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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 673829, 12 pages
A Reliable Analytical Method for Solving Higher-Order Initial Value Problems
1Department of Mathematics, Faculty of Science, Al Balqa Applied University, Salt 19117, Jordan
2Department of Mechatronics Engineering, Faculty of Engineering, University of Jordan, Amman 11942, Jordan
3Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa 13115, Jordan
4Department of Mathematics, Faculty of Science, University of Jordan, Amman 11942, Jordan
Received 2 July 2013; Accepted 23 September 2013
Academic Editor: Recai Kilic
Copyright © 2013 Omar Abu Arqub et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- O. Abu Arqub, A. El-Ajou, A. Bataineh, and I. Hashim, “A representation of the exact solution of generalized Lane-Emden equations using a new analytical method,” Abstract and Applied Analysis, vol. 2013, Article ID 378593, 10 pages, 2013.
- O. Abu Arqub, “Series solution of fuzzy differential equations under strongly generalized differentiability,” Journal of Advanced Research in Applied Mathematics, vol. 5, no. 1, pp. 31–52, 2013.
- A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Direct solution of nth-order IVPs by homotopy analysis method,” Differential Equations and Nonlinear Mechanics, vol. 2009, Article ID 842094, 15 pages, 2009.
- I. H. Abdel-Halim Hassan, “Differential transformation technique for solving higher-order initial value problems,” Applied Mathematics and Computation, vol. 154, no. 2, pp. 299–311, 2004.
- K. Al-Khaled and M. N. Anwar, “Numerical comparison of methods for solving second-order ordinary initial value problems,” Applied Mathematical Modelling, vol. 31, pp. 292–301, 2007.
- A. H. Bhrawy and W. M. Abd-Elhameed, “New algorithm for the numerical solutions of nonlinear third-order differential equations using Jacobi-Gauss collocation method,” Mathematical Problems in Engineering, vol. 2011, Article ID 837218, 14 pages, 2011.
- D. Gámez, A. I. Garralda Guillem, and M. Ruiz Galán, “Nonlinear initial-value problems and Schauder bases,” Nonlinear Analysis. Theory, Methods & Applications, vol. 63, no. 1, pp. 97–105, 2005.
- M. Lakestani and M. Dehghan, “Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions,” Computer Physics Communications, vol. 181, no. 5, pp. 957–966, 2010.
- D. Gámez, A. I. Garralda Guillem, and M. R. Galán, “High-order nonlinear initial-value problems countably determined,” Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 77–82, 2009.
- F. Geng, M. Cui, and B. Zhang, “Method for solving nonlinear initial value problems by combining homotopy perturbation and reproducing kernel Hilbert space methods,” Nonlinear Analysis. Real World Applications, vol. 11, no. 2, pp. 637–644, 2010.
- M. Podisuk, U. Chundang, and W. Sanprasert, “Single step formulas and multi-step formulas of the integration method for solving the initial value problem of ordinary differential equation,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1438–1444, 2007.
- A. A. Kosti, Z. A. Anastassi, and T. E. Simos, “An optimized explicit Runge-Kutta-Nyström method for the numerical solution of orbital and related periodical initial value problems,” Computer Physics Communications, vol. 183, no. 3, pp. 470–479, 2012.
- A. El-Ajou, O. Abu Arqub, and S. Momani, “Homotopy analysis method for second-order boundary value problems of integrodifferential equations,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 365792, 18 pages, 2012.
- O. Abu Arqub, A. El-Ajou, S. Momani, and N. Shawagfeh, “Analytical solutions of fuzzy initial value problems by HAM,” Applied Mathematics and Information Sciences, vol. 7, pp. 1903–1919, 2013.
- O. Abu Arqub, M. Al-Smadi, and S. Momani, “Application of reproducing kernel method for solving nonlinear Fredholm-Volterra integrodifferential equations,” Abstract and Applied Analysis, vol. 2013, Article ID 839836, 16 pages, 2012.
- M. Al-Smadi, O. Abu Arqub, and S. Momani, “A computational method for two-point boundary value problems of fourth-order mixed integrodifferential equations,” Mathematical Problems in Engineering, vol. 2013, Article ID 832074, 10 pages, 2013.
- O. A. Arqub, M. Al-Smadi, and N. Shawagfeh, “Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method,” Applied Mathematics and Computation, vol. 219, no. 17, pp. 8938–8948, 2013.
- N. Shawagfeh, O. Abu Arqub, and S. Momani, “Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method,” Journal of Computational Analysis and Applications. In press.
- O. Abu Arqub, Z. Abo-Hammour, S. Momani, and N. Shawagfeh, “Solving singular two-point boundary value problems using continuous genetic algorithm,” Abstract and Applied Analysis, vol. 2012, Article ID 205391, 25 pages, 2012.
- O. Abu Arqub, Z. Abo-Hammour, and S. Momani, “Application of continuous genetic algorithm for nonlinear system of second-order boundary value problems,” Applied Mathematics and Information Sciences, vol. 8, pp. 1–14, 2014.
- R. Genesio and A. Tesi, “Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems,” Automatica, vol. 28, pp. 531–548, 1992.
- A. Gökdoğan, M. Merdan, and A. Yildirim, “The modified algorithm for the differential transform method to solution of Genesio systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 45–51, 2012.
- F. Jianwen, H. Ling, X. Chen, F. Austin, and W. Geng, “Synchronizing the noise-perturbed Genesio chaotic system by sliding mode control,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 9, pp. 2546–2551, 2010.
- X. Wu, Z.-H. Guan, Z. Wu, and T. Li, “Chaos synchronization between Chen system and Genesio system,” Physics Letters A, vol. 364, no. 6, pp. 484–487, 2007.
- A. Ghorbani and J. Saberi-Nadjafi, “A piecewise-spectral parametric iteration method for solving the nonlinear chaotic Genesio system,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 131–139, 2011.
- J. Guckenheimer, “Dynamics of the van der Pol equation,” IEEE Transactions on Circuits and Systems, vol. 27, no. 11, pp. 983–989, 1980.
- Z. F. Zhang, T. Ding, and H. W. Huang, Qualitative Theory of Differential Equations, Science Press, Peking, China, 1985.
- J. K. Hale, Ordinary Differential Equations, Wiley, New York, NY, USA, 1969.
- Z. Feng, “On explicit exact solutions for the Lienard equation and its applications,” Physics Letters A, vol. 293, no. 1-2, pp. 50–56, 2002.
- D. X. Kong, “Explicit exact solutions for the Liénard equation and its applications,” Physics Letters A, vol. 196, no. 5-6, pp. 301–306, 1995.
- J. Sun, W. Wang, and L. Wu, “On explicit exact solutions for the Liénard equation and its applications,” Physics Letters A, vol. 318, no. 1-2, pp. 93–101, 2003.
- M. Matinfar, H. Hosseinzadeh, and M. Ghanbari, “A numerical implementation of the variational iteration method for the Lienard equation,” World Journal of Modelling and Simulation, vol. 4, pp. 205–210, 2008.
- D. Kaya and S. M. El-Sayed, “A numerical implementation of the decomposition method for the Lienard equation,” Applied Mathematics and Computation, vol. 171, no. 2, pp. 1095–1103, 2005.
- R. Buckmire, “Investigations of nonstandard, Mickens-type, finite-difference schemes for singular boundary value problems in cylindrical or spherical coordinates,” Numerical Methods for Partial Differential Equations, vol. 19, no. 3, pp. 380–398, 2003.
- Y. Aregbesola, “Numerical solution of Bratu problem using the method of weighted residual,” Electronic Journal of Southern African Mathematical Sciences Association, vol. 3, pp. 1–7, 2003.
- D. A. F. Kamenetski, Diffusion and Heat Exchange in Chemical Kinetics, Princeton University Press, Princeton, NJ, USA, 1955.
- I. H. Abdel-Halim Hassan and V. S. Ertürk, “Applying differential transformation method to the one-dimensional planar Bratu problem,” International Journal of Contemporary Mathematical Sciences, vol. 2, no. 29–32, pp. 1493–1504, 2007.
- H. Caglar, N. Caglar, M. Özer, A. Valaristos, A. N. Miliou, and A. N. Anagnostopoulos, “Dynamics of the solution of Bratu's equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 12, pp. e672–e678, 2009.
- S. G. Venkatesh, S. K. Ayyaswamy, and G. Hariharan, “Haar wavelet method for solving initial and boundary value problems of Bratu-type,” World Academy of Science, Engineering and Technology, vol. 67, pp. 565–568, 2010.
- S. Abbasbandy, M. S. Hashemi, and C.-S. Liu, “The Lie-group shooting method for solving the Bratu equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4238–4249, 2011.
- A.-M. Wazwaz, “Adomian decomposition method for a reliable treatment of the Bratu-type equations,” Applied Mathematics and Computation, vol. 166, no. 3, pp. 652–663, 2005.
- T. Öziş and A. Yıldırım, “Comparison between Adomian's method and He's homotopy perturbation method,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1216–1224, 2008.
- B. Batiha, “Numerical solution of Bratu-type equations by the variational iteration method,” Hacettepe Journal of Mathematics and Statistics, vol. 39, no. 1, pp. 23–29, 2010.
- Y. Aksoy and M. Pakdemirli, “New perturbation-iteration solutions for Bratu-type equations,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2802–2808, 2010.