Iterative methods for a fractional-order Volterra population model. (English) Zbl 1435.45002

Summary: We prove an existence and uniqueness theorem for a fractional-order Volterra population model via an efficient monotone iterative scheme. By coupling a spectral method with the proposed iterative scheme, the fractional-order integrodifferential equation is solved numerically. The numerical experiments show that the proposed iterative scheme is more efficient than an existing iterative scheme in the literature, the convergence of which is very sensitive to various parameters, including the fractional order of the derivative. The spectral method based on our proposed iterative scheme shows greater flexibility with respect to various parameters. Sufficient conditions are provided to select the initial guess that ensures the quadratic convergence of the quasilinearization scheme.


45D05 Volterra integral equations
45J05 Integro-ordinary differential equations
47J25 Iterative procedures involving nonlinear operators
34A08 Fractional ordinary differential equations
92D25 Population dynamics (general)
65R20 Numerical methods for integral equations
Full Text: DOI Euclid


[1] C. Canuto, A. Quarteroni, M.Y. Hussaini and T. A. Zang, Spectral methods: fundamentals in single domains, Springer (2006). · Zbl 1093.76002
[2] G. Chandhini, K.S. Prashanthi and V. Antony Vijesh, A radial basis function method for fractional Darboux problems, Eng. Anal. Bound. Elem. \bf86 (2018), 1-18. · Zbl 1403.65155
[3] V.S. Erturk, A. Yildirim, S. Momanic and Y. Khan, The differential transform method and Padé approximants for a fractional population growth model, Internat. J. Numer. Methods Heat Fluid Flow 22 (2012), no. 6, 791-802. · Zbl 1356.92073
[4] X. Hang, Analytical approximations for a population growth model with fractional order, Comm. Nonlinear Sci. and Numer. Simulat. \bf14 (2009), no. 5, 1978-1983. · Zbl 1221.65210
[5] M.H. Heydari, M.R. Hooshmandasl, C. Cattani and L. Ming, Legendre wavelets method for solving fractional population growth model in a closed system, Math. Probl. Eng. (2013), art. id. 161030. · Zbl 1296.65108
[6] T. Jankowski, Fractional equations of Volterra type involving a Riemann-Liouville derivative, Appl. Math. Lett. \bf26 (2013), 344-350. · Zbl 1259.45007
[7] K. Krishnarajulu, K. Krithivasan and R.B. Sevugan, Fractional polynomial method for solving fractional order population growth model, Commun. Korean Math. Soc. \bf31 (2016), 869-878. · Zbl 1372.34014
[8] M. Maleki and M.T. Kajani, Numerical approximations for Volterra’s population growth model with fractional order via a multi-domain pseudospectral method, Appl. Math. Model. \bf39 (2015), no. 15, 4300-4308.
[9] S. Momani and R. Qaralleh, Numerical approximations and Padé approximants for a fractional population growth model, Appl. Math. Model. \bf31 (2007), no. 9, 1907-1914. · Zbl 1167.45300
[10] S.S. Motsa, V.M. Magagula and P. Sibanda, A bivariate Chebyshev spectral collocation quasilinearization method for nonlinear evolution parabolic equations, Scient. World J. (2014), art. id. 581987, doi 10.1155/2014/581987.
[11] K. Parand and M. Delkhosh, Solving Volterra’s population growth model of arbitrary order using the generalized fractional order of the chebyshev functions, Ricerche Mat. \bf65 (2016), 307-328. · Zbl 1355.65180
[12] J.D. Ramirez and A. Vatsala, Generalized monotone iterative techniques for Caputo fractional integro-differential equations with initial condition, Neural Parallel Sci. Comput. \bf23 (2015), 219-238.
[13] K.G. TeBeest, Classroom note: numerical and analytical solutions of Volterra’s population model, SIAM Review \bf39 (1997), no. 3, 484-493, doi 10.1137/S0036144595294850. · Zbl 0892.92020
[14] G. Wang, S. Liu and L. Zhang, Neutral fractional integro-differential equation with nonlinear term depending on lower order derivative, J. Comp. Appl. Math. \bf260 (2014), 167-172. · Zbl 1293.45005
[15] Y. Wang and L. Zhu, Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method, Adv. Diff. Eqn. (2017), paper 27. · Zbl 1422.45001
[16] S. Yuzbasi, A numerical approximation for Volterra’s population growth model with fractional order, Appl. Math. Model. 37 (2013), no. 5, 3216-3227. · Zbl 1352.65657
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.