International Journal of Nonlinear Analysis and Applications

Laplace optimized decomposition method for solving fractional order logistic growth in a population

Articles in Press

Document Type : Research Paper

Author

1 University of Science & Technology, College of Engineering, Sudan

2 AlMughtaribeen University, College of Engineering, Department of General Sciences, Sudan

Abstract

In this paper, we introduce a semi-analytical method called the Laplace optimized decomposition method, abbreviated as LODM, for solving a model of a nonlinear ordinary differential equation describing the growth of population, the so-called Logistic equation with the fractional-order type, using the Caputo fractional derivative sense. The proposed technique combines the Laplace transform (LT) with a new technique called the optimized decomposition method (ODM). The results obtained by this method have been compared with those obtained by other methods. Finally, we demonstrate our numerical results with the help of tables and figures.

Keywords

[1] K.S. Aboodh and A. Ahmed, On the application of homotopy analysis method to fractional differential equations, J. Faculty Sci. Technol. 7 (2020), 1–18.
[2] I. Abu Irwaq, M. Alquran, M. Ali, I. Jaradat, and M.S.M. Noorani, Attractive new fractional-integer power series method for solving singular perturbed differential equations involving mixed fractional and integer derivatives, Results Phys. 20 (2021), 103780.
[3] B. Acay, R. Ozarslan, and E. Bas, Fractional physical models based on falling body problem, AIMS Math. 5 (2020), 2608–2628.
[4] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl. 135 (1988), 501–544.
[5] G. Adomian, A review of the decomposition method and some recent results for nonlinear equations, Comput. Math. Appl. 21 (1991), 101–127.
[6] A. Ahmadian, S. Salahshour, and M. Salimi, A robust numerical approximation of advection diffusion equations with nonsingular kernel derivative, Phys. Scripta 96 (2021), no. 12, Article ID 124015.
[7] A. Ahmed, Laplace transform method for logistic growth in a population and predator models with fractional order, Open J. Math. Sci. 7 (2023), 239–245.
[8] S. Ahmad, K. Shah, T. Abdeljawad, and B. Abdalla, On the approximation of fractal fractional differential equations using numerical inverse Laplace transform methods, CMES 135 (2023), no. 3.
[9] M. Alquran, F. Yousef, F. Alquran, T.A. Sulaiman, and Y.A. Dualwave, Solutions for the quadratic-cubic conformable-Caputo time-fractional Klein-Fock-Gordon equation, Math. Comput. Simul. 185 (2021), 62–76.
[10] S. Alshammari, M. Al-Smadi, M. Al Shammari, I. Hashim, and M.A. Alias, Advanced analytical treatment of fractional logistic equations based on residual error functions, Int. J. Differ. Equ. 2019 (2019), Article ID 7609879, 1–11.
[11] I. Area, K.A. Lazopoulos, and J.J. Nieto, Γ−fractional logistic equation, Prog. Fract. Differ. Appl. 9 (2023), 345–350.
[12] I. Area and J.J. Nieto, Fractional-order logistic differential equation with Mittag–Leffler type kernel, Fractal Fractional 5 (2021), no. 4, 273.
[13] I. Area and J.J. Nieto, Power series solution of the fractional logistic equation, Physica A 573 (2021), 125947.
[14] A. Atangana and A. Akgul, On solutions of fractal fractional differential equations, Discrete Continuous Dyn. Syst. Ser. S 14 (2021), no. 10, 3441–3457.
[15] A. Atangana and D. Baleanu, New fractional derivatives with non-local and nonsingular kernel: Theory and application to heat transfer model. Therm. Sci. 20 (2016), 763–769.
[16] A. Atangana, E. Bonyah, and A.A. Elsadany, A fractional order optimal 4D chaotic financial model with Mittag-Leffler law, Chinese J. Phys. 65 (2020), 38–53.
[17] N. Attia, A. Akgul, D. Seba, and A. Nour, On solutions of fractional logistic differential equations, Progr. Fract. Differ. Appl. 9 (2023), 351–362.
[18] C. Balzotti, M. D’Ovidio, and P. Loreti, Fractional SIS epidemic models, Fractal Fractional 4 (2020), 18 pages.
[19] S. Bhalekar and V. Daftardar-Gejji, Solving fractional-order logistic equation using a new iterative method, Int. J. Diff. Equ. 2012 (2012), Article ID 975829, 12 pages.
[20] F. Brauer, C. Castillo-Chavez, and Z. Feng, Mathematical Models in Epidemiology, Springer-Verlag, New York, 2019.
[21] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fractional Diff. Appl. 1 (2015), no. 2, 73–85.
[22] V. Daftardar-Gejji and H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 2006, 753–763.
[23] L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. 54 (2003), 3413–3442.
[24] K. Diethelm, N. Ford, and A. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29 (2002), 3–22.
[25] M. D’Ovidio and P. Loreti, Solutions of fractional logistic equations by Euler’s numbers, Physical A 506 (2018), 1081–1092.
[26] J.D. do Nascimento, R.L.C. Damasceno, G.L. de Oliveira, and R.V. Ramos, Quantum-chaotic key distribution in optical networks: From secrecy to implementation with logistic map, Quantum Inf. Process. 17 (2018), 329.
[27] V.P. Dubey, S. Dubey, D. Kumar, and J. Singh, Computational study of fractional model of atmospheric dynamics of carbon dioxide gas, Chaos Solitons Fractals 142 (2021), 279–312.
[28] V.P. Dubey, D. Kumar, and S. Dubey, A modified computational scheme and convergence for fractional order hepatitis E virus model, Advanced Numerical Methods for Differential Equations, CRC Press, 2021, pp. 279–312.
[29] V.P. Dubey, J. Singh, A.M. Alshehri, S. Dubey, and D. Kumar, Numerical investigation of fractional model of phytoplankton-toxic Phytoplankton-Zooplankton system with convergence analysis, Int. J. Biomath. 15 (2022), no. 4, 2250006.
[30] A.M.A. El-Sayed, A.E.M. El-Mesiry, H.A.A. and El-Saka, on the fractional-order logistic equations, Appl. Math. Lett. 20 2007 817-823.
[31] J.H. He, A new approach to nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. Simul. 2 (1997), 203–205.
[32] J.H. He, Homotopy perturbation method: A new nonlinear analytical technique. Appl. Math. Comput. 135 (2003), 73–79.
[33] M. Ichise, Y. Nagayanagi, and T. Kojima, An analog simulation of non-integer order transfer functions analysis of electrode process, J. Electroanal. Chem. Interfacial Electrochem. 33 (1971), 253–265.
[34] H.K. Jassim and M. Abdulshareef Hussein, A new approach for solving nonlinear fractional ordinary differential equations, Mathematics. 11 (2023), no. 7, 1565.
[35] R. Kamal, Kamran, G. Rahmat, A. Ahmadian, N.I. Arshad, and S. Salahshour, Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel, Adv. Diff. Equ. 1 (2021), 317—415.
[36] Kamran, A. Ali, and J.F. Gomez-Aguilar, A transform based local RBF method for 2D linear PDE with Caputo-Fabrizio derivative, Comptes Rendus Math. 358 (2020), no. 7, 831–842.
[37] S. Kazem, Exact solution of some linear fractional differential equations by Laplace transform, Int. J. Nonlinear Sci. 16 (2013), no. 1, 3–11.
[38] A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, The organization, Elsevier, Amsterdam, Netherlands, 2006.
[39] I. Koca, Modelling the spread of Ebola virus with Atangana Baleanu fractional operators, Eur. Phys. J. Plus 133 (2018), 100–111.
[40] R.C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51 (1984), 299–307.
[41] M. Laoubi, Z. Odibat, and B. Maayah, Effective optimized decomposition algorithms for solving nonlinear fractional differential equations, J. Comput. Nonlinear Dyn. 18 (2023), no. 2, 021001.
[42] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press: Boca Raton, FL, USA, 2003.
[43] F. Liu, V. Anh, and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math. 166 (2004), 209–219.
[44] J. Lu and G. Chen, A note on the fractional-order Chen system, Chaos Solitons Fractals 27 (2006), no. 3, 685–688.
[45] B. Maayah, S. Bushnaq, and A. Moussaoui, Numerical solution of fractional order SIR model of dengue fever disease via Laplace optimized decomposition method, J. Math. Comput. Sci. 32 (2024), 86–93.
[46] J.J. Nieto, Solution of a fractional logistic ordinary differential equation, Appl. Math. Lett. 123 (2022), 107568.
[47] Z. Odibat, An Optimized decomposition method for nonlinear ordinary and partial differential equations, Phys. A 541 (2020), 13 pages.
[48] Z. Odibat, A universal predictor–corrector algorithm for numerical simulation of generalized fractional differential equations, Nonlinear Dyn. 105 (2021), no. 3, 2363–2374.
[49] Z. Odibat, The optimized decomposition method for a reliable treatment of IVPs for second order differential equations, Phys. Scr. 96 (2021).
[50] E. Pelinovsky, A. Kurkin, O. Kurkina, M. Kokoulina, and A. Epifanova, Logistic equation and COVID-19, Chaos Solitons Fractals 140 (2020), 110241.
[51] X. Qiang, Kamran, A. Mahboob, and Y.M. Chu, Numerical approximation of fractional-order Volterra integrodifferential equation, J. Funct. Spaces 2020 (2020), Article ID 8875792, 12 pages.
[52] D. Rani and V. Mishra, Modification of Laplace Adomian decomposition method for solving nonlinear Volterra integral and integro-differential equations based on Newton Raphson formula, Eur. J. Pure Appl. Math. 11 (2018), 202–214.
[53] T. Saito and K. Shigemoto, A logistic curve in the SIR model and its application to deaths by COVID-19 in Japan, Eur. J. Appl. Sci. 10 (2022), no. 5.
[54] K. Shah, T. Abdeljawad, F. Jarad, and Q. Al-Mdallal, On nonlinear conformable fractional order dynamical system via differential transform method, CMES 136 (2023), no. 2, 1457–1472.
[55] H.H. Sun, A.A. Abdelwahad, and B. Onaral, Linear approximation of transfer function with a pole of fractional order, IEEE Trans. Autom. Control. 29 (1984), 441–444.
[56] H.R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2003.
[57] V. Tarasov, Exact solutions of Bernoulli and logistic fractional differential equations with power law confidents, Mathematics 8 (2020), no. 12, 2231.
[58] V.V. Tarasova and V.E. Tarasov, Logistic map with memory from economic model, Chaos Solitons Fractals 95 (2017), 84–91.
[59] C.A. Valentim Jr, J.A. Oliveira, S.A. Rabi, and N.A. David, Can fractional calculus help improve tumor growth models?, J. Comput. Appl. Math. 379 (2020), 112964.
[60] D. Vivek, K. Kanagarajan, and S. Harikrishnan, Numerical solution of fractional-order logistic equations by fractional Euler’s method, IJRASET 4 (2016), 775–780.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 17 April 2025
  • Receive Date: 16 October 2023
  • Revise Date: 25 April 2024
  • Accept Date: 19 August 2024