# A novel solution for fractional chaotic Chen system

Volume 8, Issue 5, pp 478--488
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### Authors

A. K. Alomari - Department of Mathematics, Faculty of Science, Yarmouk University, 211-63 Irbid, Jordan.

### Abstract

A novel solution to the fraction chaotic Chen system is presented in this paper by using the step homotopy analysis method. This method yields a continuous solution in terms of a rapidly convergent infinite power series with easily computable terms. Moreover, the residual error of the SHAM solution is defined and computed for each time interval. Via the computing of the residual error we observe that the accuracy of the present method tends to $10^{-11}$ which is very high.

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##### ISRP Style

A. K. Alomari, A novel solution for fractional chaotic Chen system, Journal of Nonlinear Sciences and Applications, 8 (2015), no. 5, 478--488

##### AMA Style

Alomari A. K., A novel solution for fractional chaotic Chen system. J. Nonlinear Sci. Appl. (2015); 8(5):478--488

##### Chicago/Turabian Style

Alomari, A. K.. "A novel solution for fractional chaotic Chen system." Journal of Nonlinear Sciences and Applications, 8, no. 5 (2015): 478--488

### Keywords

• Chaotic system
• fractional Chen system
• homotopy analysis method
• step homotopy analysis method
• residual error.

•  65P20
•  26A33
•  34A08

### References

• [1] S. Abbasbandy, The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation, Phys. Lett. A, 361 (2007), 478–483.

• [2] A. K. Alomari, M. S. M. Noorani, R. Nazar, On the homotopy analysis method for the exact solutions of Helmholtz equation, Chaos Solitons Fractal, 41 (2009), 1873–1879.

• [3] A. K. Alomari, F. Awawdeh, N. Tahat, F. Bani Ahmad, W . Shatanawi , Multiple solutions for fractional differential equations: Analytic approach, Appl. Math. Comput., 219 (2013), 8893–8903.

• [4] A. K. Alomari, M. S. N. Noorani, R. Nazar, Adaptation of homotopy analysis method for the numeric-analytic solution of Chen system, Comm. Nonlinear Sci. Numer. Simul., 14 (2009), 2336–2346.

• [5] A. K. Alomari, M. S. N. Noorani, R. Nazar, C. P. Li, Homotopy analysis method for solving fractional Lorenz system, Comm. Nonlinear Sci. Numer. Simul., 15 (2010), 1864–1872.

• [6] M. A. F. Araghi, A. Fallahzadeh, Discrete Homotopy Analysis Method for Solving Linear Fuzzy Differential Equations, Adv. Environ. Biol., 9 (2015), 195–201.

• [7] A. S. Bataineh, M. S. N. Noorani, I. Hashim, Solving systems of ODEs by homotopy analysis method , Comm. Nonlinear Sci. Numer. Simul., 13 (2008), 2060–2070.

• [8] M. S. H. Chowdhury, I. Hashim, Application of multistage homotopy-perturbation method for the solutions of the Chen system , Nonlinear Anal. Real. World Appl., 10 (2009), 381–391.

• [9] K. Diethelm, N. J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), 229–248.

• [10] K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3–22.

• [11] A. Fallahzadeh, K. Shakibi, A method to solve Convection-Diffusion equation based on homotopy analysis method, J. Interpolat. Approx. Sci. Comput., 2015 (2015), 8 pages.

• [12] A. Golbabai, K. Sayevand , An efficient applications of hes variational iteration method based on a reliable modification of Adomian algorithm for nonlinear boundary value problems , J. Nonlinear Sci. Appl., 3 (2010), 152 – 156.

• [13] T. Hayat, M. Sajid, On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder, Phys. Lett. A, 361 (2007), 316–322.

• [14] T. Hayat, M. Khan, Homotopy solutions for a generalized second-grade fluid past a porous plate, Nonlinear Dyn., 42 (2005), 395–405.

• [15] H. Jafari, V. Daftardar-Gejji , Solving a system of nonlinear fractional differential equations using Adomian decomposition, J. Comput. Appl. Math., 196 (2006), 644–651.

• [16] C. Li C, G. Peng, Chaos in Chen’s system with a fractional order, Chaos Solitons Fractals, 22 (2004), 443–450.

• [17] S. Liao, The proposed homotopy analysis techniques for the solution of nonlinear problems, Ph.D. Dissertation. Shanghai Jiao Tong University, Shanghai (in English) (1992)

• [18] S. Liao, Beyond perturbation: Introduction to the homotopy analysis method, CRC Press, Chapman and Hall, Boca Raton (2003)

• [19] S. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 983–997.

• [20] Y. Liu Y, H. Shi, Existence of unbounded positive solutions for BVPs of singular fractional differential equations, J. Nonlinear Sci. Appl., 5 (2012), 281–293.

• [21] S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A., 365 (2007), 345–350.

• [22] S. Momani, Z. Odibat, Numerical comparison of methods for solving linear differential equations of fractional order, Chaos Solitons Fractals, 31 (2007), 1248–1255.

• [23] M. Inc, On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method, Phys. Lett. A., 365 (2007), 412–415.

• [24] Z. Odibat, S. Momani, Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals, 36 (2008), 167–174.

• [25] Z. Odibat, S. Momani, Application of variational iteration method to nonlinear differential equation of fractional order , Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 27–34.

• [26] I. Podlubny, Fractional differential equations, Academic Press, New York (1999)

• [27] T. Qiu, Z. Bai , Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Nonlinear Sci. Appl., 1 (2008), 123–131.

• [28] M. Sajid, T. Javed, T. Hayat, MHD rotating flow of a viscous fluid over a shrinking surface, Nonlinear Dyn., 51 (2008), 259–265.

• [29] N. T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., 131 (2002), 517–529.

• [30] T. Wang, F. Xie, Existence and uniqueness of fractional differential equations with integral boundary conditions, J. Nonlinear Sci. Appl., 1 (2008), 206–212.

• [31] Y. Wang, Y. Yang, Positive solutions for Caputo fractional differential equations involving integral boundary conditions, J. Nonlinear Sci. Appl., 8 (2015), 99–109.

• [32] J. Wanga, X. Xionga, Y. Zhang, Extending synchronization scheme to chaotic fractional-order Chen systems, Physica A, 370 (2006), 279–285.

• [33] W. Yang, Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations, J. Nonlinear Sci. Appl., 8 (2015), 110–129.

• [34] T. S. Zhou, C. Li, Synchronization in fractional-order differential systems, Phys. D., 212 (2005), 111–125.

• [35] H. Zhu, S. Zhou, Z. He, Chaos synchronization of the fractional-order Chen’s system, Chaos Solitons Fractals, 41 (2009), 2733–2740.