Topological Methods in Nonlinear Analysis

Simulation of the predator-prey problem by the homotopy-perturbation method revised

M. S. H. Chowdhury, Ishak Hashim, and R. Roslan

Full-text: Open access

Abstract

In this paper, the predator-prey problem is revisited. Previous solution by homotopy-perturbation method (HPM) is improved by treating the homotopy-perturbation method as an algorithm in a sequence of intervals (i.e. time steps) called the multistage homotopy-perturbation method (shortly MHPM). Numerical results show that the multistage homotopy-perturbation method and the classical fourth-order Rungge-Kutta (RK4) methods are in complete agreement.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 31, Number 2 (2008), 263-270.

Dates
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1463150270

Mathematical Reviews number (MathSciNet)
MR2432084

Zentralblatt MATH identifier
1157.34004

Citation

Chowdhury, M. S. H.; Hashim, Ishak; Roslan, R. Simulation of the predator-prey problem by the homotopy-perturbation method revised. Topol. Methods Nonlinear Anal. 31 (2008), no. 2, 263--270. https://projecteuclid.org/euclid.tmna/1463150270


Export citation

References

  • A. Blendez, A. Hernandez and T. Blendez, Application of He's homotopy-perturbation method to the Duffing-harmonic oscillator , Internat. J. Nonlinear Sci. Num. Simulation, 8 (2007), 79–88 \ref \key 2
  • J. D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell Publishing Company, Waltham Massachusetts (1968) \ref\key 3
  • D. D. Ganji, The application of He's homotopy-perturbation method to nonlinear equations arising in heat transfer , Phys. Lett. A, 355 (2006), 337–341 \ref\key 4
  • D. D. Ganji and A. Sadighi, Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations , J. Comput. Appl. Math., 207 (2006), 24–34 \ref\key 5 ––––, Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations , Internat. J. Nonlinear Sci. Num. Simulation, 7 (2006), 413–420 \ref\key 6 ––––, Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations , Internat. J. Nonlinear Sci., 7 , 411–418 (2006) \ref\key 7
  • M. Gorji, D. D. Ganji and S. Soleimani, New application of He's homotopy perturbation method , Internat. J. Nonlinear Sci., 8 , 319–328 (2007) \ref\key 8
  • J. H. He, Variational iteration method –- a kind of non-linear analytical technique: Some examples , Internat. J. Non-Linear Mech., 34 (1999), 699–708 \ref\key 9 ––––, Variational iteration method for autonomous ordinary differential systems , Appl. Math. Comput., 114 (2000), 115–123 \ref\key 10 ––––, A coupling method of a homotopy technique and a perturbation technique for non-linear problems , Internat. J. Non-Linear Mech., 35 (2000), 37–43 \ref\key 11 ––––, New Interpretation of homotopy-perturbation method , Internat. J. Modern Phys. B, 20 (2006), 2561–2568 \ref\key 12 ––––, Homotopy perturbation method for bifurcation of nonlinear problems , Internat. J. Nonlinear Sci. Num. Simulation, 6 (2005), 207–208 \ref\key 13 ––––, Some asymptotic methods for strongly nonlinear equations , Internat. J. Modern Phys. B, 20 (2006), 1141–1199 \ref\key 14 ––––, Non-perturbative methods for strongly nonlinear problems , Berlin, dissertation. de–Verlag im Internet GmbH (2006) \ref\key 15
  • J. H. He and M. A. Abdou, New periodic solutions for nonlinear evolution equations using exp-function method , Chaos Solitons Fractals. in press \ref\key 16
  • J. H. He and X.-H. Wu, Exp-function method for nonlinear wave equations , Chaos Solitons Fractals, 30 (2006), 700–708 \ref\key 17
  • H. Khaleghi, D. D. Ganji and A. Sadighi, Application of variational iteration and homotopy-perturbation methods to nonlinear heat transfer equations with variable coefficients , Numer. Heat Trans. Part A, 52 (2007), 25–42 \ref\key 18
  • D. Lesnic, Blow-up solutions obtained using the decomposition method , Chaos Solitons Fractals, 28 (2006), 776–787 \ref\key 19
  • T. Ozis and A. Yildirim, A comparative study of He's homotopy-perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities , Internat. J. Nonlinear Sci. Mumer. Simulation, 8 (2007), 243–248 \ref\key 20
  • M. Rafei and D. D. Ganji, Explicit solutions of Helmholtz equation and fifth-order Kdv equation using homotopy perturbation method , Internat. J. Nonlinear Sci. Num. Simulation, 7 (2006), 321–328 \ref\key 21 –––– Explicit solutions of Helmholtz equation and fifth-order KdV equation using homotopy perturbation method, Internat. J. Nonlinear Sci., 7 , 321–328 (2006) \ref\key 22
  • H. Tari, D. D. Ganji and M. Rostamian, Approximate solutions of $K (2,2)$, KdV and modified KdV equations by variational iteration method, homotopy perturbation method and homotopy analysis method , Internat. J. Non-linear Sci., 8 , 203–210 (2007) \ref\key 23
  • J. Wang, J.-K. Chen and S. Liao, An explicit solution of the large deformation of a cantilever beam under point load at the free tip , J. Comput. Appl. Math.. in press (2006) \ref\key 24
  • A. M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators , Appl. Math. Comput., 111 (2000), 53–69 \ref\key 25
  • A. M. Wazwaz and A. Gorguis, Exact solutions for heat-like and wave-like equations with variable coefficients , Appl. Math. Comput., 149 (2004), 15–29