International Journal of Nonlinear Analysis and Applications
Solutions of fractional Lotka-Volterra and Lorenz-Stenflo equations by Sumudu decomposition method
Document Type : Research Paper
Author
Department of Mathematical Sciences, Adekunle Ajasin University, P.M.B 001, Akungba Akoko, Ondo State, Nigeria
Abstract
In this paper, we obtain approximate analytical solutions for the fractional predator-prey and the generalized fractional Lorenz-Stenflo systems using the Sumudu decomposition method. The fractional derivative is described in the Caputo sense. The results show that the method gives an easy-to-implement procedure and accurate approximate solutions.
Keywords
1] A.K. Alomari, M.S.M. Noorani, R. Nazar and C.P. Li, Homotopy analysis method for solving fractional Lorenz system, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 1864–1872.
[2] A. Atangana and A. Secer, A note on fractional order derivatives and table of fractional derivatives of some special functions, Abstr. Appl. Anal.ysis 2013 (2013), 1–8.
[2] A. Atangana and A. Secer, A note on fractional order derivatives and table of fractional derivatives of some special functions, Abstr. Appl. Anal.ysis 2013 (2013), 1–8.
[3] R.O. Awonusika, Analytical solution of a class of fractional Lane–Emden equation: a power series method, Int. J. Appl. Comput. Math, 8 (2022), 155.
[4] R.O Awonusika, O.A Mogbojuri, Approximate analytical solution of fractional Lane-Emden equation by MittagLeffler function method, J. Nigerian Soc. Phys. Sci. 4 (2022), no. 2, 265–280.
[5] F.B.M. Belgacem and A.A. Karaballi, Sumudu transform fundamental properties investigations and applications, Int. J. Appl. Math. Stoch. Anal. 2006 (2006), 1–23.
[6] W.E. Boyce and R.C. DiPrima Elementary differential equations and boundary value problems, John Wiley & Sons, Inc. New York, 2001.
[7] N. Bildik and S. Deniz The use of Sumudu decomposition method for solving predator-prey systems, Math. Sci. Lett. 5 (2016), no. 3, 285–289.
[8] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2015), 73–85.
[9] Y.-M. Chen and H.-H. Liang, Zero-zero-Hopf bifurcation and ultimate bound estimation of a generalized LorenzStenflo hyperchaotic system, Math. Meth. Appl. Sci. 40 (2016), no. 10, 3424—3432.
[10] S. Das and P.K. Gupta, A fractional predator-prey model and its solution, Int. J. Nonlinear Sci. Numer. Simul. 10 (2009), no. 7, 873–876.
[11] B. Ghanbari, On approximate solutions for a fractional prey–predator model involving the Atangana–Baleanu derivative, Adv. Differ. Equ. 2020 (2020), 679.
[12] P. Goswamia and R.T. Alqahtanib, Solutions of fractional differential equations by Sumudu transform and variational iteration method, J. Nonlinear Sci. Appl. 9 (2016), 1944–1951.
[13] I. Hashim, M.S.M. Noorani, R. Ahmad, S.A. Bakar, E.S. Ismail and A.M. Zakaria Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos Solitons Fractals 28 (2006), 1149-–1158.
[14] Q.D. Katatbeh, F.B.M. Belgacem, Applications of the Sumudu transform to fractional differential equations, Nonlinear Stud. 18 (2011), no. 1, 99–112.
[15] M.M. Khader, N.H. Swetlam and A.M.S. Mahdy, The Chebyshev collection method for solving fractional order Klein-Gordon equation, WSEAS Trans. Math. 13 (2014), 32–38.
[16] A. Kili¸cman, H. Eltayeb and P.R. Agarwal, On Sumudu transform and system of differential equations, Abstr. Appl. Anal. 2010 (2010), Article ID 598702.
[17] D. Kumar, J. Singh and S. Rathore, Sumudu decomposition method for nonlinear equations, Int. Math. Forum 7 (2012), no. 11, 515–521.
[18] A.M.S. Mahdy and M. Higazy Numerical different methods for solving the nonlinear biochemical reaction model, Int. J. Appl. Comput. Math. 5 (2019), 1–17.
[19] A.M.S. Mahdy, Kh. Lotfy and A.A. El-Bary, Use of optimal control in studying the dynamical behaviors of fractional financial awareness model, Soft Comput. 26 (2022), 3401–3409.
[20] A.M.S. Mahdy and G.M.A. Marai, Sumudu decomposition method for solving fractional Riccati equation, J. Abstr. Comput Math. 3 (2018), no. 1, 42-50.
[21] A.M.S. Mahdy, A.S. Mohamed and A.A.H. Mtawa, Sumudu decomposition method for solving fractional-order logistic differential equation, J. Adv. Math. 10 (2015), no. 7, 3642–3649.
[22] C. Milici, G. Drˇagˇanescu and J.T. Machado, Introduction to Fractional Differential Equations, Nonlinear Systems and Complexity, Springer, 2019.
[23] A.S. Mohamed, A.M.S. Mahdy and A.A. H. Mtawa Aproximate analytical solution to a time-fractional FokkerPlanck equation, Bothalia 45 (2015), no. 4, 57–69.
[24] M. Mossa Al-Sawalha and M.S.M. Noorani, On solving the Lorenz system by differential transformation method, Chin. Phys. Lett. 25 (2008), no. 4, 1217.
[25] I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999.
[4] R.O Awonusika, O.A Mogbojuri, Approximate analytical solution of fractional Lane-Emden equation by MittagLeffler function method, J. Nigerian Soc. Phys. Sci. 4 (2022), no. 2, 265–280.
[5] F.B.M. Belgacem and A.A. Karaballi, Sumudu transform fundamental properties investigations and applications, Int. J. Appl. Math. Stoch. Anal. 2006 (2006), 1–23.
[6] W.E. Boyce and R.C. DiPrima Elementary differential equations and boundary value problems, John Wiley & Sons, Inc. New York, 2001.
[7] N. Bildik and S. Deniz The use of Sumudu decomposition method for solving predator-prey systems, Math. Sci. Lett. 5 (2016), no. 3, 285–289.
[8] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2015), 73–85.
[9] Y.-M. Chen and H.-H. Liang, Zero-zero-Hopf bifurcation and ultimate bound estimation of a generalized LorenzStenflo hyperchaotic system, Math. Meth. Appl. Sci. 40 (2016), no. 10, 3424—3432.
[10] S. Das and P.K. Gupta, A fractional predator-prey model and its solution, Int. J. Nonlinear Sci. Numer. Simul. 10 (2009), no. 7, 873–876.
[11] B. Ghanbari, On approximate solutions for a fractional prey–predator model involving the Atangana–Baleanu derivative, Adv. Differ. Equ. 2020 (2020), 679.
[12] P. Goswamia and R.T. Alqahtanib, Solutions of fractional differential equations by Sumudu transform and variational iteration method, J. Nonlinear Sci. Appl. 9 (2016), 1944–1951.
[13] I. Hashim, M.S.M. Noorani, R. Ahmad, S.A. Bakar, E.S. Ismail and A.M. Zakaria Accuracy of the Adomian decomposition method applied to the Lorenz system, Chaos Solitons Fractals 28 (2006), 1149-–1158.
[14] Q.D. Katatbeh, F.B.M. Belgacem, Applications of the Sumudu transform to fractional differential equations, Nonlinear Stud. 18 (2011), no. 1, 99–112.
[15] M.M. Khader, N.H. Swetlam and A.M.S. Mahdy, The Chebyshev collection method for solving fractional order Klein-Gordon equation, WSEAS Trans. Math. 13 (2014), 32–38.
[16] A. Kili¸cman, H. Eltayeb and P.R. Agarwal, On Sumudu transform and system of differential equations, Abstr. Appl. Anal. 2010 (2010), Article ID 598702.
[17] D. Kumar, J. Singh and S. Rathore, Sumudu decomposition method for nonlinear equations, Int. Math. Forum 7 (2012), no. 11, 515–521.
[18] A.M.S. Mahdy and M. Higazy Numerical different methods for solving the nonlinear biochemical reaction model, Int. J. Appl. Comput. Math. 5 (2019), 1–17.
[19] A.M.S. Mahdy, Kh. Lotfy and A.A. El-Bary, Use of optimal control in studying the dynamical behaviors of fractional financial awareness model, Soft Comput. 26 (2022), 3401–3409.
[20] A.M.S. Mahdy and G.M.A. Marai, Sumudu decomposition method for solving fractional Riccati equation, J. Abstr. Comput Math. 3 (2018), no. 1, 42-50.
[21] A.M.S. Mahdy, A.S. Mohamed and A.A.H. Mtawa, Sumudu decomposition method for solving fractional-order logistic differential equation, J. Adv. Math. 10 (2015), no. 7, 3642–3649.
[22] C. Milici, G. Drˇagˇanescu and J.T. Machado, Introduction to Fractional Differential Equations, Nonlinear Systems and Complexity, Springer, 2019.
[23] A.S. Mohamed, A.M.S. Mahdy and A.A. H. Mtawa Aproximate analytical solution to a time-fractional FokkerPlanck equation, Bothalia 45 (2015), no. 4, 57–69.
[24] M. Mossa Al-Sawalha and M.S.M. Noorani, On solving the Lorenz system by differential transformation method, Chin. Phys. Lett. 25 (2008), no. 4, 1217.
[25] I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999.
[26] L. Stenflo, Generalized Lorenz equations for acoustic-gravity waves in the atmosphere, Phys. Scripta 53 (1996), no. 1, 83–84.
[27] S.H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering, CRC Press, Boca Raton, FL, 2018.
[28] G.K. Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, Int. J. Math. Educ. Sci. Technol. 24 (1993) no. 1, 35–43.
[29] A.M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput. 111 (2000), 33–51.
[30] F. Zhang, R. Chen and X. Chen, Analysis of a generalized Lorenz-Stenflo equation, Complexity 2017 (2017).
[27] S.H. Strogatz, Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineering, CRC Press, Boca Raton, FL, 2018.
[28] G.K. Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, Int. J. Math. Educ. Sci. Technol. 24 (1993) no. 1, 35–43.
[29] A.M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Appl. Math. Comput. 111 (2000), 33–51.
[30] F. Zhang, R. Chen and X. Chen, Analysis of a generalized Lorenz-Stenflo equation, Complexity 2017 (2017).
)
Volume 14, Issue 5
May 2023Pages 325-336
- Receive Date: 05 October 2022
- Revise Date: 01 February 2023
- Accept Date: 05 February 2023