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Analytical and numerical results of fractional differential-difference equations. (English) Zbl 1332.37066

Summary: In this paper, we examine the fractional differential-difference equation (FDDE) by employing the proposed sensitivity approach (SA) and Adomian transformation method (ADTM). In SA, the nonlinear differential-difference equation is converted to infinite linear equations which have a wide criterion to solve for the analytical solution. By ADTM, the FDDE is converted into ordinary differential-difference equation that can be solved. We test both the techniques through some test problems which are arising in nonlinear dynamical systems and found that ADTM is equivalently appropriate and simpler method to handle than SA.

MSC:

37N30 Dynamical systems in numerical analysis
34A08 Fractional ordinary differential equations
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[1] E. Fermi, J. Pasta, S. Ulam, Collected Papers of Enrico Fermi, Chicago Press, volume 2, page 978, Chicago Press, 1965.
[2] P. Marquié, J. M. Bilbault, M. Rernoissnet, Observation of nonlinear localized modes in an electrical lattice, Phys. Rev. E 51 (1995), 6127-6133.
[3] A. S. Davydov, The theory of contraction of proteins under their excitation, J. Theoret. Biol., 38 (1973), 559-569.
[4] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego - New York - London - Tokyo - Toronto, 1999. · Zbl 0924.34008
[5] J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1140-1153. · Zbl 1221.26002
[6] J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado, Advances in Fractional Calculus, Springer, Netherland, 2007. · Zbl 1116.00014
[7] L. Decreusefond, A. S. Ustunel, Stochastic analysis of the fractional Brownian motion, Potential Anal., 10 (1999), 177-214. · Zbl 0924.60034
[8] T. E. Duncan, Y. Hu, B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion, I. Theory, SIAM J. Control Optim., 38 (2000), 582-612. · Zbl 0947.60061
[9] M. S. El Naschie, A review of E infinity theory and the mass spectrum of high energy particle physics, Chaos Solitons Fractals, 19 (2004), 209-236. · Zbl 1071.81501
[10] G. Jumarie, A Fokker-Planck equation of fractional order with respect to time, J. Math. Phys., 33 (10) (1992), 3536-3542. · Zbl 0761.60071
[11] E. Barkai, Fractional Fokker-Planck equation, solutions and applications, Phys. Rev. E, 63 (2001), 1-17.
[12] V. V. Anh, N. N. Leonenko, Scaling laws for fractional diffusion-wave equations with singular initial data, Statist. Probab. Lett., 48 (2000), 239-252. · Zbl 0970.35174
[13] A. Hanygad, Multidimensional solutions of time-fractional diffusion-wave equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. A 458 (2002), 933-957.
[14] S. Abbasbandy, Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomians decomposition method, Appl. Math. Comput., 172 (1) (2006), 485-490. · Zbl 1088.65063
[15] Z. Odibat, S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals, 36 (1) (2008), 167-174. · Zbl 1152.34311
[16] N. A. Khan, A. Ara, M. Jamil, An efficient approach for solving the Riccati equation with fractional orders, Comput. Math. Appl., 61 (2011), 2683-2689. · Zbl 1221.65205
[17] N. A. Khan, M. Jamil, A. Ara, N. U. Khan, On efficient method for system of fractional differential equations, Adv. Differential Equations, 2011 Article ADE/303472. · Zbl 1217.65134
[18] J. Cang, Y. Tan, H. Xu, S. Liao, Series solutions of non-linear Riccati differential equations with fractional order, Chaos Solitons Fractals, 40 (2009), 1-9. · Zbl 1197.34006
[19] Y. Tan, S. Abbasbany, Homotopy analysis method for quadratic Riccati differential equation, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 539-546. · Zbl 1132.34305
[20] M. Gülsu, M. Sezer, On the solution of the Riccati equation by the Taylor matrix method, Appl. Math. Comput., 176 (2) (2006), 414-421. · Zbl 1093.65072
[21] G. Darmani, S. Setayeshi, H. Ramezanpour, Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach, Commun. Theor. Phys. (Beijing), 57 (1) (2012), 5-9. · Zbl 1247.34019
[22] G. Y. Tang, Z. W. Luo, Suboptimal control of linear systems with state time-delay, In: Systems, Man, and Cybernetics, 1999. IEEE SMC’99 Conference Proceedings, 99 (5) (1999), 104-109.
[23] M. Malek-Zavarei, M. Jamshidi, Time-Delay Systems: Analysis, Optimization and Applications, North-Holland Systems and Control Series, Elsevier Science Ltd, 1987. · Zbl 0658.93001
[24] J. H. He, Z. B., Li, Applications of the franctional complex transform to franctional differential equations, Nonlinear Science Letters A, 2 (2011), 121-126.
[25] J. H. He, S. K. Elagan, Z. B. Li, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376 (2012), 257-259. · Zbl 1255.26002
[26] J. S. Duan, An efficient algoritm for the multivariable adomian polynomials, Appl. Math. Comput., 217 (2010), 2456-2467. · Zbl 1204.65022
[27] J. S. Duan, Convenienet analytic recurrence algorithms for the adomian polynomials, Appl. Math. Comput., 217 (2011), 6337-6348. · Zbl 1214.65064
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