Abstract
In this paper, under investigation is a nonlinear chemistry system of ordinary differential equations, whose mechanism is exemplified by certain radioactive series, hydrolyses, and reaction of potassium permanganate, oxalic, and manganous sulfate. Via symbolic computation, an analytic solution for the system is obtained, which has higher accuracy and can describe the mechanism more completely than those in the previous results. Moreover, analysis is performed on the system and figures are plotted for understanding the reaction mechanism. Difference between the analytic solution obtained in this paper and the approximate one obtained previously is analyzed.
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Zhang, S.: A generalized auxiliary equation method and its application to (2+1)-dimensional Korteweg–de Vries equations. Comput. Math. Appl. 54, 1028–1038 (2007)
Lü, X., Zhu, H.W., Meng, X.H., Yang, Z.C., Tian, B.: Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communications. J. Math. Anal. Appl. 336, 1305–1315 (2007)
Rocha, W., Verreault, A.: Clothing up DNA for all seasons: Histone chaperones and nucleosome assembly pathways. FEBS Lett. 582, 1938–1949 (2008)
Nitsch, C.: A nonlinear parabolic system arising in damage mechanics under chemical aggression. Nonlinear Anal. 61, 695–713 (2005)
Yang, J.: Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics. Phys. Rev. E 59, 2393–2405 (1999)
Wang, H.Q.: Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations. Appl. Math. Comput. 170, 17–35 (2005)
Whitcomb, J.D., Raju, I.S., Goree, J.G.: Reliability of the finite element method for calculating free edge stresses in composite laminates. Comput. Struct. 15, 23–37 (1982)
Kaya, D., Inan, E.I.: Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation. Appl. Math. Comput. 151, 775–787 (2004)
He, J.H., Wu, X.H.: Construction of solitary solution and compacton-like solution by variational iteration method. Chaos Solitons Fractals 29, 108–113 (2006)
He, J.H.: Homotopy perturbation method for solving boundary value problems. Phys. Lett. A 350, 87–88 (2006)
Hirota, R.: The Direct Method in Soliton Theory. Cambridge Univ. Press, Cambridge (2004)
Yao, Y.Q., Chen, D.Y., Zeng, Y.B.: N-soliton solutions for a (2+1)-dimensional breaking soliton equation with self-consistent sources. Nonlinear Anal. 72, 57–64 (2010)
Batiha, B., Noorani, M.S.M., Hashim, I.: Variational iteration method for solving multispecies Lotka–Volterra equations. Comput. Math. Appl. 54, 903–909 (2007)
Ganji, D.D., Nourollahi, M., Mohseni, E.: Application of He’s methods to nonlinear chemistry problems. Comput. Math. Appl. 54, 1122–1132 (2007)
Xie, F.D.: Exact solutions of some systems of nonlinear partial differential equations using symbolic computation. Comput. Math. Appl. 44, 711–716 (2002)
Grujić, Z., Kalisch, H.: Gevrey regularity for a class of water-wave models. Nonlinear Anal. 71, 1160–1170 (2009)
Frost, A.A., Pearson, R.G.: Kinetics and Mechanism. Wiley, New York (1961), rev. ed.
Daniel, M., Lathab, M.M.: Soliton in alpha helical proteins with interspine coupling at higher order. Phys. Lett. A 302, 94–104 (2002)
Harcourt, A.V., Esson, W.: On the laws of connexion between the conditions of a chemical change and its amount. Philos. Trans. R. Soc. Lond. 156, 193–221 (1866)
Moore, J.W., Pearson, R.G.: Kinetics and Mechanism, 3rd edn. Canada (1981)
Hull, T.E., Enright, W.H., Fellen, B.M., Sedgwick, A.E.: Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9, 603–636 (1972)
Sweilam, N.H., Khader, M.M.: Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method. Comput. Math. Appl. 58, 2134–2141 (2009)
Gatto, M.A., Seery, J.B.: Numerical evaluation of the modified Bessel functions I and K. Comput. Math. Appl. 7, 203–209 (1981)
Ling, Z.: Jordan type inequalities involving the Bessel and modified Bessel functions. Comput. Math. Appl. 59, 724–736 (2010)
Wazwaz, A.M.: A comparison between Adomian decomposition method and Taylor series method in the series solution. Appl. Math. Comput. 97, 37–44 (1998)
Yu, X., Gao, Y.T., Sun, Z.Y., Liu, Y.: Solitonic propagation and interaction for a generalized variable-coefficient forced Korteweg–de Vries equation in fluids. Phys. Rev. E 83, 056601 (2011)
Yu, X., Gao, Y.T., Sun, Z.Y., Liu, Y.: N-soliton solutions, Backlund transformation and Lax pair for a generalized variable-coefficient fifth-order Korteweg–de Vries equation. Phys. Scr. 81, 045402 (2010)
Sun, Z.Y., Gao, Y.T., Yu, X., Liu, W.J., Liu, Y.: Bound vector solitons and soliton complexes for the coupled nonlinear Schrödinger equations. Phys. Rev. E 80, 066608 (2009)
Sun, Z.Y., Gao, Y.T., Yu, X., Meng, X.H., Liu, Y.: Inelastic interactions of the multiple-front waves for the modified Kadomtsev–Petviashvili equation in fluid dynamics, plasma physics and electrodynamics. Wave Motion 46, 511 (2009)
Wang, L., Gao, Y.T., Gai, X.L., Sun, Z.Y.: Inelastic interactions and double Wronskian solutions for the Whitham–Broer–Kaup model in shallow water. Phys. Scr. 80, 065017 (2009)
Wang, L., Gao, Y.T., Gai, X.L.: Odd-soliton-like solutions for the variable-coefficient variant Boussinesq model in the long gravity waves. Z. Naturforsch. A 65, 1 (2010)
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Liu, LC., Tian, B., Xue, YS. et al. Analytic solution for a nonlinear chemistry system of ordinary differential equations. Nonlinear Dyn 68, 17–21 (2012). https://doi.org/10.1007/s11071-011-0200-6
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DOI: https://doi.org/10.1007/s11071-011-0200-6