Optimal Parameters for Nonlinear Hirota-Satsuma Coupled KdV System by Using Hybrid Firefly Algorithm with Modified Adomian Decomposition


  • Omar Saber Qasim Department of Mathematics, University of Mosul, Mosul
  • Karam Adel Abed Department of Mathematics, University of Mosul, Mosul
  • Ahmed F. Qasim Department of Mathematics, University of Mosul, Mosul




Hirota-Satsuma coupled kdv, Adomian decomposition method, firefly algorithm


In this paper, several parameters of the non-linear Hirota-Satsuma coupled KdV system were estimated using a hybrid between the Firefly Algorithm (FFA) and the Modified Adomian decomposition method (MADM). It turns out that optimal parameters can significantly improve the solutions when using a suitably selected fitness function for this problem. The results obtained show that the approximate solutions are highly compatible with the exact solutions and that the hybrid method FFA_MADM gives higher efficiency and accuracy compared to the classic MADM method.


G. Adomian, Nonlinear stochastic operator equations: Academic press, 2014.

M. Almazmumy, F. Hendi, H. Bakodah, and H. Alzumi, "Recent modifications of Adomian decomposition method for initial value problem in ordinary differential equations," American journal of computational mathematics, vol. 2, p. 228, 2012.

H. O. Bakodah, "Some Modifications of Adomian Decomposition Method Applied to Nonlinear System of Fredholm Integral Equations of the Second Kind," International Journal of Contemporary Mathematical Sciences, vol. 7, 2012.

I. L. El-Kalla, "Error analysis of Adomian series solution to a class of nonlinear differential equations," Appl. Math. E-Notes, vol. 7, pp. 214-221, 2007.

S. El-Wakil and M. Abdou, "New applications of Adomian decomposition method," Chaos, Solitons & Fractals, vol. 33, pp. 513-522, 2007.

R. Hirota and J. Satsuma, "Soliton solutions of a coupled Korteweg-de Vries equation," Physics Letters A, vol. 85, pp. 407-408, 1981.

A. Halim, S. Kshevetskii, and S. Leble, "Numerical integration of a coupled Korteweg-de Vries system," Computers & Mathematics with Applications, vol. 45, pp. 581-591, 2003.

A. Halim and S. Leble, "Analytical and numerical solution of a coupled KdV-MKdV system," Chaos, Solitons & Fractals, vol. 19, pp. 99-108, 2004.

K. Raslan, "The decomposition method for a Hirota-Satsuma coupled KdV equation and a coupled MKdV equation," International Journal of Computer Mathematics, vol. 81, pp. 1497-1505, 2004.

S. Jimenez and O. Vera, "Smoothing properties for Hirota-Satsuma systems," Electronic Journal of Qualitative Theory of Differential Equations, vol. 2011, pp. 1-30, 2011.

M. Alquran and R. Al-Omary, "Soliton and periodic solutions to the generalized Hirota-Satsuma coupled system using trigonometric and hyperbolic function methods," International Journal of Nonlinear Science, vol. 14, pp. 150-159, 2012.

M. Ismail and H. Ashi, "A numerical solution for Hirota-Satsuma coupled KdV equation," in Abstract and Applied Analysis, 2014.

M. A. Z. Raja, "Solution of the one-dimensional Bratu equation arising in the fuel ignition model using ANN optimised with PSO and SQP," Connection Science, vol. 26, pp. 195-214, 2014.

T. Apostolopoulos and A. Vlachos, "Application of the firefly algorithm for solving the economic emissions load dispatch problem," International journal of combinatorics, vol. 2011, 2010.

X.-S. Yang, "Firefly algorithm, stochastic test functions and design optimisation," arXiv preprint arXiv:1003.1409, 2010.

X.-S. Yang, S. S. S. Hosseini, and A. H. Gandomi, "Firefly algorithm for solving non-convex economic dispatch problems with valve loading effect," Applied soft computing, vol. 12, pp. 1180-1186, 2012.

X.-S. Yang, "Firefly algorithm, Levy flights and global optimization," in Research and development in intelligent systems XXVI, ed: Springer, 2010, pp. 209-218.

X.-S. Yang and X. He, "Firefly algorithm: recent advances and applications," arXiv preprint arXiv:1308.3898, 2013.

X.-S. Yang, Cuckoo search and firefly algorithm: Theory and applications vol. 516: Springer, 2013.

N. A. Al-Thanoon, O. S. Qasim, and Z. Y. Algamal, "Tuning parameter estimation in SCAD-support vector machine using firefly algorithm with application in gene selection and cancer classification," Computers in biology and medicine, vol. 103, pp. 262-268, 2018.

O. S. Qasim and Z. Y. Algamal, "Feature selection using particle swarm optimization-based logistic regression model," Chemometrics and Intelligent Laboratory Systems, vol. 182, pp. 41-46, 2018.

S. Aukasik and S. A>>ak, "Firefly algorithm for continuous constrained optimization tasks," in International conference on computational collective intelligence, 2009, pp. 97-106.

S. Padhan, R. K. Sahu, and S. Panda, "Application of firefly algorithm for load frequency control of multi-area interconnected power system," Electric power components and systems, vol. 42, pp. 1419-1430, 2014.

S. Carbas, "Design optimization of steel frames using an enhanced firefly algorithm," Engineering Optimization, vol. 48, pp. 2007-2025, 2016.