Caputo-Based Numerical Modeling of Fractional Logistic Growth with Fractal Structures

Authors

https://doi.org/10.48314/anowa.v2i2.73

Abstract

Transportation is one of the most important aspects of human activity, supporting various social and economic transactions. Meanwhile, in order to remain competitive, freight transportation businesses and logistics providers must deliver high-quality, dependable, and effective services. The effective design of the service network in this industry necessitates strategic and tactical decisions regarding service frequency, optimal route selection, and market share allocation among companies. In this study, we have examined the static competition between two transportation companies by utilizing a Mixed-Integer Nonlinear Programming (MINLP) model. The competition is studied by calculating entrant’s service frequency and each company’s market share using a logit function. In this type of competition, the incumbent’s decisions about route selection and frequency determination are known beforehand, and our goal is to maximize profits for the new market entrant. Additionally, a number of constraints have been taken into account, including route capacities, the maximum allowable frequency on each link, and penalty costs associated with the incomplete utilization of route capacities. To evaluate and validate the model, real-world data from the Iranian Road Maintenance and Transportation Organization has been employed. Furthermore, in the sensitivity analysis phase, the impacts of changing the values of important parameters on the model's outputs were evaluated. This investigation aims to improve understanding of the system's dynamics and clarify how these factors influence optimal decision-making processes. 

Keywords:

Fractal-fractional calculus, Caputo-type derivative, Logistic growth, Soft computing, Fractional numerical methods, Memory effect

References

  1. [1] Verhulst., P. F. (1838). Notice sur la loi que la population suit dans son accroissement. Correspondence mathematique et physique, 10, 113–129. https://cir.nii.ac.jp/crid/1570009749935841536
  2. [2] Diethelm, K., & Ford, N. J. (2010). The analysis of fractional differential equations. Lecture notes in mathematics, 2004. https://doi.org/10.1007/978-3-642-14574-2
  3. [3] Podlubny, I. (1999). Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press. https://books.google.com/books/about/Fractional_Differential_Equations.html?id=K5FdXohLto0C
  4. [4] Atangana, A., & Akgül, A. (2020). Analysis of new trends of fractional differential equations. In Fractional order analysis (pp. 91–111). John Wiley & Sons. https://doi.org/10.1002/9781119654223.ch3
  5. [5] Karaagac, B., Owolabi, K. M., & Pindza, E. (2023). A computational technique for the Caputo fractal-fractional diabetes mellitus model without genetic factors. International journal of dynamics and control, 11(5), 2161–2178. https://doi.org/10.1007/s40435-023-01131-7
  6. [6] Sun, H., Zhang, Y., Baleanu, D., Chen, W., & Chen, Y. (2018). A new collection of real world applications of fractional calculus in science and engineering. Communications in nonlinear science and numerical simulation, 64, 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019
  7. [7] Ahmed, S., Ullah, S., Riaz, M., Ahmed, W., & Sheikh, M. (2025). Numerical analysis of fractional logistic growth models using computational methods with MATLAB codes. International journal of applied and computational mathematics, 11(5), 198. https://doi.org/10.1007/s40819-025-02003-8
  8. [8] Moumen, A., Saleh, H. N., Albala, H., Aldwoah, K., Saber, H., Hassan, E. I., & Hassan, T. S. (2025). On polynomial φ-contractions with applications to fractional logistic growth equations. Fractal and fractional, 9(6), 366. https://doi.org/10.3390/fractalfract9060366
  9. [9] Stempin, P., & Sumelka, W. (2025). Approximation of fractional caputo derivative of variable order and variable terminals with application to initial/boundary value problems. Fractal and fractional, 9(5), 269. https://doi.org/10.3390/fractalfract9050269
  10. [10] Adak, S., Barman, S., Jana, S., Majee, S., & Kar, T. K. (2025). Modelling and analysis of a fractional-order epidemic model incorporating genetic algorithm-based optimization. Journal of applied mathematics and computing, 71(1), 901–925. https://doi.org/10.1007/s12190-024-02224-y
  11. [11] Armstrong, S., & Vicol, V. (2025). Anomalous diffusion by fractal homogenization. Annals of pde, 11(1), 2. https://doi.org/10.1007/s40818-024-00189-6
  12. [12] Alinei-Poiana, T., Dulf, E. H., & Kovacs, L. (2023). Fractional calculus in mathematical oncology. Scientific reports, 13(1), 10083. https://doi.org/10.1038/s41598-023-37196-9
  13. [13] Tarasov, V. E. (2010). Fractional dynamics: Applications of fractional calculus to dynamics of particles, fields and media. Springer Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14003-7
  14. [14] Gunasekar, T., Manikandan, S., Haque, S., Suba, M., & Mlaiki, N. (2025). Fractal-fractional mathematical modeling of monkeypox disease and analysis of its Ulam–Hyers stability. Boundary value problems, 2025(1), 20. https://doi.org/10.1186/s13661-025-02013-x
  15. [15] Atangana, A. (2017). Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system. Chaos, solitons & fractals, 102, 396–406. https://doi.org/10.1016/j.chaos.2017.04.027
  16. [16] Atangana, A., & Baleanu, D. (2016). New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Thermal science, 20(3), 763–769. https://doi.org/10.2298/TSCI160111018A
  17. [17] Samko, S. G. (1993). Fractional integrals and derivatives: Theory and applications. Gordan and Breach Science. Gordon & Breach Science Pub.
  18. [18] Caputo, M., & Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel. Progress in fractional differentiation & applications, 1(2), 73–85. http://dx.doi.org/10.12785/pfda/010201
  19. [19] El-Sayed, A. M. A., El-Mesiry, A. E. M., & El-Saka, H. A. A. (2007). On the fractional-order logistic equation. Applied mathematics letters, 20(7), 817–823. https://doi.org/10.1016/j.aml.2006.08.013

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Published

2026-06-18

Issue

Vol. 2 No. 2 (2026): Annals of Optimization With Applications (ANOWA)

Section

Articles

How to Cite

Akhavan Ghassabzade, F. ., & Bagherpoorfard, M. . (2026). Caputo-Based Numerical Modeling of Fractional Logistic Growth with Fractal Structures. Annals of Optimization With Applications, 2(2), 125-133. https://doi.org/10.48314/anowa.v2i2.73

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