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The Mathematical Dynamics of the Caputo Fractional Order Social Media Addiction Design

Yıl 2025, , 1 - 10, 27.03.2025

Bahar Karaman , Emrah Karaman

https://doi.org/10.33434/cams.1559808
Cited By: 1

Öz

The paper presents the mathematical dynamics and numerical simulations for a fractional-order social media addiction (FSMA) model. This addiction structure is replaced by involving the Caputo fractional (CF) derivative to get the FSMA model. In this study, our main goal is to understand how the fractional derivative impresses the dynamics of the model. Thus, the theoretical properties are first examined. Afterward, the stability properties of the mentioned model are discussed. Besides, the fractional backward differentiation formula (FBDF) displays numerical simulations of the model. Observing both theoretical and numerical results, the two equilibrium points' stability is not impacted by the order of fractional derivatives. However, each solution converges more quickly to its stationary state for higher values of the fractional-order derivative. Finally, we would like to say that the acquired numerical results are compatible with our theoretical outcomes.

Anahtar Kelimeler

Fractional backward differentiation formula(FBDF) Social media addiction Stability analysis

Kaynakça

  • [1] M. Drahosova, P. Balco, The analysis of advantages and disadvantages of use of social media in European Union, Proc. Comput. Sci., 109 (2017), 1005-1009.
  • [2] R. Faizi, A. E. Afia, R. Chiheb, Exploring the potential benefits of using social media in education, IJEP, 3(4) (2013), 50-53.
  • [3] A. Simsek, K. Elciyar, T. Kizilhan, A comparative study on social media addiction of high school and university students, Contemporary Edu. Tech., 10(2) (2019), 106-119.
  • [4] S. Zivnuska, J. R. Carlson, D. S. Carlson, R. B. Harris, K. J. Harris, Social media addiction and social media reactions: The implications for job performance, J. Soc. Psychol., 159(6) (2019), 746-760.
  • [5] C. S. Andreassen, T. Torsheim, G. S. Burnborg, S. Pallesen, Development of a Facebook addiction scale, Psychol. Rep., 110(2) (2012), 501–517.
  • [6] J. R. Carlson, S. Zivnuska, D. S. Carlson, R. Harris, K. J. Harris, Social media use in the workplace: A study of dual effects, J. Organ. End User Comput., 28(1) (2016), 15–28.
  • [7] H. T. Alemneh, N. Y. Alemu, Mathematical modeling with optimal control analysis of social media addiction, Infect. Dis. Model., 6 (2021), 405-419.
  • [8] J. Kongson, W. Sudsutad, C. Thaiprayoon, J. Alzabut, C. Tearnbucha, On analysis of a nonlinear fractional system for social media addiction involving Atangana-Balenau-Caputo derivative, Adv. Differ. Equ., 2021 (2021), 356-385.
  • [9] Shutaywi, Meshal, et al. Modeling and analysis of the addiction of social media through fractional calculus, Front. Appl. Math. Stat., 9 (2023), 1210404.
  • [10] I. K. Adu, A. L. Mojeeb, C. Yang, Mathematical model of drinking epidemic, J. Adv. Math. Comput., 22(5) (2017), 1-10.
  • [11] Y. Guo, T. Li, Optimal control and stability analysis of an online game addiction model with two stages, Math. Methods Appl. Sci., 43(7) (2020), 4391-4408.
  • [12] S. H. Ma, H. F. Huo, X. Y. Meng, Modelling alcoholism as a contagious disease: A mathematical model with awareness programs and time delay, Discrete Dyn Nat. Soc., 2015 (2015) Article ID 260195.
  • [13] S. A. Samad, M. T. Islam,S. T. H. Tomal, M., Biswas, Mathematical assessment of the dynamical model of smoking tobacco epidemic in Bangladesh, Int. J. Sci. Manag. Stud., 3(2) (2020), 36-48.
  • [14] E. Demirci, A fractional order model of hepatitis B transmission under the effect of vaccination, Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat., 71(2) (2022), 566-580. https://doi.org/10.31801/cfsuasmas.1103630
  • [15] B. Karaman, The global stability investigation of the mathematical design of a fractional-order HBV infection, J. Appl. Math. Comput., 68 (2022), 4759–4775. https://doi.org/10.1007/s12190-022-01721-2.
  • [16] S. S. Askar, G. Dipankar, P. K. Santra, A. A. Elsadany, G. S. Mahapatra, A fractional order SITR mathematical model for forecasting of transmission of COVID-19 of India with lockdown effect, Results Phys., 24 (2021), 104067.
  • [17] I. Owusu-Mensah, L. Akinyemi, B. Oduro, O. S. Iyiola, A fractional order approach to modeling and simulations of the novel COVID-19, Adv. Differ. Equ., 2020 (2020), 683.
  • [18] V. F. Morales-Delgado, J. F. Gomez-Aguilar, M. A. Taneco-Hernandez, Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense, Int. J. Electron. Commun., 85 (2018), 61-81.
  • [19] R. Garrappa, Numerical solution of fractional differential equations: A survey and software Tutorial, Mathematics 6(2) (2018), 16.
  • [20] R. Garrappa, Trapezoidal methods for fractional differential equations: Theoretical and computational aspects, Math. Comput. Simul., 110 (2015), 96-112.
  • [21] H. L. Li, Z. Long, H. Cheng, J. Yao-Lin, T. Zhidong, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, J. Appl. Math. Comput., 54 (2017), 435-449.
  • [22] Z. M. Odibat, N. T. Shawaghef, Generalized Taylor’s formula, Appl. Math. Comput., 186 (2007), 286-293.
  • [23] G. M. Mittag-Leffler, Sur l’integrable de Laplace-Abel, Comptes Rendus de l’Academie des Sciences Series II, 136 (1903), 937–939.
  • [24] K. Diethelm, Tha Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010.
  • [25] I. Podlubny, Fractional Differential Equations, Academie Press, New York, 1999.
  • [26] K. B. Oldham, J. Spanier, The Fractional Calculus, New York London, Academic Press, 1974.
  • [27] S. K. Choi, B. Kang, N. Koo, Stability for Caputo fractional differential systems, Abstr. Appl. Anal., 2014 (2014), 1-6.
  • [28] Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810–1821.
  • [29] C. Castillo Chavez, Z. Feng, W. Huang, On the Computation of R0 and Its Role on Global Stability, In Mathematical Approaches for Emerging and Remerging Infectious Diseases: An introduction. IMA, Springer, Berlin, 2002.
  • [30] C. V. Leon, Volterra Lyapunov functions for fractional-order epidemic systems, Commun. Nonlinear Sci. Numer. Simul., 24 (2015), 75-85.
  • [31] L. C. Cardoso, R. F. Camargo, F. L. P. Santos, J. P. C. Santos, Global stability analysis of a fractional differential system in hepatitis B, Chaos, Solitons and Fractals, 143 (2021), 110619.
  • [32] C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986), 704-719.
  • [33] C. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comput., 45(163) (1983), 87-102.
  • [34] L. Galeone, R. Garrappa, On multistep methods for differential equations of fractional order, Mediterr. J. Math., 3 (2006), 565-580.

Yıl 2025, , 1 - 10, 27.03.2025

Bahar Karaman , Emrah Karaman

https://doi.org/10.33434/cams.1559808
Cited By: 1

Öz

Kaynakça

  • [1] M. Drahosova, P. Balco, The analysis of advantages and disadvantages of use of social media in European Union, Proc. Comput. Sci., 109 (2017), 1005-1009.
  • [2] R. Faizi, A. E. Afia, R. Chiheb, Exploring the potential benefits of using social media in education, IJEP, 3(4) (2013), 50-53.
  • [3] A. Simsek, K. Elciyar, T. Kizilhan, A comparative study on social media addiction of high school and university students, Contemporary Edu. Tech., 10(2) (2019), 106-119.
  • [4] S. Zivnuska, J. R. Carlson, D. S. Carlson, R. B. Harris, K. J. Harris, Social media addiction and social media reactions: The implications for job performance, J. Soc. Psychol., 159(6) (2019), 746-760.
  • [5] C. S. Andreassen, T. Torsheim, G. S. Burnborg, S. Pallesen, Development of a Facebook addiction scale, Psychol. Rep., 110(2) (2012), 501–517.
  • [6] J. R. Carlson, S. Zivnuska, D. S. Carlson, R. Harris, K. J. Harris, Social media use in the workplace: A study of dual effects, J. Organ. End User Comput., 28(1) (2016), 15–28.
  • [7] H. T. Alemneh, N. Y. Alemu, Mathematical modeling with optimal control analysis of social media addiction, Infect. Dis. Model., 6 (2021), 405-419.
  • [8] J. Kongson, W. Sudsutad, C. Thaiprayoon, J. Alzabut, C. Tearnbucha, On analysis of a nonlinear fractional system for social media addiction involving Atangana-Balenau-Caputo derivative, Adv. Differ. Equ., 2021 (2021), 356-385.
  • [9] Shutaywi, Meshal, et al. Modeling and analysis of the addiction of social media through fractional calculus, Front. Appl. Math. Stat., 9 (2023), 1210404.
  • [10] I. K. Adu, A. L. Mojeeb, C. Yang, Mathematical model of drinking epidemic, J. Adv. Math. Comput., 22(5) (2017), 1-10.
  • [11] Y. Guo, T. Li, Optimal control and stability analysis of an online game addiction model with two stages, Math. Methods Appl. Sci., 43(7) (2020), 4391-4408.
  • [12] S. H. Ma, H. F. Huo, X. Y. Meng, Modelling alcoholism as a contagious disease: A mathematical model with awareness programs and time delay, Discrete Dyn Nat. Soc., 2015 (2015) Article ID 260195.
  • [13] S. A. Samad, M. T. Islam,S. T. H. Tomal, M., Biswas, Mathematical assessment of the dynamical model of smoking tobacco epidemic in Bangladesh, Int. J. Sci. Manag. Stud., 3(2) (2020), 36-48.
  • [14] E. Demirci, A fractional order model of hepatitis B transmission under the effect of vaccination, Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat., 71(2) (2022), 566-580. https://doi.org/10.31801/cfsuasmas.1103630
  • [15] B. Karaman, The global stability investigation of the mathematical design of a fractional-order HBV infection, J. Appl. Math. Comput., 68 (2022), 4759–4775. https://doi.org/10.1007/s12190-022-01721-2.
  • [16] S. S. Askar, G. Dipankar, P. K. Santra, A. A. Elsadany, G. S. Mahapatra, A fractional order SITR mathematical model for forecasting of transmission of COVID-19 of India with lockdown effect, Results Phys., 24 (2021), 104067.
  • [17] I. Owusu-Mensah, L. Akinyemi, B. Oduro, O. S. Iyiola, A fractional order approach to modeling and simulations of the novel COVID-19, Adv. Differ. Equ., 2020 (2020), 683.
  • [18] V. F. Morales-Delgado, J. F. Gomez-Aguilar, M. A. Taneco-Hernandez, Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense, Int. J. Electron. Commun., 85 (2018), 61-81.
  • [19] R. Garrappa, Numerical solution of fractional differential equations: A survey and software Tutorial, Mathematics 6(2) (2018), 16.
  • [20] R. Garrappa, Trapezoidal methods for fractional differential equations: Theoretical and computational aspects, Math. Comput. Simul., 110 (2015), 96-112.
  • [21] H. L. Li, Z. Long, H. Cheng, J. Yao-Lin, T. Zhidong, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, J. Appl. Math. Comput., 54 (2017), 435-449.
  • [22] Z. M. Odibat, N. T. Shawaghef, Generalized Taylor’s formula, Appl. Math. Comput., 186 (2007), 286-293.
  • [23] G. M. Mittag-Leffler, Sur l’integrable de Laplace-Abel, Comptes Rendus de l’Academie des Sciences Series II, 136 (1903), 937–939.
  • [24] K. Diethelm, Tha Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010.
  • [25] I. Podlubny, Fractional Differential Equations, Academie Press, New York, 1999.
  • [26] K. B. Oldham, J. Spanier, The Fractional Calculus, New York London, Academic Press, 1974.
  • [27] S. K. Choi, B. Kang, N. Koo, Stability for Caputo fractional differential systems, Abstr. Appl. Anal., 2014 (2014), 1-6.
  • [28] Y. Li, Y. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810–1821.
  • [29] C. Castillo Chavez, Z. Feng, W. Huang, On the Computation of R0 and Its Role on Global Stability, In Mathematical Approaches for Emerging and Remerging Infectious Diseases: An introduction. IMA, Springer, Berlin, 2002.
  • [30] C. V. Leon, Volterra Lyapunov functions for fractional-order epidemic systems, Commun. Nonlinear Sci. Numer. Simul., 24 (2015), 75-85.
  • [31] L. C. Cardoso, R. F. Camargo, F. L. P. Santos, J. P. C. Santos, Global stability analysis of a fractional differential system in hepatitis B, Chaos, Solitons and Fractals, 143 (2021), 110619.
  • [32] C. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986), 704-719.
  • [33] C. Lubich, Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comput., 45(163) (1983), 87-102.
  • [34] L. Galeone, R. Garrappa, On multistep methods for differential equations of fractional order, Mediterr. J. Math., 3 (2006), 565-580.
Toplam 34 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Adi Diferansiyel Denklemler, Fark Denklemleri ve Dinamik Sistemler, Uygulamalı Matematik (Diğer)
Bölüm Makaleler
Yazarlar

Bahar Karaman 0000-0001-6631-8562

Emrah Karaman 0000-0002-0466-3827

Erken Görünüm Tarihi 25 Şubat 2025
Yayımlanma Tarihi 27 Mart 2025
Gönderilme Tarihi 2 Ekim 2024
Kabul Tarihi 31 Ocak 2025
Yayımlandığı Sayı Yıl 2025

Kaynak Göster

APA Karaman, B., & Karaman, E. (2025). The Mathematical Dynamics of the Caputo Fractional Order Social Media Addiction Design. Communications in Advanced Mathematical Sciences, 8(1), 1-10. https://doi.org/10.33434/cams.1559808
AMA Karaman B, Karaman E. The Mathematical Dynamics of the Caputo Fractional Order Social Media Addiction Design. Communications in Advanced Mathematical Sciences. Mart 2025;8(1):1-10. doi:10.33434/cams.1559808
Chicago Karaman, Bahar, ve Emrah Karaman. “The Mathematical Dynamics of the Caputo Fractional Order Social Media Addiction Design”. Communications in Advanced Mathematical Sciences 8, sy. 1 (Mart 2025): 1-10. https://doi.org/10.33434/cams.1559808.
EndNote Karaman B, Karaman E (01 Mart 2025) The Mathematical Dynamics of the Caputo Fractional Order Social Media Addiction Design. Communications in Advanced Mathematical Sciences 8 1 1–10.
IEEE B. Karaman ve E. Karaman, “The Mathematical Dynamics of the Caputo Fractional Order Social Media Addiction Design”, Communications in Advanced Mathematical Sciences, c. 8, sy. 1, ss. 1–10, 2025, doi: 10.33434/cams.1559808.
ISNAD Karaman, Bahar - Karaman, Emrah. “The Mathematical Dynamics of the Caputo Fractional Order Social Media Addiction Design”. Communications in Advanced Mathematical Sciences 8/1 (Mart 2025), 1-10. https://doi.org/10.33434/cams.1559808.
JAMA Karaman B, Karaman E. The Mathematical Dynamics of the Caputo Fractional Order Social Media Addiction Design. Communications in Advanced Mathematical Sciences. 2025;8:1–10.
MLA Karaman, Bahar ve Emrah Karaman. “The Mathematical Dynamics of the Caputo Fractional Order Social Media Addiction Design”. Communications in Advanced Mathematical Sciences, c. 8, sy. 1, 2025, ss. 1-10, doi:10.33434/cams.1559808.
Vancouver Karaman B, Karaman E. The Mathematical Dynamics of the Caputo Fractional Order Social Media Addiction Design. Communications in Advanced Mathematical Sciences. 2025;8(1):1-10.

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