ArticleOpen Access

SOME RECENT DEVELOPMENTS ON DYNAMICAL $ℏ$ -DISCRETE FRACTIONAL TYPE INEQUALITIES IN THE FRAME OF NONSINGULAR AND NONLOCAL KERNELS

Department of Mathematics, Government College University, Faisalabad 38000, Pakistan

Celestial Mechanics and Space Dynamics Research Group, Astronomy Department National Research, Institute of Astronomy and Geophysics (NRIAG), Helwan 11421, Cairo, Egypt

E-mail Address: elbaz.abouelmagd@nriag.sci.eg

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AASMA KHALID

Department of Mathematics, Government College Women University, Faisalabad 38000, Pakistan

E-mail Address: aasmakhalid@gcwuf.edu.pk

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FOZIA BASHIR FAROOQ

Imam Mohammad Ibn Saud Islamic University, Riyadh, KSA, Saudi Arabia

E-mail Address: Ffarooq@imamu.edu.sa

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, and

YU-MING CHU

Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China

Institute for Advanced Study Honoring Chen Jian Gong, Hangzhou Normal University, Hangzhou 311121, P. R. China

E-mail Address: chuyuming@zjhu.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0218348X22401107Cited by:97 (Source: Crossref)

This article is part of the issue:

Special Issue on Dynamical Systems and Their Applications to Engineering, Economy and Health Sciences
Guest Editors: Juan L. G. Guirao, Elbaz I. Abouelmagd and Bruce A. Wade

Abstract

Discrete fractional calculus ( $𝒟 ℱ 𝒞$ ) is significant for neural networks, complex dynamic systems and frequency response analysis approaches. In contrast with the continuous-time frameworks, fewer outcomes are accessible for discrete fractional operators. This study investigates some major consequences of two sorts of inequalities by considering discrete Atangana–Baleanu $(𝒜 ℬ)$ -fractional operator having $ℏ$ -discrete generalized Mittag-Leffler kernels in the sense of Riemann type ( $𝒜 ℬ ℛ$ ). Certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete $𝒜 ℬ$ -fractional operators having $ℏ$ -discrete generalized Mittag-Leffler kernels are given. Moreover, several other generalizations can be generated for nabla $ℏ$ -fractional sums. The proposing discretization is a novel form of the existing operators that can be provoked by some intriguing features of chaotic systems to design efficient dynamics description in short time domains. Furthermore, by combining two mechanisms, numerous new special cases are introduced.

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References

1. G.-C. Wu, Z. G. Deng, D. Baleanu and D. Q. Zeng , New variable-order fractional chaotic systems for fast image encryption, Chaos 29 (2019) 083103. Crossref, Web of Science, Google Scholar
2. F. M. Atici and S. Sengul , Modeling with fractional difference equations, J. Math. Anal. Appl. 369 (2010) 1–9. Crossref, Web of Science, Google Scholar
3. M. Nazeer, F. Hussain, M. I. Khan, Asad-ur-Rehman, E. R. El-Zahar, Y.-M. Chu and M. Y. Malik , Theoretical study of MHD electro-osmotically flow of third-grade fluid in micro channel, Appl. Math. Comput. 420 (2022) 126868. Crossref, Web of Science, Google Scholar
4. N. H. Can, H. Jafari and M. N. Ncub , Fractional calculus in data fitting, Alexandria Eng. J. 59(5) (2020) 3269–3274. Crossref, Web of Science, Google Scholar
5. S. Sadeghi, H. Jafari and S. Nematia , Operational matrix for Atangana–Baleanu derivative based on Genocchi polynomials for solving FDEs, Chaos Solitons Fractals 135 (2020) 109736. Crossref, Web of Science, Google Scholar
6. R. M. Ganji, H. Jafari and D. Baleanu , A new approach for solving multi variable orders differential equations with Mittag-Leffler kernel, Chaos Solitons Fractals 130 (2020) 109405. Crossref, Web of Science, Google Scholar
7. Y.-M. Chu, B. M. Shankaralingappa, B. J. Gireesha, F. Alzahrani, M. I. Khan and S. U. Khan , Combined impact of Cattaneo–Christov double diffusion and radiative heat flux on bio-convective flow of Maxwell liquid configured by a stretched nano-material surface, Appl. Math. Comput. 419 (2022) 126883. Crossref, Web of Science, Google Scholar
8. H. F. Ismael, H. Bulut and H. M. Baskonus , W-shaped surfaces to the nematic liquid crystals with three nonlinearity laws, Soft Comput. 25 (2021) 4513–4524, https://doi.org/10.1007/s00500-020-05459-6. Crossref, Web of Science, Google Scholar
9. H. Rezazadeh, M. Odabasi, K. U. Tariq, R. Abazari and H. M. Baskonus , On the conformable nonlinear Schrödinger equation with second order spatiotemporal and group velocity dispersion coefficients, Chin. J. Phys. 72 (2021) 403–414. Crossref, Web of Science, Google Scholar
10. K. Hosseini, K. M. Mirzazadeh and H. M. Baskonus , 1-Soliton solutions of the $(2 + 1)$ -dimensional Heisenberg ferromagnetic spin chain model with the beta time derivative, Optic. Quantum Elect. 53(2) (2021) 1–10. Web of Science, Google Scholar
11. A. Kumar, E. Ilhan, A. Ciancio, G. Yel and H. M. Baskonus , Extractions of some new travelling wave solutions to the conformable Date–Jimbo–Kashiwara–Miwa equation, AIMS Math. 6(5) (2021) 4238–4264. Crossref, Web of Science, Google Scholar
12. Y.-M. Chu, U. Nazir, M. Sohali, M. M. Selim and J.-R. Lee , Enhancement in thermal energy and solute particles using hybrid nanoparticles by engaging activation energy and chemical reaction over a parabolic surface via finite element approach, Fractal Fract. 5 (2021) 119. Crossref, Web of Science, Google Scholar
13. A. Atangana and D. Baleanu , New fractional derivative with non-local and non-singular kernel, Therm. Sci. 20 (2016) 757–763. Crossref, Web of Science, Google Scholar
14. M. Caputo and M. Fabrizio , A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2015) 73–85. Google Scholar
15. S. Rashid, S. Sultana, Y. Karaca, A. Khalid and Y.-M. Chu , Some further extensions considering discrete proportional fractional operators, Fractals 30(1) (2022) 2240026. Link, Web of Science, Google Scholar
16. X. F. Wang and G. Chen , Synchronization in small-world dynamical networks, Int. J. Bifurc. Chaos 12 (2002) 187–192. Link, Web of Science, Google Scholar
17. H. Günerhan and E. Çelik , Analytical and approximate solutions of fractional partial differential-algebraic equations, Appl. Math. Nonlinear Sci. 5(1) (2020) 109–120. Crossref, Google Scholar
18. F. Evirgen, S. Uçar and N. Ozdemir , System analysis of HIV infection model with CD4+ T under non-singular kernel derivative, Appl. Math. Nonlinear Sci. 5(1) (2020) 139–146. Crossref, Google Scholar
19. A. Yokus and S. Gülbahar , Numerical solutions with linearization techniques of the fractional Harry Dym equation, Appl. Math. Nonlinear Sci. 4(1) (2019) 35–42. Crossref, Google Scholar
20. E. İlhan and O. Kymaz , A generalization of truncated M-fractional derivative and applications to fractional differential equations, Appl. Math. Nonlinear Sci. 5(1) (2020) 171–188. Crossref, Google Scholar
21. K. S. Al-Ghafri and H. Rezazadeh , Solitons and other solutions of $(2 + 1)$ -dimensional space-time fractional modified KdV–Zakharov–Kuznetsov equation, Appl. Math. Nonlinear Sci. 4(2) (2019) 289–304. Crossref, Google Scholar
22. M. A. Balci , Fractional interaction of financial agents in a stock market network, Appl. Math. Nonlinear Sci. 5(1) (2020) 317–336. Crossref, Google Scholar
23. M. S. M. Selvi and L. Rajendran , Application of modified wavelet and homotopy perturbation methods to nonlinear oscillation problems, Appl. Math. Nonlinear Sci. 4(2) (2019) 351–364. Crossref, Google Scholar
24. S. Sengul, Discrete fractional calculus and its applications to tumor growth, Master Thesis, Western Kentucky University (2010). Google Scholar
25. F. Dusunceli , New exact solutions for generalized $(3 + 1)$ -shallow water-like (SWL) equation, Appl. Math. Nonlinear Sci. 4(2) (2019) 365–370. Crossref, Google Scholar
26. R. A. Mundewadi and S. Kumbinarasaiah , Numerical solution of Abel’s integral equations using Hermite Wavelet, Appl. Math. Nonlinear Sci. 4(2) (2019) 395–406, https://doi.org/10.2478/AMNS.2019.2.00037. Crossref, Google Scholar
27. T. A. Sulaiman and H. Bulut , The new extended rational SGEEM for construction of optical solitons to the (2+1)-dimensional Kundu–Mukherjee–Naskar model, Appl. Math. Nonlinear Sci. 4(2) (2019) 513–52H12. Crossref, Google Scholar
28. A. Jajarmi and D. Baleanu , Suboptimal control of fractional-order dynamic systems with delay argument, J. Vib. Control 24 (2018) 2430–2446. Crossref, Web of Science, Google Scholar
29. N. R. O. Bastos, R. A. C. Ferreira and D. F. M. Torres , Necessary optimality conditions for fractional difference problems of the calculus of variations, Discrete Contin. Dyn. Syst. 29(2) (2011) 417–437. Crossref, Web of Science, Google Scholar
30. C. Goodrich and A. C. Peterson , Discrete Fractional Calculus (Springer, 2015). Crossref, Google Scholar
31. M. T. Holm , The Laplace transform in discrete fractional calculus, Comput. Math. Appl. 62 (2011) 1591–1601. Crossref, Web of Science, Google Scholar
32. Y. M. Chu, S. Rashid, F. Jarad, M. A. Noor and H. Kalsoom , More new results on integral inequalities for generalized K-fractional conformable integral operators, Discrete Contin. Dyn. Syst. Ser. S 14(7) (2021) 2119–2135. Web of Science, Google Scholar
33. S. S. Zhou, S. Rashid, A. Rauf, F. Jarad, Y. S. Hamed and K. M. Abualnaja , Efficient computations for weighted generalized proportional fractional operators with respect to a monotone function, AIMS Math. 6 (2021) 8001–8029. Crossref, Web of Science, Google Scholar
34. S. Rashid, S. Sultana, F. Jarad, H. Jafari and Y. S. Hamed , More efficient estimates via h-discrete fractional calculus theory and applications, Chaos Solitons Fractals 147 (2021) 110981. Crossref, Web of Science, Google Scholar
35. H. G. Jile, S. Rashid, F. B. Farooq and S. Sultana , Some inequalities for a new class of convex functions with applications via local fractional integral, J. Funct. Spaces 2021 (2021) 6663971. Web of Science, Google Scholar
36. S. Rashid, S. Parveen, H. Ahmad and Y. M. Chu , New quantum integral inequalities for some new classes of generalized $ψ$ -convex functions and their scope in physical systems, Open Phys. 19 (2021) 35–50. Crossref, Web of Science, Google Scholar
37. A. A. El-Deeb and S. Rashid , On some new double dynamic inequalities associated with Leibniz integral rule on time scales, Adv. Differ. Equs. 2021 (2021) 125. Crossref, Web of Science, Google Scholar
38. S. S. Zhou, S. Rashid, S. Parveen, A. O. Akdemir and Z. Hammouch , New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operator, AIMS Math. 6 (2021) 4507–4525. Crossref, Web of Science, Google Scholar
39. F. M. Atici and P. W. Eloe , Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ. Spec. 3 (2009) 1–12. Crossref, Google Scholar
40. B. Zheng , Some new discrete fractional inequalities and their applications in fractional difference equations, J. Math. Inequal. 9(3) (2019) 823–839. Crossref, Web of Science, Google Scholar
41. M. Bohner and R. A. C. Ferreira , Some discrete fractional inequalities of Chebyshev type, African Diaspora. J. Math. 11(2) (2011) 132–137. Google Scholar
42. G. A. Anastassiou , About discrete fractional calculus with inequalities, Intelligent Mathematics, Computational Analysis (Springer, 2011), pp. 575–585. Crossref, Google Scholar
43. S. B. Chen, S. Rashid, M. A. Noor, R. Ashraf and Y. M. Chu , A new approach on fractional calculus and probability density function, AIMS Math. 5 (2020) 7041–7054. Crossref, Web of Science, Google Scholar
44. M. Al-Qurashi, S. Rashid, S. Sultana, H. Ahmad and K. A. Gepreel , New formulation for discrete dynamical type inequalities via $h$ -discrete fractional operator pertaining to nonsingular kernel, Math. Bioscis. Eng. 18 (2021) 1794–1812, https://doi.org/10.3934/mbe.2021093. Crossref, Web of Science, Google Scholar
45. Y. M. Chu, S. Rashid and J. Singh , A novel comprehensive analysis on generalized harmonically $Ψ$ -convex with respect to Raina’s function on fractal set with applications, Math. Methods Appl. Sci. (2021), https://doi.org/10.1002/mma.7346. Crossref, Web of Science, Google Scholar
46. S. Rashid, F. Jarad and Z. Hammouch , Some new bounds analogous to generalized proportional fractional integral operator with respect to another function, Discrete Contin. Dyn. Syst.-Ser. S 14(10) (2021) 3703–3718. Crossref, Web of Science, Google Scholar
47. S. Rashid, S. I. Butt, S. Kanwal, H. Ahmad and M. K. Wang , Quantum integral inequalities with respect to Raina’s function via coordinated generalized $ψ$ -convex functions with applications, J. Fun. Spaces 2021 (2021) 6631474. Web of Science, Google Scholar
48. S. Rashid, Y. M. Chu, J. Singh and D. Kumar , A unifying computational framework for novel estimates involving discrete fractional calculus approaches, Alexandria Eng. J. 60 (2021) 2677–2685. Crossref, Web of Science, Google Scholar
49. M. Al-Qurashi, S. Rashid, Y. Karaca, Z. Hammouch, D. Baleanu and Y. M. Chu , Achieving more precise bounds based on double and triple integral as proposed by generalized proportional fractional operators in the Hilfer sense, Fractals 29(5) (2021) 2140027. Link, Web of Science, Google Scholar
50. M. K. Wang, S. Rashid, Y. Karaca, Z. Hammouch, D. Baleanu and Y. M. Chu , New multi-functional approach for kth-order differentiability governed by fractional calculus via approximately generalized $(ψ, ℏ)$ -convex functions in Hilbert space, Fractals 29(5) (2021) 2140019. Link, Web of Science, Google Scholar
51. M. Al Qurashi, S. Rashid, A. Khalid and Y. Karaca, Y. M. Chu , New computations of Ostrowski type inequality pertaining to fractal style with applications, Fractals 29(5) (2021) 2140026. Link, Web of Science, Google Scholar
52. E. Set, M. E. Özdemir and S. Dragomir , On the Hermite–Hadamard inequality and other integral inequalities involving two functions, J. Inequal. Appl. 2010 (2010) 148102. Crossref, Web of Science, Google Scholar
53. Y.-M. Chu, M. Adil Khan, T. Ali and S. S. Dragomir , Inequalities for $𝜗$ -fractional differentiable functions. J. Inequal. Appl. 2017 (2017) 93. Crossref, Web of Science, Google Scholar
54. H. Khan, C. Tunç, A. Alkhazan, B. Ameen and A. Khan , A generalization of Minkowski’s inequality by Hahn integral operator, J. Taibah Univ. Sci. 12(5) (2018) 506–513. Crossref, Web of Science, Google Scholar
55. T. Abdeljawad , Different type kernel h-fractional differences and their fractional $ℏ$ -sums, Chaos Solitons Fractals 116 (2018) 146–156. Crossref, Web of Science, Google Scholar
56. T. Abdeljawad , On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc. 2013 (2013) 1–12. Web of Science, Google Scholar
57. I. Suwan, T. Abdeljawad and F. Jarad , Monotonicity analysis for nabla h-discrete fractional Atangana–Baleanu differences, Chaos Soliton Fractals 117 (2018) 50–59. Crossref, Web of Science, Google Scholar
58. I. Suwan, S. Owies and T. Abdeljawad , Monotonicity results for h-discrete fractional operators and application, Adv Differ. Equ 2018 (2018) 207. Crossref, Web of Science, Google Scholar
59. T. Abdeljawad, F. Jarad and J. Alzabut , Fractional proportional differences with memory, Eur. Phys. J. Special Top. 226 (2017) 3333–3354. Crossref, Web of Science, Google Scholar
60. T. Abdeljawad and D. Baleanu , On fractional derivatives with generalized Mittag-Leffler kernels, Adv. Differ. Eqs. 2018 (2018) 468. Crossref, Web of Science, Google Scholar
61. H. Khan, T. Abdeljawad, C. Tunç, A. Alkhazzan and A. Khan , Minkowski’s inequality for the AB-fractional integral operator, Adv. Differ. Eqs. 2019 (2019) 96. Google Scholar

Vol. 30, No. 02

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History

Received 18 December 2020

Accepted 13 July 2021

Published: February 7, 2022

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This is an Open Access article in the “Special Issue on Dynamical Systems and Their Applications to Engineering, Economy and Health Sciences”, edited by Juan L.G. Guirao (Technical University of Cartagena, Spain), Elbaz I. Abouelmagd (National Research Institute of Astronomy and Geophysics (NRIAG), Cairo, Egypt) & Bruce A. Wade (University of Louisiana at Lafayette, USA) published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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SOME RECENT DEVELOPMENTS ON DYNAMICAL ℏ<math altimg="eq-00001.gif" display="inline"><mi>ℏ</mi></math>-DISCRETE FRACTIONAL TYPE INEQUALITIES IN THE FRAME OF NONSINGULAR AND NONLOCAL KERNELS

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SOME RECENT DEVELOPMENTS ON DYNAMICAL $ℏ$ -DISCRETE FRACTIONAL TYPE INEQUALITIES IN THE FRAME OF NONSINGULAR AND NONLOCAL KERNELS