International Journal of Differential Equations

Advanced Analytical Treatment of Fractional Logistic Equations Based on Residual Error Functions

Saleh Alshammari, Mohammed Al-Smadi, Mohammad Al Shammari, Ishak Hashim, and Mohd Almie Alias

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this article, an analytical reliable treatment based on the concept of residual error functions is employed to address the series solution of the differential logistic system in the fractional sense. The proposed technique is a combination of the generalized Taylor series and minimizing the residual error function. The solution methodology depends on the generation of a fractional expansion in an effective convergence formula, as well as on the optimization of truncated errors, Resqjt, through the use of repeated Caputo derivatives without any restrictive assumptions of system nature. To achieve this, some logistic patterns are tested to demonstrate the reliability and applicability of the suggested approach. Numerical comparison depicts that the proposed technique has high accuracy and less computational effect and is more efficient.

Article information

Source
Int. J. Differ. Equ., Volume 2019 (2019), Article ID 7609879, 11 pages.

Dates
Received: 3 June 2019
Revised: 28 July 2019
Accepted: 25 August 2019
First available in Project Euclid: 17 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1571277678

Digital Object Identifier
doi:10.1155/2019/7609879

Mathematical Reviews number (MathSciNet)
MR4015790

Citation

Alshammari, Saleh; Al-Smadi, Mohammed; Al Shammari, Mohammad; Hashim, Ishak; Alias, Mohd Almie. Advanced Analytical Treatment of Fractional Logistic Equations Based on Residual Error Functions. Int. J. Differ. Equ. 2019 (2019), Article ID 7609879, 11 pages. doi:10.1155/2019/7609879. https://projecteuclid.org/euclid.ijde/1571277678


Export citation

References

  • K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  • M. Al-Smadi and O. A. Arqub, “Computational algorithm for solving fredholm time-fractional partial integrodifferential equations of dirichlet functions type with error estimates,” Applied Mathematics and Computation, vol. 342, pp. 280–294, 2019.
  • O. Abu Arqub and M. Al-Smadi, “Atangana-Baleanu fractional approach to the solutions of Bagley-Torvik and Painlevé equations in Hilbert space,” Chaos, Solitons and Fractals, vol. 117, pp. 161–167, 2018.
  • M. Al-Smadi, A. Freihat, H. Khalil, S. Momani, and R. A. Khan, “Numerical multistep approach for solving fractional partial differential equations,” International Journal of Computational Methods, vol. 14, Article ID 1750029, pp. 1–15, 2017.
  • K. Moaddy, A. Freihat, M. Al-Smadi, E. Abuteen, and I. Hashim, “Numerical investigation for handling fractional-order Rabinovich-Fabrikant model using the multistep approach,” Soft Computing, vol. 22, no. 3, pp. 773–782, 2018.
  • S. Momani, O. Abu Arqub, A. Freihat, and M. Al-Smadi, “Analytical approximations for Fokker-Planck equations of fractional order in multistep schemes,” Applied and Computational Mathematics, vol. 15, no. 3, pp. 319–330, 2016.
  • M. Al-Smadi, “Simplified iterative reproducing kernel method for handling time-fractional BVPs with error estimation,” Ain Shams Engineering Journal, vol. 9, no. 4, pp. 2517–2525, 2018.
  • Z. Altawallbeh, M. Al-Smadi, I. Komashynska, and A. Ateiwi, “Numerical solutions of fractional systems of two-point BVPs by using the iterative reproducing Kernel algorithm,” Ukrainian Mathematical Journal, vol. 70, no. 5, pp. 687–701, 2018.
  • M. Al-Smadi, A. Freihat, M. m. A. Hammad, S. Momani, and O. A. Arqub, “Analytical approximations of partial differential equations of fractional order with multistep approach,” Journal of Computational and Theoretical Nanoscience, vol. 13, no. 11, pp. 7793–7801, 2016.
  • A. K. Golmankhaneh and C. Cattani, “Fractal logistic equation,” Fractal and Fractional, vol. 3, no. 3, p. 41, 2019.
  • S. Bhalekar and V. Daftardar-Gejji, “Solving fractional-order logistic equation using a new iterative method,” International Journal of Differential Equations, vol. 2012, Article ID 975829, pp. 1–12, 2012.
  • M. M. Khader and M. M. Babatin, “On approximate solutions for fractional logistic differential equation,” Mathematical Problems in Engineering, vol. 2013, Article ID 391901, pp. 1–7, 2013.
  • N. H. Sweilam, M. M. Khader, and A. M. S. Mahdy, “Numerical studies for fractional-order Logistic differential equation with two different delays,” Journal of Applied Mathematics, vol. 2012, Article ID 764894, pp. 1–14, 2012.
  • M. Hamarsheh and A. I. Ismail, “Analytical approximation for fractional order logistic equation,” International Journal of Pure and Applied Mathematics, vol. 115, no. 2, pp. 225–245, 2017.
  • M. Al Shammari, M. Al-Smadi, O. Abu Arqub, I. Hashim, and M. A. Alias, “Adaptation of residual power series method to solve Fredholm fuzzy integro-differential equations,” AIP Conference Proceedings, vol. 2111, Article ID 020002, 2019.
  • K. Moaddy, M. Al-Smadi, and I. Hashim, “A novel representation of the exact solution for differential algebraic equations system using residual power-series method,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 205207, pp. 1–12, 2015.
  • M. Alaroud, M. Al-Smadi, R. R. Ahmad, and U. K. Salma Din, “Computational optimization of residual power series algorithm for certain classes of fuzzy fractional differential equations,” International Journal of Differential Equations, vol. 2018, Article ID 8686502, pp. 1–11, 2018.
  • A. El-Ajou, O. Abu Arqub, and M. Al-Smadi, “A general form of the generalized Taylor's formula with some applications,” Applied Mathematics and Computation, vol. 256, pp. 851–859, 2015.
  • S. Hasan, M. Al-Smadi, A. Freihet, and S. Momani, “Two computational approaches for solving a fractional obstacle system in Hilbert space,” Advances in Difference Equations, vol. 2019, no. 55, 2019.
  • A. Freihet, S. Hasan, M. Al-Smadi, M. Gaith, and S. Momani, “Construction of fractional power series solutions to fractional stiff system using residual functions algorithm,” Advances in Difference Equations, vol. 2019, p. 95, 2019.
  • M. Alaroud, M. Al-Smadi, R. R. Ahmad, and U. K. Salma Din, “An analytical numerical method for solving fuzzy fractional Volterra integro-differential equations,” Symmetry, vol. 11, no. 2, p. 205, 2019.
  • S. Alshammari, M. Al-Smadi, I. Hashim, and M. A. Alias, “Applications of fractional power series approach in solving fractional Volterra integro-differential equations,” AIP Conference Proceedings, vol. 2111, Article ID 020003, 2019.
  • A. Akgül, Y. Khan, E. K. Akgül, D. Baleanu, and M. M. Al Qurashi, “Solutions of nonlinear systems by reproducing kernel method,” The Journal of Nonlinear Sciences and Applications, vol. 10, no. 8, pp. 4408–4417, 2017.
  • D. Baleanu, A. K. Golmankhaneh, and A. K. Golmankhaneh, “Solving of the fractional non-linear and linear Schrodinger equations by homotopy perturbation method,” Romanian Journal of Physics, vol. 54, pp. 823–832, 2009.
  • M. Al-Smadi, O. Abu Arqub, and S. Momani, “A computational method for two-point boundary value problems of fourth-order mixed integrodifferential equations,” Mathematical Problems in Engineering, vol. 2013, Article ID 832074, pp. 1–10, 2013.
  • M. Al-Smadi, O. Abu Arqub, and A. El-Ajou, “A numerical iterative method for solving systems of first-order periodic boundary value problems,” Journal of Applied Mathematics, vol. 2014, Article ID 135465, pp. 1–10, 2014. \endinput