A new piecewise-quasilinearization method for solving chaotic systems of initial value problems

• Authors: Sandile Motsa
• Languages of publication: EN

Abstracts

EN

In this paper, a modification of the successive linearization method (SLM) for solving nonlinear initial value problems is introduced for the first time. The proposed method is based on a novel technique of extending the standard SLM and adapting it to a sequence of multiple intervals. In this new application the method is referred to as the piecewise successive linearization method(PSLM). This new algorithm is applied to chaotic and non-chaotic differential equations that model the Lotka-Volterra, Lorenz, Rössler and Genesio-Tesi systems. A comparative study between the new algorithm and the MATLAB Runge-Kutta based in-built solver (ode45) method is presented. The results demonstrate accuracy and reliability of the proposed PSLM algorithm.

Keywords

EN

successive linearization method; Lotka-Voltera; Rössler attactor; Genisio-Tesi system; Lorenz system; chaotic system

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2012
• Volume: 10
• Issue: 4
• Pages: 936-946

Dates

published

1 - 8 - 2012

online

17 - 7 - 2012

Contributors

author

Sandile Motsa

• School of Mathematical Sciences, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg, 3209, South Africa

References

• [1] O. Abdulaziz, N. F. M. Noor, I. Hashim, M. S. M. Noorani, Chaos Soliton. Fract. 36, 1405 (2008) http://dx.doi.org/10.1016/j.chaos.2006.09.007[Crossref]
• [2] A. K Alomari, M.S.M. Noorani, R. Nazar, Phys. Scr. 81, 045005 (2010) http://dx.doi.org/10.1088/0031-8949/81/04/045005[Crossref]
• [3] A. K. Alomari, Comput. Math. Appl. 61, 2528 (2011) http://dx.doi.org/10.1016/j.camwa.2011.02.043[Crossref]
• [4] A. K. Alomari, M. S. M. Noorani, R. Nazar, Commun. Nonlinear Sci. 14, 2336 (2009) http://dx.doi.org/10.1016/j.cnsns.2008.06.011[Crossref]
• [5] A. K. Alomari, M. S. M. Noorani, R. Nazar, C. P. Li, Commun. Nonlinear Sci. 15, 1864 (2010) http://dx.doi.org/10.1016/j.cnsns.2009.08.005[Crossref]
• [6] F. G. Awad, P. Sibanda, S. S. Motsa, O. D. Makinde, Comput. Math. Appl. 61, 1431 (2011) http://dx.doi.org/10.1016/j.camwa.2011.01.015[Crossref]
• [7] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics, (Springer-Verlag, Berlin, 1988)
• [8] G. Chen, T. Ueta, Int. J. Bifurcat. Chaos 9(7), 1465 (1999) http://dx.doi.org/10.1142/S0218127499001024[Crossref]
• [9] M. S. H. Chowdhury, I. Hashim, Nonlinear Anal. Real. 10, 381 (2009) http://dx.doi.org/10.1016/j.nonrwa.2007.09.014[Crossref]
• [10] J. R. Dormand, P. J. Prince, J. Comput. Appl. Math. 6(1), 19 (1980) http://dx.doi.org/10.1016/0771-050X(80)90013-3[Crossref]
• [11] M. R. Faieghi, H. Delavari, Commun. Nonlinear Sci. 17, 731 (2012) http://dx.doi.org/10.1016/j.cnsns.2011.05.038[Crossref]
• [12] R. Genesio, A. Tesi, Automatica 28, 531 (1992) http://dx.doi.org/10.1016/0005-1098(92)90177-H[Crossref]
• [13] A. Ghorbani, J. Saberi-Nadjafi, Math. Comput. Model. 54, 131 (2011) http://dx.doi.org/10.1016/j.mcm.2011.01.044[Crossref]
• [14] S.M. Goh, M. S. M. Noorani, I. Hashim, Numer. Algorithms 54, 245 (2010) http://dx.doi.org/10.1007/s11075-009-9333-9[Crossref]
• [15] A. Gökdogan, M. Merdan, A Yildirim, Commun. Nonlinear Sci. 17, 45 (2012) http://dx.doi.org/10.1016/j.cnsns.2011.03.039[Crossref]
• [16] I. Hashim, M. S. M. Noorani, R. Ahmad, S. A. Bakar, E. S. Ismail, A. M. Zakaria, Chaos Soliton. Fract. 28, 1149 (2006) http://dx.doi.org/10.1016/j.chaos.2005.08.135[Crossref]
• [17] E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963) http://dx.doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2[Crossref]
• [18] Z. G. Makukula, P. Sibanda, S. S. Motsa, Bound. Value Probl., 471793 (2010)
• [19] M. Merdan, A. Gokdogan and V. S. Erturk, Iran. J. Sci. Technol. A1, 9 (2011)
• [20] M. Mossa Al-Sawalha, M. S. M. Noorani, I. Hashim, Chaos Soliton. Fract. 40, 1801 (2009) http://dx.doi.org/10.1016/j.chaos.2007.09.062[Crossref]
• [21] S. S. Motsa, Int. J. Mod. Sim. Sci. Comp. 2, 355 (2011) http://dx.doi.org/10.1142/S1793962311000499[Crossref]
• [22] S. S. Motsa, P. Sibanda, S. Shateyi, Math. Method. Appl. Sci. 34, 1406 (2011) http://dx.doi.org/10.1002/mma.1449[Crossref]
• [23] S. S. Motsa, P. Sibanda, Comput. Math. Appl. 63, 1197 (2012) http://dx.doi.org/10.1016/j.camwa.2011.12.035[Crossref]
• [24] Z. M. Odibat, C. Bertelle, M. A. Aziz-Alaoui, G. H. E. Duchamp, Comput. Math. Appl. 59, 1462 (2010) http://dx.doi.org/10.1016/j.camwa.2009.11.005[Crossref]
• [25] J. I. Ramos, Appl. Math. Comput. 198, 92 (2008) http://dx.doi.org/10.1016/j.amc.2007.08.030[Crossref]
• [26] O. E. Rössler, Phys. Lett. A 57, 397 (1976) http://dx.doi.org/10.1016/0375-9601(76)90101-8[Crossref]
• [27] S. Shateyi, S.S. Motsa, Bound. Value Probl., Article ID 257568 (2010)
• [28] N. T. Shawagfeh, G. Adomian, Appl. Math. Comput 76, 251 (1996) http://dx.doi.org/10.1016/0096-3003(95)00162-X[Crossref]
• [29] L. N. Trefethen, Spectral Methods in MATLAB, (SIAM, Philadelphia, 2000) http://dx.doi.org/10.1137/1.9780898719598[Crossref]

Identifiers

DOI

10.2478/s11534-011-0124-2