Numerical computation of exponential functions in frame of Nabla fractional calculus

Document Type : Research Paper

Authors

1 Department of Mathematics, Birla Institute of Technology and Science Pilani, Hyderabad - 500078, Telangana, India.

2 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 12435, Saudi Arabia.

3 Department of Industrial Engineering, OST_IM Technical University, Ankara 06374, Turkiye.

Abstract

Exponential functions play an essential role in describing the qualitative properties of solutions of nabla fractional difference equations. In this article, we illustrate their asymptotic behavior. We know that these functions involve infinite series of ratios of gamma functions, and it is challenging to compute them. For this purpose, we propose a novel matrix technique to compute the addressed functions numerically. The results are supported by illustrative examples. The proposed method can be extended to obtain numerical solutions for non-homogeneous nabla fractional difference equations.

Keywords

References

  • [1] T. Abdeljawad, On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc.,2013 (2013), 1-12.
  • [2] T. Abdeljawad and J. Alzabut, On Riemann–Liouville fractional q-difference equations and their application to retarded logistic type model, Math. Methods Appl. Sci., 41(18) (2018), 8953–8962.
  • [3] T. Abdeljawad and F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), 1-13 .
  • [4] N. Acar and F. M. Atıcı, Exponential functions of discrete fractional calculus, Appl. Anal. Discrete Math., 7(2) (2013), 343–353.
  • [5] K. Ahrendt, L. Castle, M. Holm, and K. Yochman, Laplace transforms for the nabla-difference operator and a fractional variation of parameters formula, Commun. Appl. Anal., 16(3) (2012), 317–347.
  • [6] A. Alkhazzan, P. Jiang, D. Baleanu, H. Khan, and A. Khan, Stability and existence results for a class of nonlinear fractional differential equations with singularity, Math. Methods Appl. Sci., 41(18) (2018), 9321–9334.
  • [7] J. Alzabut, S. Tyagi, and S. Abbas, Discrete fractional-order BAM neural networks with leakage delay: existence and stability results, Asian J. Control, 22(1) (2020), 143–155.
  • [8] J. Alzabut and T. Abdeljawad, A generalized discrete fractional Gronwall inequality and its application on the uniqueness of solutions for nonlinear delay fractional difference system, Appl. Anal. Discrete Math., 12(1) (2018), 36–48.
  • [9] G. A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Modelling, 51(5-6) (2010), 562–571.
  • [10] F. M. Atıcı and P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., Special Edition I, 1(3) (2009), 1-12.
  • [11] F. M. Atıcı and P. W. Eloe, Linear systems of fractional nabla difference equations, Rocky Mountain J. Math., 41(2) (2011), 353–370.
  • [12] M. Bohner and A. Peterson, Dynamic equations on time scales. An introduction with applications, Birkh¨auser, Boston, MA, 2001.
  • [13] J. Cˇerm´ak, T. Kisela, and L. Nechv´atal, Stability and asymptotic properties of a linear fractional difference equation, Adv. Difference Equ., 122 (2012), 1-14.
  • [14] S. Elaydi, An introduction to difference equations, Third edition, Springer, New York, 2005.
  • [15] P. Eloe and J. Jonnalagadda, Mittag–Leffler stability of systems of fractional nabla difference equations, Bull. Korean Math. Soc., 56(4) (2019), 977–992.
  • [16] S. R. Grace, H. Adıguzel, J. Alzabut, and J. M. Jonnalagadda, Asymptotic behavior of positive solutions for three types of fractional difference equations with forcing term, Vietnam J. Math., 49(4) (2021), 1151–1164.
  • [17] S. R. Grace, J. Alzabut, S. Punitha, V. Muthulakshmi, and H. Adıguzel, On the nonoscillatory behavior of solutions of three classes of fractional difference equations, Opuscula Math., 40(5) (2020), 549–568.
  • [18] C. Goodrich and A. C. Peterson, Discrete fractional calculus, Springer, Cham, 2015.
  • [19] H. L. Gray and N. F. Zhang, On a new definition of the fractional difference, Math. Comp., 50(182) (1988), 513–529.
  • [20] B. Jia, L. Erbe, and A. Peterson, Comparison theorems and asymptotic behavior of solutions of discrete fractional equations, Electron. J. Qual. Theory Differ. Equ., 89 (2015), 1-18.
  • [21] A. Khan, H. M. Alshehri, T. Abdeljawad, Q. M. Al-Mdallal, and H. Khan, Stability analysis of fractional nabla difference COVID-19 model, Results in Physics, 22 (2021), 1-8.
  • [22] A. Khan, H. M. Alshehri, J. F. Gomez-Aguilar, Z. A. Khan, and G. A. Fernandez-Anaya, Predator-prey model involving variable-order fractional differential equations with Mittag–Leffler kernel, Adv. Difference Equ., 189 (2021), 1-18.
  • [23] A. Khan, Z. A. Khan, T. Abdeljawad, and H. Khan, Analytical analysis of fractional-order sequential hybrid system with numerical application, Adv. Contin. Discrete Models, 12 (2022), 1-19.
  • [24] Z. A. Khan, A. Khan, T. Abdeljawad, and H. Khan, Computational analysis of fractional order imperfect testing infection disease model, Fractals, 30(5) (2022), 1-17.
  • [25] H. Khan, C. Tunc, and A. Khan, Green function’s properties and existence theorems for nonlinear singular-delay- fractional differential equations, Discrete Contin. Dyn. Syst. Ser. S, 13(9) (2020), 2475–2487.
  • [26] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B. V., Amsterdam, 2006.
  • [27] K. S. Miller and B. Ross, Fractional difference calculus. Univalent functions, fractional calculus, and their appli- cations, K¯oriyama, 1988, 139–152, Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989.
  • [28] A. Nagai, An integrable mapping with fractional difference, J. Phys. Soc. Japan, 72(9) (2003), 2181–2183.
  • [29] I. Podlubny, Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, San Diego, CA, 1999.
  • [30] I. Podlubny, Matrix approach to discrete fractional calculus, Fract. Calc. Appl. Anal., 3(4) (2000), 359–386.
  • [31] A. G. M. Selvam, D. Baleanu, J. Alzabut, D. Vignesh, and S. Abbas, On Hyers–Ulam Mittag–Leffler stability of discrete fractional Duffing equation with application on inverted pendulum, Adv. Difference Equ., 456 (2020), 1-16.
  • [32] A. G. M. Selvam, J. Alzabut, R. Dhineshbabu, S. Rashid and M. Rehman, Discrete fractional order two-point boundary value problem with some relevant physical applications, J. Inequal. Appl., (2020), 1-19.
  • [33] K. Shah, Z. A. Khan, A. Ali, R. Amin, H. Khan, and A. Khan, Haar wavelet collocation approach for the solution of fractional order COVID-19 model using Caputo derivative, Alexandria Engineering Journal, 59 (2020), 3221– 3231.
  • [34] I. Ullah, S. Ahmad, Q. Al-Mdallal, Z. A. Khan, H. Khan, and A. Khan, Stability analysis of a dynamical model of tuberculosis with incomplete treatment, Adv. Difference Equ., (2020), 1-14.
  • [35] G. Wu, D. Baleanu, W. Luo, and H. Wei, Lyapunov functions for Riemann–Liouville-like fractional difference equations, Appl. Math. Comput., 314 (2017), 228–236.
Computational Methods for Differential Equations
Volume 11, Issue 2
April 2023
Pages 291-302

Files

History

  • Receive Date: 29 March 2022
  • Revise Date: 11 August 2022
  • Accept Date: 15 September 2022

Share

How to cite

Statistics