Sturm-Liouville problem in multiplicative fractional calculus

Tuba Gulsen; Sertac Goktas; Thabet Abdeljawad; Yusuf Gurefe; Tuba Gulsen; Sertac Goktas; Thabet Abdeljawad; Yusuf Gurefe

doi:10.3934/math.20241109

AIMS Mathematics

2024, Volume 9, Issue 8: 22794-22812. doi: 10.3934/math.20241109

Previous Article Next Article

Research article

Sturm-Liouville problem in multiplicative fractional calculus

1.
Department of Mathematics, Faculty of Science, Fırat University, Elazığ, Türkiye
2.
Department of Mathematics, Faculty of Science, Mersin University, Mersin, Türkiye
3.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh, Saudi Arabia
4.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea
6.
Department of Mathematics and Applied Mathematics, School of Science and Technology, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa

Received: 13 February 2024 Revised: 26 June 2024 Accepted: 03 July 2024 Published: 23 July 2024
MSC : 11N05, 26A33, 34B24, 34L05

Multiplicative calculus, or geometric calculus, is an alternative to classical calculus that relies on division and multiplication as opposed to addition and subtraction, which are the basic operations of classical calculus. It offers a geometric interpretation that is especially helpful for simulating systems that degrade or expand exponentially. Multiplicative calculus may be extended to fractional orders, much as classical calculus, which enables the analysis of systems having fractional scaling properties. So, in this paper, the well-known Sturm-Liouville problem in fractional calculus is reformulated in multiplicative fractional calculus. The considered problem consists of the Sturm-Liouville operator using multiplicative conformable derivatives on the equation and on boundary conditions. This research aimed to explore some of the problem's spectral aspects, like being self-adjointness of the operator, orthogonality of different eigenfunctions, and reality of all eigenvalues. In this specific situation, Green's function is also recreated.
- geometric calculus,
- multiplicative conformable fractional calculus,
- non-Newtonian Calculus,
- Sturm-Liouville equation
Citation: Tuba Gulsen, Sertac Goktas, Thabet Abdeljawad, Yusuf Gurefe. Sturm-Liouville problem in multiplicative fractional calculus[J]. AIMS Mathematics, 2024, 9(8): 22794-22812. doi: 10.3934/math.20241109

Related Papers:

Abstract

Multiplicative calculus, or geometric calculus, is an alternative to classical calculus that relies on division and multiplication as opposed to addition and subtraction, which are the basic operations of classical calculus. It offers a geometric interpretation that is especially helpful for simulating systems that degrade or expand exponentially. Multiplicative calculus may be extended to fractional orders, much as classical calculus, which enables the analysis of systems having fractional scaling properties. So, in this paper, the well-known Sturm-Liouville problem in fractional calculus is reformulated in multiplicative fractional calculus. The considered problem consists of the Sturm-Liouville operator using multiplicative conformable derivatives on the equation and on boundary conditions. This research aimed to explore some of the problem's spectral aspects, like being self-adjointness of the operator, orthogonality of different eigenfunctions, and reality of all eigenvalues. In this specific situation, Green's function is also recreated.

References

[1]	T. Abdeljawad, M. Grossman, On geometric fractional calculus, J. Semigroup Theory Appl., 2016 (2016), 2.
[2]	D. Baleanu, Z. B. Guvenlç, J. A. T. Machado, New trends in nanotechnology and fractional calculus applications, New York: Springer, 2010. https://doi.org/10.1007/978-90-481-3293-5
[3]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, New York: Elsevier, 2006.
[4]	K. S. Miller, B. Ross, An introduction to fractional calculus and fractional differential equations, New York: Wiley, 1993.
[5]	K. B. Oldham, J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, New York: Academic Press, 1974. https://doi.org/10.1016/S0076-5392(09)60219-8
[6]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, New York: Academic Press, 1998.
[7]	T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
[8]	R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
[9]	A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889–898. https://doi.org/10.1515/math-2015-0081 doi: 10.1515/math-2015-0081
[10]	T. Gülşen, E. Yilmaz, H. Kemaloǵlu, Conformable fractional Sturm-Liouville equation and some existence results on time scales, Turk. J. Math., 42 (2018), 1348–1360. https://doi.org/10.3906/mat-1704-120 doi: 10.3906/mat-1704-120
[11]	J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1140–1153. https://doi.org/10.1016/j.cnsns.2010.05.027 doi: 10.1016/j.cnsns.2010.05.027
[12]	M. D. Ortigueira, J. A. T. Machado, What is a fractional derivative?, J. Comput. Phys., 293 (2015), 4–13. https://doi.org/10.1016/j.jcp.2014.07.019 doi: 10.1016/j.jcp.2014.07.019
[13]	R. Kumar, S. Kumar, S. Kaur, S. Jain, Time fractional generalized Korteweg-de Vries equation: Explicit series solutions and exact solutions, J. Frac. Calc. Nonlinear Sys., 2 (2021), 62–77. https://doi.org/10.48185/jfcns.v2i2.315 doi: 10.48185/jfcns.v2i2.315
[14]	R. Ferreira, Generalized discrete operators, J. Frac. Calc. Nonlinear Sys., 2 (2021), 18–23. https://doi.org/10.48185/jfcns.v2i1.279 doi: 10.48185/jfcns.v2i1.279
[15]	M. Grossman, An introduction to Non-Newtonian calculus, Int. J. Math. Educ. Sci. Technol., 10 (1979), 525–528. https://doi.org/10.1080/0020739790100406 doi: 10.1080/0020739790100406
[16]	M. Grossman, R. Katz, Non-Newtonian calculus, Pigeon Cove, MA: Lee Press, 1972.
[17]	A. E. Bashirov, E. M. Kurpinar, A. Ozyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (2008), 36–48. https://doi.org/10.1016/j.jmaa.2007.03.081 doi: 10.1016/j.jmaa.2007.03.081
[18]	A. E. Bashirov, M. Riza, On complex multiplicative differentiation, TWMS J. Appl. Eng. Math., 1 (2011), 75–85.
[19]	K. Boruah, B. Hazarika, G-calculus, TWMS J. Appl. Eng. Math., 8 (2018), 94–105.
[20]	D. A. Stanley, A multiplicative calculus, Primus IX, 9 (1999), 310–326.
[21]	D. Aniszewska, Multiplicative Runge-Kutta methods, Nonlinear Dyn., 50 (2007), 265–272. https://doi.org/10.1007/s11071-006-9156-3 doi: 10.1007/s11071-006-9156-3
[22]	D. Aniszewska, M. Rybaczuk, Chaos in multiplicative systems, Chaotic Syst., 2010, 9–16. https://doi.org/10.1142/9789814299725_0002
[23]	A. E. Bashirov, G. Bashirova, Dynamics of literary texts and diffusion, Online J. Commun. Media Technol., 1 (2011), 60–82.
[24]	A. E. Bashirov, E. Misirli, Y. Tandogdu, A. Ozyapici, On modeling with multiplicative differential equations, Appl. Math. J. Chin. Univ., 26 (2011), 425–438. https://doi.org/10.1007/s11766-011-2767-6 doi: 10.1007/s11766-011-2767-6
[25]	A. Benford, The Law of anomalous numbers, Proc. Am. Phil. Soc., 78 (1938), 551–572.
[26]	M. Cheng, Z. Jiang, A new class of production function model and its application, J. Syst. Sci. Inf., 4 (2016), 177–185. https://doi.org/10.21078/JSSI-2016-177-09 doi: 10.21078/JSSI-2016-177-09
[27]	D. Filip, C. Piatecki, A non-Newtonian examination of the theory of exogenous economic growth, Math. Aeterna, 4 (2014), 101–117.
[28]	L. Florack, H. van Assen, Multiplicative calculus in biomedical image analysis, J. Math. Imaging Vis., 42 (2012), 64–75. https://doi.org/10.1007/s10851-011-0275-1 doi: 10.1007/s10851-011-0275-1
[29]	H. Özyapıcı, İ. Dalcı, A. Özyapıcı, Integrating accounting and multiplicative calculus: An effective estimation of learning curve, Comput. Math. Org. Theory, 23 (2017), 258–270. https://doi.org/10.1007/s10588-016-9225-1 doi: 10.1007/s10588-016-9225-1
[30]	N. Yalcin, The solutions of multiplicative Hermite diferential equation and multiplicative Hermite polynomials, Rend. Circ. Mat. Palermo II Ser., 70 (2021), 9–21. https://doi.org/10.1007/s12215-019-00474-5 doi: 10.1007/s12215-019-00474-5
[31]	N. Yalcin, E. Celik, Solution of multiplicative homogeneous linear differential equations with constant exponentials, New Trend Math. Sci., 6 (2018), 58–67. http://dx.doi.org/10.20852/ntmsci.2018.270 doi: 10.20852/ntmsci.2018.270
[32]	N. Yalcin, M. Dedeturk, Solutions of multiplicative ordinary differential equations via the multiplicative differential transform method, AIMS Mathematics, 6 (2021), 3393–3409. https://doi.org/10.3934/math.2021203 doi: 10.3934/math.2021203
[33]	E. Yilmaz, Multiplicative Bessel equation and its spectral properties, Ricerche Mat., 2021. https://doi.org/10.1007/s11587-021-00674-1
[34]	Z. Zhao, T. Nazir, Existence of common coupled fixed points of generalized contractive mappings in ordered multiplicative metric spaces, Electron. J. Appl. Math., 1 (2023), 1–15. https://doi.org/10.61383/ejam.20231341 doi: 10.61383/ejam.20231341
[35]	B. Meftah, H. Boulares, A. Khan, T. Abdeljawad, Fractional multiplicative Ostrowski-type inequalities for multiplicative differentiable convex functions, Jordan J. Math. Stat., 17 (2024), 113–128. https://doi.org/10.47013/17.1.7 doi: 10.47013/17.1.7
[36]	S. Goktas, A new type of Sturm-Liouville equation in the Non-Newtonian calculus, J. Funct. Spaces., 2021 (2021), 5203939. https://doi.org/10.1155/2021/5203939 doi: 10.1155/2021/5203939
[37]	S. Goktas, E. Yilmaz, A. C. Yar, Multiplicative derivative and its basic properties on time scales, Math. Methods Appl. Sci., 45 (2022), 2097–2109. https://doi.org/10.1002/mma.7910 doi: 10.1002/mma.7910
[38]	S. Goktas, H. Kemaloglu, E. Yilmaz, Multiplicative conformable fractional Dirac system, Turk. J. Math., 46 (2022), 973–990. https://doi.org/10.55730/1300-0098.3136 doi: 10.55730/1300-0098.3136
[39]	M. Al-Refai, T. Abdeljawad, Fundamental results of conformable Sturm-Liouville eigenvalue problems, Complexity, 2017 (2017), 3720471. https://doi.org/10.1155/2017/3720471 doi: 10.1155/2017/3720471
[40]	B. P. Allahverdiev, H. Tuna, Y. Yalçinkaya, Conformable fractional Sturm‐Liouville equation, Math. Methods Appl. Sci., 42 (2019), 3508–3526. https://doi.org/10.1002/mma.5595 doi: 10.1002/mma.5595
[41]	M. Klimek, O. P. Agrawal, Fractional Sturm-Liouville problem, Comput. Math. Appl., 66 (2013), 795–812. https://doi.org/10.1016/j.camwa.2012.12.011 doi: 10.1016/j.camwa.2012.12.011
[42]	M. Rivero, J. J. Trujillo, M. P. Velasco, A fractional approach to the Sturm-Liouville problem, Cent. Eur. J. Phys., 11 (2013), 1246–1254. https://doi.org/10.2478/s11534-013-0216-2 doi: 10.2478/s11534-013-0216-2
[43]	U. Kadak, Y. Gurefe, A generalization on weighted means and convex functions with respect to the Non-Newtonian calculus, Int. J. Anal., 2016 (2016), 5416751. https://doi.org/10.1155/2016/5416751 doi: 10.1155/2016/5416751
[44]	S. Goktas, Multiplicative conformable fractional differential equations, Turk. J. Sci. Tech., 17 (2022), 99–108. https://doi.org/10.55525/tjst.1065429 doi: 10.55525/tjst.1065429

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)