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HADAMARD FRACTIONAL CALCULUS ON TIME SCALES

School of Mathematical Sciences, Bohai University, Jinzhou 121013, P. R. China

Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, P. R. China

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GUO-CHENG WU

https://orcid.org/0000-0002-1946-6770

Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, P. R. China

E-mail Address: wuguocheng@gmail.com

Corresponding author.

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, and

JIA-LI WEI

School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, Jiangsu Province, P. R. China

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https://doi.org/10.1142/S0218348X22501456Cited by:28 (Source: Crossref)

Abstract

This study defines a Hadamard fractional sum by use of the time-scale theory. Then a $h$ $h$ -fractional difference is given and fundamental theorems are proved. Initial value problems of fractional difference equations are presented and their equivalent fractional sum equations are provided. The discrete Mittag-Leffler function solutions of linear fractional difference equations are obtained. It can be concluded that the new discrete fractional calculus of Hadamard type is well defined.

Keywords:

References

1. A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier Amsterdam, 2006). Google Scholar
2. V. Kiryakova, Generalized Fractional Calculus and Applications (Longman Science and Technical Harlow, UK, 1994). Google Scholar
3. L. Ma and C. P. Li, On Hadamard fractional calculus, Fractals 25 (2017) 1750033. Link, Web of Science, Google Scholar
4. M. Gohar, C. P. Li and C. T. Yin, On Caputo–Hadamard fractional differential equations, Int. J. Comput. Math. 97 (2020) 1459–1483. Crossref, Web of Science, Google Scholar
5. C. P. Li, Z. Q. Li and Z. Wang, Mathematical analysis and the local discontinuous Galerkin method for Caputo–Hadamard fractional partial differential equation, J. Sci. Comput. 85(2) (2020) 1–27. Crossref, Web of Science, Google Scholar
6. E. Y. Fan, C. P. Li and Z. Q. Li, Numerical approaches to Caputo–Hadamard fractional derivatives with applications to long-term integration of fractional differential systems, Commun. Nonlinear Sci. Numer. Simul. 106 (2022) 106096. Crossref, Web of Science, Google Scholar
7. C. P. Li and Z. Q. Li, The Blow–up and global existence of solution to Caputo–Hadamard fractional partial differential equation with fractional Laplacian, J. Nonlinear Sci. 31 (2021) 80. Crossref, Web of Science, Google Scholar
8. C. P. Li and Z. Q. Li, Stability and logarithmic decay of the solution to Hadamard-type fractional differential equation, J. Nonlinear Sci. 31(2) (2021) 1–60. Crossref, Web of Science, Google Scholar
9. G. T. Wang, K. Pei and Y. Q. Chen, Stability analysis of nonlinear Hadamard fractional differential system, J. Franklin Inst. 356 (2019) 6538–6546. Crossref, Web of Science, Google Scholar
10. Y. Liu, Y. G. Zhu and Z. Q. Lu, On Caputo–Hadamard uncertain fractional differential equations, Chaos Soliton Fractals 146 (2021) 110894. Crossref, Web of Science, Google Scholar
11. T. Abdeljawad, On Riemann and Caputo fractional differences, Comput. Math. Appl. 62 (2011) 1602–1611. Crossref, Web of Science, Google Scholar
12. T. Abdeljawad, Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ. 2013 (2013) 36. Crossref, Web of Science, Google Scholar
13. C. Goodrich and A. C. Peterson, Discrete Fractional Calculus (Springer, 2015). Crossref, Google Scholar
14. R. Abu–Saris and Q. Al–Mdallal, On the asymptotic stability of linear system of fractional–order difference equations, Fract. Calc. Appl. Anal. 16 (2013) 613–629. Crossref, Web of Science, Google Scholar
15. J. Cermak, I. Gyori and L. Nechvatal, On explicit stability conditions for a linear fractional difference system, Fract. Calc. Appl. Anal. 18 (2105) 651–672. Crossref, Web of Science, Google Scholar
16. F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009) 981–989. Crossref, Web of Science, Google Scholar
17. M. T. Holm, The Laplace transform in discrete fractional calculus, Comput. Math. Appl. 62 (2011) 1591–1601. Crossref, Web of Science, Google Scholar
18. G. C. Wu and D. Baleanu, Discrete fractional logistic map and its chaos, Nonlinear Dyn. 75 (2014) 283–287. Crossref, Web of Science, Google Scholar
19. 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
20. F. M. Atıcı, M. Atıcı, N. Nguyen, T. Zhoroev and G. Koch, A study on discrete and discrete fractional pharmacokinetics-pharmacodynamics models for tumor growth and anti–cancer effects, Comput. Math. Biophys. 7 (2019) 10–24. Crossref, Google Scholar
21. H. Kong, G. Yang and C. Luo, Modeling aftershocks by fractional calculus: Exact discretization versus approxinmate discretization, Fractals 29 (2021) 2140038. Link, Web of Science, Google Scholar
22. K. S. Miller and B. Ross, Fractional difference calculus, in Proceedings International Symposium Univalent Functions, Fractional Calculus and Their Applications (Nihon University, Koriyama, Japan, 1989), pp. 139–152. Google Scholar
23. T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ. 2016 (2016) 232. Crossref, Web of Science, Google Scholar
24. T. Abdeljawad, S. Banerjee and G. C. Wu, Discrete tempered fractional calculus for new chaotic systems with short memory and image encryption, Optik 203 (2020) 163698. Crossref, Web of Science, Google Scholar
25. H. Fu, L. L. Huang, T. Abdeljawad and C. Luo, Tempered fractional calculus on time scale for discrete–time systems, Fractals 29 (2021) 2140033. Link, Web of Science, Google Scholar
26. R. A. C. Ferreira, Discrete weighted fractional calculus and applications, Nonlinear Dyn. 104 (2021) 2531–2536. Crossref, Web of Science, Google Scholar
27. D. R. Anderson and M. Bohner, A multivalued logarithm on time scales, Appl. Math. Comput. 397 (2021) 125954. Web of Science, Google Scholar
28. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications (Birkhäuser, Boston, 2001). Crossref, Google Scholar
29. S. D. Zeng, D. Baleanu, Y. R. Bai and G. C. Wu, Fractional differential equations of Caputo–Katugampola type and numerical solutions, Appl. Math. Comput. 315 (2017) 549–554. Crossref, Web of Science, Google Scholar
30. H. Kong, G. C. Wu, H. Fu and K. T. Wu, Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory, Int. J. Nonlinear Sci. Numer. Simul. (2022), https://doi.org/10.1515/ijnsns-2021-0232. Web of Science, Google Scholar
31. G. C. Wu, H. Kong, M. Luo, H. Fu and L. L. Huang, Unified predictor–corrector method for fractional differential equations with general kernel functions, Fract. Calc. Appl. Anal. 25 (2022) 648–667. Crossref, Web of Science, Google Scholar
32. D. Mozyrska and E. Girejko, Overview of fractional $h$ $h$ -difference operators, in Advances in Harmonic Analysis and Operator Theory (Springer, Basel, 2013), pp. 253–268. Crossref, Google Scholar
33. M. T. Holm, The Theory of Discrete Fractional Calculus: Development and Application, Dissertations & Thesis, Gradworks (2011). Google Scholar
34. H. Fu, G. C. Wu, G. Yang and L. L. Huang, Continuous-time random walk to a general fractional Fokker–Planck equation on fractal media, Eur. Phys. J. Spec. Top. 230 (2021) 3927–3933. Crossref, Web of Science, Google Scholar
35. Q. Fan, G. C. Wu and H. Fu, A note on function space and boundedness of a general fractional integral in continuous time random walk, J. Nonlinear Math. Phys. 29 (2022) 95–102. Crossref, Web of Science, Google Scholar