential-type functions on a uniform grid by shifts of a basis function
A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators
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 |
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.
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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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