×

Numerical studies for fractional-order logistic differential equation with two different delays. (English) Zbl 1251.65118

Summary: A numerical method for solving the fractional-order logistic differential equation with two different delays (FOLE) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FOLE to a system of algebraic equations. Special attention is given to study the convergence and the error estimate of the presented method. Numerical illustrations are presented to demonstrate utility of the proposed method. Chaotic behavior is observed and the smallest fractional order for the chaotic behavior is obtained. Also, FOLE is studied using variational iteration method (VIM) and the fractional complex transform is introduced to convert the fractional logistic equation to its differential partner, so that its variational iteration algorithm can be simply constructed. Numerical experiment is presented to illustrate the validity and the great potential of both proposed techniques.

MSC:

65L99 Numerical methods for ordinary differential equations
34A08 Fractional ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] I. R. Epstein and Y. Luo, “Differential delay equations in chemical kinetics. Nonlinear models: the cross-shaped phase diagram and the Oregonator,” The Journal of Chemical Physics, vol. 95, no. 1, pp. 244-254, 1991.
[2] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993. · Zbl 0777.34002
[3] E. Fridman, L. Fridman, and E. Shustin, “Steady modes in relay control systems with time delay and periodic disturbances,” Journal of Dynamic Systems, Measurement and Control, vol. 122, no. 4, pp. 732-737, 2000.
[4] L. C. Davis, “Modifications of the optimal velocity traffic model to include delay due to driver reaction time,” Physica A, vol. 319, pp. 557-567, 2003. · Zbl 1008.90007
[5] S. Bhalekar and V. Daftardar-Gejji, “A predictor-corrector scheme for solving nonlinear delay dierential equations of fracctional order,” Journal of Fractional Calculus and Applications, vol. 1, no. 5, pp. 1-9, 2011. · Zbl 1238.34095
[6] R. L. Bagley and P. J. Torvik, “On the appearance of the fractional derivative in the behavior of real materials,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 294-298, 1984. · Zbl 1203.74022
[7] K. Diethelm, “An algorithm for the numerical solution of differential equations of fractional order,” Electronic Transactions on Numerical Analysis, vol. 5, pp. 1-6, 1997. · Zbl 0890.65071
[8] A. M. A. El-Sayed, H. A. A. El-Saka, and E. M. El-Maghrabi, “On the fractional-order logistic equation with two different delays,” Zeitschrift fur Naturforschung. Section A, vol. 66, no. 3-4, pp. 223-227, 2011.
[9] J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57-68, 1998. · Zbl 0942.76077
[10] Ch. Lubich, “Discretized fractional calculus,” SIAM Journal on Mathematical Analysis, vol. 17, no. 3, pp. 704-719, 1986. · Zbl 0624.65015
[11] N. H. Sweilam, M. M. Khader, and A. M. Nagy, “Numerical solution of two-sided space-fractional wave equation using finite difference method,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2832-2841, 2011. · Zbl 1209.65089
[12] N. H. Sweilam, M. M. Khader, and A. M. S. Mahdy, “Crank-Nicolson nite dierence method for solving time-fractional diusion equation,” Journal of Fractional Calculus and Applications, vol. 2, no. 2, pp. 1-9, 2012.
[13] J. H. He, “Variational iteration method-a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699-708, 1999. · Zbl 1342.34005
[14] J. H. He, “Variational iteration method for delay dierential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 2, pp. 235-236, 1997.
[15] J.-H. He, “A short remark on fractional variational iteration method,” Physics Letters. A, vol. 375, no. 38, pp. 3362-3364, 2011. · Zbl 1252.49027
[16] M. Inc, “The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 476-484, 2008. · Zbl 1146.35304
[17] N. H. Sweilam and M. M. Khader, “Variational iteration method for one dimensional nonlinear thermoelasticity,” Chaos, Solitons and Fractals, vol. 32, no. 1, pp. 145-149, 2007. · Zbl 1131.74018
[18] N. H. Sweilam and M. M. Khader, “On the convergence of variational iteration method for nonlinear coupled system of partial differential equations,” International Journal of Computer Mathematics, vol. 87, no. 5, pp. 1120-1130, 2010. · Zbl 1201.65165
[19] N. H. Sweilam, M. M. Khader, and F. T. Mohamed, “On the numerical solutions of two dimensional Maxwell’s equations,” Studies in Nonlinear Sciences, vol. 1, no. 3, pp. 82-88, 2010.
[20] N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numerical studies for a multi-order fractional differential equation,” Physics Letters. A, vol. 371, no. 1-2, pp. 26-33, 2007. · Zbl 1209.65116
[21] H. Jafari and V. Daftardar-Gejji, “Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 488-497, 2006. · Zbl 1102.65135
[22] I. Hashim, O. Abdulaziz, and S. Momani, “Homotopy analysis method for fractional IVPs,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 674-684, 2009. · Zbl 1221.65277
[23] Z.-B. Li and J.-H. He, “Fractional complex transform for fractional differential equations,” Mathematical & Computational Applications, vol. 15, no. 5, pp. 970-973, 2010. · Zbl 1215.35164
[24] Z. B. Li and J. H. He, “Application of the fractional complex transform to fractional dierential equations,” Nonlinear Science Letters A, vol. 2, pp. 121-126, 2011.
[25] M. M. Khader, “On the numerical solutions for the fractional diffusion equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 6, pp. 2535-2542, 2011. · Zbl 1221.65263
[26] M. M. Khader, “Numerical solution of nonlinear multi-order fractional dierential equations by implementation of the operational matrix of fractional derivative,” Studies in Nonlinear Sciences, vol. 2, no. 1, pp. 5-12, 2011.
[27] M. M. Khader, “Introducing an efficient modication of the variational iteration method by using Chebyshev polynomials,” Application and Applied Mathematics, vol. 7, no. 1, pp. 283-299, 2012. · Zbl 1245.65088
[28] M. M. Khader, “Introducing an efficient modification of the homotopy perturbation method by using Chebyshev polynomials,” Arab Journal of Mathematical Sciences, vol. 18, no. 1, pp. 61-71, 2012. · Zbl 1236.65079
[29] M. M. Khader, N. H. Sweilam, and A. M. S. Mahdy, “An efficient numerical method for solving the fractional diffusion equation,” Journal of Applied Mathematics and Bioinformatics, vol. 1, pp. 1-12, 2011. · Zbl 1270.65048
[30] M. M. Khader and A. S. Hendy, “The approximate and exact solutions of the fractional-order delay dierential equations using Legendre pseudospectral method,” International Journal of Pure and Applied Mathematics, vol. 74, no. 3, pp. 287-297, 2012. · Zbl 1246.34064
[31] E. A. Rawashdeh, “Numerical solution of fractional integro-differential equations by collocation method,” Applied Mathematics and Computation, vol. 176, no. 1, pp. 1-6, 2006. · Zbl 1106.65111
[32] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[33] S. Das, Functional Fractional Calculus, Springer, New York, NY, USA, 2nd edition, 2011. · Zbl 1243.91052
[34] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, London, UK, 1993. · Zbl 0818.26003
[35] M. A. Snyder, Chebyshev Methods in Numerical Approximation, Prentice-Hall, Englewood Cliffs, NJ, USA, 1966. · Zbl 0173.44102
[36] E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2364-2373, 2011. · Zbl 1231.65126
[37] G. Jumarie, “Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution,” Journal of Applied Mathematics & Computing, vol. 24, no. 1-2, pp. 31-48, 2007. · Zbl 1145.26302
[38] G. Jumarie, “Cauchy’s integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order,” Applied Mathematics Letters, vol. 23, no. 12, pp. 1444-1450, 2010. · Zbl 1202.30068
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.