Sains Malaysiana 42(3)(2013): 347–358

A Step Variational Iteration Method for Solving Non-chaotic and Chaotic Systems

(Kaedah Lelaran Ubahan Langkah bagi Menyelesaikan Sistem Kalut dan Tak Kalut)

R. Yulita Molliq* *

Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Negeri Medan

20221 Medan, Sumatera Utara, Indonesia

M.S.M. Noorani & R.R. Ahmad

School of Mathematical Sciences, Faculty of Science and Technology

Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor D.E. Malaysia

A.K. Alomari,

Department of Mathematics, Faculty of Science, Hashemite University 13115 Zarqa, Jordan

Received: 25 May 2010/Accepted: 17 September 2012

ABSTRACT

In this paper, a new reliable method called the step variational iteration method (SVIM) based on an adaptation of the variational iteration method (VIM) is presented to solve non–chaotic and chaotic systems. The SVIM uses the general Lagrange multipliers for constructing the correction functional for the problems. The SVIM yields a step analytical solution of the form of a rapidly convergent infinite power series with easily computable terms and obtain a good approximate solution for larger intervals. The accuracy of the presented solution obtained is in an excellent agreement with the previously published solutions.

Keywords: Chaotic and non-chaotic systems; Lagrange multiplier; multistage variational iteration method; step variational iteration method; variational iteration method

ABSTRAK

Dalam kertas ini, kaedah baru dinamakan kaedah lelaran ubahan langkah (KLUL) berasaskan satu adaptasi kaedah lelaran ubahan digunakan untuk sistem tak-kalut dan kalut. KLUL menggunakan pendarab Lagrange umum untuk membina fungsian pembetulan bagi mengatasi masalah berkenaan. KLUL menghasilkan penyelesaian analisis dalam bentuk siri kuasa tak terhingga yang menumpu pantas dengan sebutan yang mudah dikira. Penyelesaian penghampiran diperoleh adalah baik untuk selang yang lebih besar. Ketepatan penyelesaian yang diperoleh adalah sangat baik bila dibandingkan dengan penyelesaian yang terdahulu.

Kata kunci: Kaedah lelaran ubahan; kaedah lelaran ubahan langkah; kaedah lelaran ubahan multitahap; sistem tak-kalut dan kalut; pendarab Lagrange

REFERENCES

Alomari, A.K., Noorani, M.S.M. & Nazar, R. 2009. Adaptation of homotopy analysis method for the numeric-analytic solution of Chen system. Commun. Nonlinear Sci. Numer. Simulat. 14: 2336-2346.

Alomari, A.K., Noorani, M.S.M. & Nazar, R. & Li, C.P. 2010. Homotopy analysis method for solving fractional Lorenz system. Commun. Nonlinear Sci. Numer. Simulat. 15(7): 1864-1872.

Batiha, B., Noorani, M.S.M. & Hashim, I. 2007a. Variational iteration method for solving multispecies Lotka–Volterra equations. Comput. Math. Appl. 54: 903-909.

Batiha, B., Noorani, M.S.M. & Hashim, I. 2007b. The multistage variational iteration method for class of nonlinear system of ODEs. Phys. Scr. 78: 388-392.

Chowdhury, M.S.H. & Hashim, I. 2009. Application of multistage homotopy-perturbation method for the solutions of the Chen system. Nonlinear Anal. Real World Anal. 10(1): 381-391.

Chowdhury, M.S.H., Hashim, I. & Momani, S. 2009. The multistage homotopy-perturbation method: A powerful scheme for handling the Lorenz system. Chaos Solitons & Fractals 40(4): 1929-1937.

Faraz, N., Khan, Y. & Austin, F. 2010. An alternative approach to differential-difference equations using the variational iteration method, Zeitschrift fur Naturforschung - Section A J. Phys. Sci. 65(12): 1055-1059.

Faraz, N., Khan, Y. & Yildirim, A. 2011. Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II. J. King Saud University (Sci.) 23: 77-81.

Finlayson, B.A. 1972. The Method of Weighted Residuals and Variational Principles. New York: Academic Press.

Genesio, R. & Tesi, A. 1992. A harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28: 531-548.

Goh, S.M., Noorani, M.S.M. & Hashim, I. 2009a. Efficiency of variational iteration method for chaotic Genesio system-Classical and multistage approach. Chaos Solitons & Fractals 40: 2152-2159.

Goh, S.M., Noorani, M.S.M. & Hashim, I. 2009b. A new application of variational iteration method for the chaotic Rössler system. Chaos Solitons Fractals 42(3): 1604-1610.

Goh, S.M., Noorani, M.S.M. & Hashim, I. 2008. Prescribing a multistage analytical method to a prey-predator dynamical system. Phys. Lett. A 373: 107-110.

He, J.H. 1997a. Semi-inverse method of establishing generalized principles for fluid mechanics with emphasis on turbomachinery aerodynamics. Int. J. Turbo Jet–Engines 14: 23-28.

He, J.H. 1997b. Variational iteration method for delay differential equations. Commun. Nonlinear Sci. Numer. Simulat. 2: 235-236.

He, J.H. 1998. Approximate analytical solutions for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Engrg. 167(1-2): 57-68.

He, J.H. & Wu, X.H. 2007. Variational iteration method: New developments and applications. Comput. Math. Appl. 54: 881-894.

He, J.H. & Wu, X.H. 2006. Construction of solitary solution and compaction-like solution by variational iteration method. Chaos, Solitons & Fractals 29(1): 108-113.

He, J.H. 2007. The variational iteration method for eighth-order initial boundary value problems, Phys. Scr. 76(6): 680-682.

Inokuti, M., Sekine, H. & Mura, T. 1978. General use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Naseer, S. (ed.) Variational Method in the Mechanics of Solids New York: Pergamon Press. pp. 156-162.

Ivancevic, V.G. & Ivancevic, T.T. 2008. Complex Nonlinearity: Chaos, Phase Transitions, Topology Change, and Path Integrals. New York: Springer.

Khan, Y., Faraz, N., Yildirim, A. & Wu, Q. 2011. Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science. Comput. Math. Appl. 62: 2273-2278.

Noorani, M.S.M., Hashim, I., Ahmad, R., Bakar, S.A., Ismail, E.S. & Zakaria, A.M. 2007. Comparing numerical methods for the solutions of the Chen system. Chaos Solitons & Fractals 32: 1296-304.

Rangkuti, Y.M. & Noorani, M.S.M. 2012. The exact solution of delay differential equations using coupling variational iteration with taylor series and small term. Bull. Math. 4(1): 1-15.

Rössler, O.E. 1976. An equation for continuous chaos. Phys Lett A. 573: 97-398.

Serletis, A. & Gogas, P. 1997. Chaos in east European black market exchange rates. Research Economics 51(4): 359-385.

Serletis, A. & Gogas, P. 2000. Purchasing power parity nonlinearity and chaos. Appl. Financial Economics 10(6): 615-622.

Serletis, A. & Gogas, P. 1999. The north American gas markets are chaotic. Energy J. 20: 83-103.

Werndl, C. 2009. What are the new implications of chaos for unpredictability? The British J. Phil. Sci. 60(1): 195-220.

Yulita Molliq, R., Noorani, M.S.M. & Hashim, I. 2009a. Variational iteration method for fractional heat–and wave–like equations. Nonlinear Anal. Real World Appl. 10(3): 1854-1869.

Yulita Molliq, R., Noorani, M.S.M., Hashim, I. & Ahmad, R.R. 2009b. Variational iteration method for fractional Zakhanov–Kuznetsov equations. J. Comput. Appl. Math. 233: 103-108.

Yulita Molliq, R. & Batiha, B. 2012. Approximate analytic solutions of fractional Zakharov-Kuznetsov equations by fractional complex transform. Int. J. Engin. Tech. 1(1): 1-13.

*Corresponding author; email: yulitamolliq@yahoo.com