Approximate analytical solution of linear and nonlinear fractional delay differential equations using new variational iteration method

Vineet Srivastava


In this paper, an approximate analytical method, New Variational Iteration Method (NVIM) is introduced in this paper for the approximate analytical solution of Fractional Delay Differential Equations (FDDE). The algorithm is illustrated by studying initial value linear and nonlinear problems. The results obtained are presented and show that only few terms are required to get an approximate solution.


Fractional Differential Equation; Delay differential equation; Variational Iteration method

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M. Jamshidi , C.M. Wang , A computational algorithm for large-scale nonlinear time-delays systems, IEEE Trans. Syst. Man Cybern. 14 (1984) 2–9 .

M. Lakshmanan , D.V. Senthilkumar , Dynamics of nonlinear time-delay systems, Springer Series Synergetics, 2010 .

G. Adomian , R. Rach , Nonlinear stochastic differential delay equations, J. Math. Anal. Appl. 91 (1983) 94–101

J. Baranowski , Legendre polynomial approximations of time delay systems, in: Proceeding of 12th International PhD Workshop, 1, 2010, pp. 15–20 .

F.M. Asl , A.G. Ulsoy , Analysis of a system of linear delay differential equations, J. Dyn. Syst. Meas. Control 125 (2) (2003) 215–223 .

X.T. Wang , Numerical solution of delay systems containing inverse time by hybrid functions, Appl. Math. Comput. 173 (1) (2006) 535–546 .

Y. Takahashi , M.J. Rabins , D.M. Auslander , Control and dynamic systems, Addison Wesley, Massachusetts, 1970 .

J.K. Hale , Theory of functional differential equations, Springer-Verlag, New York, 1977 .

A. C. J. Luo and V. Afraimovich (Eds.), Long – range Interaction, Stochasticity and Fractional Dynamics, New York, Springer, 2010.

N. Laskin and G. M. Zaslavsky, Nonlinear fractional dynamics on a lattice with long –range interactions, Physica A 368, 38-4, 2006.

G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford, Oxford University Press, 2005.

Daftardar-Gejji V (2014) Fractional calculus: theory and applications. Narosa Publishing House, New Delhi.

Epstein IR, LuoY(1991) Differential delay equations in chemical kinetics.Nonlinear models: the cross-shaped phase diagram and the oregonator. J Chem Phys 95(1):244–254

Davis L (2003) Modifications of the optimal velocity traffic model to include delay due to driver reaction time. Phys A Stat Mech Appl 319:557–567

Fridman E, Fridman L, Shustin E (2000) Steady modes in relay control systems with time delay and periodic disturbances. J Dyn Syst Meas Control 122(4):732–737

Kuang Y (1993) Delay differential equations: with applications in population dynamics, vol 191. Academic Press, London

Bhalekar S, Daftardar-Gejji V, Baleanu D, Magin R (2011) Fractional Bloch equation with delay. Comput Math Appl 61(5):1355–1365

Baleanu D, Magin RL, Bhalekar S, Daftardar-Gejji V (2015) Chaos in the fractional order nonlinear Blochequation with delay. Commun Nonlinear Sci Numer Simul 25(1–3):41–49

Willé DR, Baker CT (1992) DELSOL—a numerical code for the solution of systems of delay-differential equations. Appl Numer Math 9(3–5):223–234

Maraaba T, Baleanu D, Jarad F (2008a) Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives. J Math Phys 49(8):083507

Maraaba TA, Jarad F,BaleanuD(2008b) On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives. Sci China Ser A Math 51(10):1775–1786

MorgadoML, Ford NJ, Lima PM(2013) Analysis and numerical methods for fractional differential equations with delay. J Comput Appl Math 252:159–168

Wang Z, Huang X, Shi G (2011) Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay. Comput Math Appl 62(3):1531–1539

J.H. He, Variational iteration method for delay differential equations, Commun. Nonlinear Sci. Numer. Simul. 2 (4) (1997) 235–236.

J.H. He, Variational iteration method—A kind of nonlinear analytical technique: Some examples, Internat. J. Non-Linear Mech. 34 (1999) 699–708.

Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul. 1 (7) (2006) 15–27.

A. M. Wazwaz, The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations, Int. J. Computers and Mathematics with Applications, 54 (2007), pp. 933–939.

A.M. Wazwaz, A comparison between the variational iteration method and Adomian decomposition method, Journal of Computational and Applied Mathematics, 207 (1) (2007) 129–136.

I. Podlubny, Fractional differential equations, Academic Press, San Diego, CA, 1999.

M. Caputo, Linear models of dissipation whose Q is almost frequency independent. Part II, J. Roy. Astr. Soc. 13 (1967) 529–539.

El-Safty, A., Salim, M. S. and El-Khatib, M. A. (2003). Convergent of the spline functions for delay dynamic system. Int. J. Comput. Math., 80(4), 509–518.

Shadia, M. (1992). Numerical solution of delay differential and neutral differential equations using spline methods. Ph. D Thesis, Assuit University.

El-Safty,A. (1993). Approximate solution of the delay differential equation y = f (x, y(x), y(α(x))) with cubic spline functions. Bull. Fac. Sci., Assuit Univ., 22(2-c), 67–73.

Ibrahim, M. A.-K., EI-Safty, A. and Abo-Hasha, S. M. (1995). 2h-step spline method for the solution of delay differential equations. Comput. Math. Appl., 29(8), 1–6.

EI-Safty, A. and Abo-Hasha, S. M. (1990). On the application of spline functions to initial value problem with retarded argument. Int. J. Comput. Math., 32, 137–179.

Daftardar-GejjiV, SukaleY, Bhalekar S (2015) Solving fractional delay differential equations: a newapproach. Fract Calc Appl Anal 18(2):400–418

Jhinga A. and Daftardar-Gejji V. (2019) A new numericalmethod for solving fractional delay differential equations, Computational and Applied Mathematics, 38(166)



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