A nonstandard finite difference method for solving the fractional logistic model

• Autorzy: Ali Samah , Bashier Eihab

Identyfikatory

DOI

10.17512/jamcm.2024.4.02

• Języki publikacji: EN

Abstrakty

EN

This paper proposes an alternative solution formula for the logistic model, which is derived by substituting the exponential function with the Mittag-Leffler function in the solution of the first-order logistic model. Then, it developed two nonstandard finite difference approaches to solve the fractional logistic model. One method employed Mickens’s concepts to construct a nonstandard finite difference scheme, under the assumption that the analytical solution is unknown. The second method relies on the proposed analytical solution of the fractional logistic model. Surprisingly the two nonstandard finite difference algorithms are exactly the same. The convergence of the nonstandard finite difference scheme is proven by establishing its consistency and stability. Furthermore, it has been proven that the proposed numerical method is unconditionally stable. The performance of the method is demonstrated through two numerical examples selected from literature.

Słowa kluczowe

EN

denominator function; exact finite difference scheme; fractional logistic model; nonstandard finite difference methods; Mittag-Leffler function

PL

funkcja mianownika; dokładny schemat skończonej różnicy; ułamek modelu logistycznego; niestandardowe metody skończonej różnicy; funkcja Mittag-Lefflera

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2024
• Tom: Vol. 23, nr 4
• Strony: 18--29
• Opis fizyczny: Bibliogr. 17 poz., rys., tab.

Twórcy

autor

Ali Samah

• Department of Modeling and Computational Mathematics-Al Neelain University Khartoum, Sudan

autor

Bashier Eihab

• Department of Mathematics, Faculty of Education and Arts, Sohar University Sohar, Oman
• Department of Applied Mathematics, Faculty of Mathematical Sciences and Informatics University of Khartoum, Khartoum, Sudan

Bibliografia

• 1. Abreu-Blaya, R., Fleitas, A., N´apoles Vald´es, J.E., Reyes, R., Rodr´ıguez, J.M., & Sigarreta, J.M. (2020). On the conformable fractional logistic models. Mathematical Methods in the Applied
• Sciences. DOI: 10.1002/mma.6180.
• 2. Khalouta, A. (2024). Existence uniqueness and convergence solution of nonlinear Caputo-Fabrizio fractional biological population model. Sahand Communications in Mathematical Analysis.
• 3. Khalouta, A. (2024). A new identification of lagrange multipliers to study solutions of nonlinear Caputo-Fabrizio fractional problems. Partial Differential Equations in Applied Mathematics, 10, 100711.
• 4. de Barros, L.C., Lopes, M.M., Pedro, F.S., Esmi, E., dos Santos, J.P.C., & S´anchez, D.E. (2021). The memory effect on fractional calculus: an application in the spread of COVID-19. Computational and Applied Mathematics, 40(3).
• 5. Podlubny, I. (1998). Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications. Elsevier Science and Technology Books.
• 6. Milici, C., Dr˘ag˘anescu, G., & Machado, J.T. (2019). Introduction to Fractional Differential Equations. Springer International Publishing.
• 7. El-Sayed, A., El-Mesiry, A., & El-Saka, H. (2007). On the fractional-order logistic equation. Applied Mathematics Letters, 20(7), 817-823.
• 8. Daftardar-Gejji, V., & Jafari, H. (2006). An iterative method for solving nonlinear functional equations. Journal of Mathematical Analysis and Applications, 316(2), 753-763.
• 9. Bhalekar, S., & Daftardar-Gejji, V. (2012). Solving fractional-order logistic equation using a new iterative method. International Journal of Differential Equations, 2012, 1-12.
• 10. Khader, M.M., & Babatin, M.M. (2013). On approximate solutions for fractional logistic differentia equation. Mathematical Problems in Engineering, 2013, 1-7.
• 11. Pitolli, F., & Pezza, L. (2017). A Fractional Spline Collocation Method for the Fractional-order Logistic Equation. Springer International Publishing, 307-318.
• 12. Arshad, M.S., Baleanu, D., Riaz, M.B., & Abbas, M. (2020). A novel 2-stage fractional Runge-Kutta method for a time-fractional logistic growth model. Discrete Dynamics in Nature and Society, 2020, 1-8.
• 13. Area, I., & Nieto, J. (2021). Power series solution of the fractional logistic equation. Physica A: Statistical Mechanics and its Applications, 573, 125947.
• 14. West, B.J. (2015). Exact solution to fractional logistic equation. Physica A: Statistical Mechanicsand its Applications, 429, 103-108.
• 15. Area, I., Losada, J., & Nieto, J.J. (2016). A note on the fractional logistic equation. Physica A: Statistical Mechanics and its Applications, 444, 182-187.
• 16. Mickens, R.E. (1994). Nonstandard Finite Difference Models of Differential Equations. World Scientific.
• 17. Kathiri, S., Bashier, E., & Hamid, N. (2024). Development of a non-standard finite difference
• method for solving a fractional decay model. Journal of Applied Mathematics and Informatics,
• 42(3), 182-187.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).