##
**Enhanced multistage differential transform method: application to the population models.**
*(English)*
Zbl 1242.65151

Summary: We present an efficient computational algorithm, namely, the enhanced multistage differential transform method (E-MsDTM) for solving prey-predator systems. Since the differential transform method (DTM) is based on the Taylor series, it is difficult to obtain accurate approximate solutions in large domain. To overcome this difficulty, the multistage differential transform method (MsDTM) has been introduced and succeeded to have reliable approximate solutions for many problems. In MsDTM, it is the key to update an initial condition in each subdomain. The standard MsDTM utilizes the approximate solution directly to assign the new initial value. Because of local convergence of the Taylor series, the error is accumulated in a large domain. In E-MsDTM, we propose the new technique to update an initial condition by using integral operator. To demonstrate efficiency of the proposed method, several numerical tests are performed and compared with ones obtained by other numerical methods such as MsDTM, multistage variational iteration method (MVIM), and fourth-order Runge-Kutta method (RK4).

### MSC:

65L99 | Numerical methods for ordinary differential equations |

34A99 | General theory for ordinary differential equations |

92D25 | Population dynamics (general) |

PDFBibTeX
XMLCite

\textit{Y. Do} and \textit{B. Jang}, Abstr. Appl. Anal. 2012, Article ID 253890, 14 p. (2012; Zbl 1242.65151)

Full Text:
DOI

### References:

[1] | F. Kangalgil and F. Ayaz, “Solitary wave solutions for the KdV and mKdV equations by differential transform method,” Chaos, Solitons and Fractals, vol. 41, no. 1, pp. 464-472, 2009. · Zbl 1198.35222 · doi:10.1016/j.chaos.2008.02.009 |

[2] | A. S. V. Ravi Kanth and K. Aruna, “Differential transform method for solving linear and non-linear systems of partial differential equations,” Physics Letters. A, vol. 372, no. 46, pp. 6896-6898, 2008. · Zbl 1227.35018 · doi:10.1016/j.physleta.2008.10.008 |

[3] | I. H. A. Hassan, “Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems,” Chaos, Solitons and Fractals, vol. 36, no. 1, pp. 53-65, 2008. · Zbl 1152.65474 · doi:10.1016/j.chaos.2006.06.040 |

[4] | F. Ayaz, “Solutions of the system of differential equations by differential transform method,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 547-567, 2004. · Zbl 1032.35011 · doi:10.1016/S0096-3003(02)00794-4 |

[5] | I. H. A. Hassan, “Different applications for the differential transformation in the differential equations,” Applied Mathematics and Computation, vol. 129, no. 2-3, pp. 183-201, 2002. · Zbl 1026.34010 · doi:10.1016/S0096-3003(01)00037-6 |

[6] | M. J. Jang, C. L. Chen, and Y. C. Liu, “Two-dimensional differential transform for partial differential equations,” Applied Mathematics and Computation, vol. 121, no. 2-3, pp. 261-270, 2001. · Zbl 1024.65093 · doi:10.1016/S0096-3003(99)00293-3 |

[7] | B. Jang, “Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 224-230, 2009. · Zbl 1175.65149 · doi:10.1016/j.cam.2009.07.012 |

[8] | B. Jang, “Solving linear and nonlinear initial value problems by the projected differential transform method,” Computer Physics Communications, vol. 181, no. 5, pp. 848-854, 2010. · Zbl 1205.65205 · doi:10.1016/j.cpc.2009.12.020 |

[9] | H. Ko\ccak and A. Yıldırım, “An efficient algorithm for solving nonlinear age-structured population models by combining homotopy perturbation and Padé techniques,” International Journal of Computer Mathematics, vol. 88, no. 3, pp. 491-500, 2011. · Zbl 1209.92048 · doi:10.1080/00207160903477159 |

[10] | A. Yildirim and Y. Cherruault, “Analytical approximate solution of a SIR epidemic model with constant vaccination strategy by homotopy perturbation method,” Kybernetes, vol. 38, no. 9, pp. 1566-1575, 2009. · Zbl 1192.65115 · doi:10.1108/03684920910991540 |

[11] | A. Gökdo\ugan, M. Merdan, and A. Yildirim, “A multistage differential transformation method for approximate solution of Hantavirus infection model,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 1-8, 2012. · doi:10.1016/j.cnsns.2011.05.023 |

[12] | A. Gökdo\vgan, M. Merdan, and A. Yildirim, “Adaptive multi-step differential transformation method to solving nonlinear differential equations,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 761-769, 2012. · Zbl 1255.65135 · doi:10.1016/j.mcm.2011.09.001 |

[13] | Z. M. Odibat, C. Bertelle, M. A. Aziz-Alaoui, and G. H. E. Duchamp, “A multi-step differential transform method and application to non-chaotic or chaotic systems,” Computers & Mathematics with Applications, vol. 59, no. 4, pp. 1462-1472, 2010. · Zbl 1189.65170 · doi:10.1016/j.camwa.2009.11.005 |

[14] | V. S. Ertürk, Z. M. Odibat, and S. Momani, “An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 996-1002, 2011. · Zbl 1228.92064 · doi:10.1016/j.camwa.2011.03.091 |

[15] | V. S. Erturk, G. Zaman, and S. Momani, “A numeric-analytic method for approximating a giving up smoking model containing fractional derivatives,” Computers & Mathematics with Applications. In press. · Zbl 1268.65107 · doi:10.1016/j.camwa.2012.02.002 |

[16] | V.S. Erturk, G. Zaman, and S. Momani, “Application of multi-step differential transform method for the analytical and numerical solutions of the density dependent Nagumo telegraph equation,” Romanian Journal of Physics. In press. |

[17] | S. M. Goh, M. S. M. Noorani, and I. Hashim, “Prescribing a multistage analytical method to a prey-predator dynamical system,” Physics Letters, Section A, vol. 373, no. 1, pp. 107-110, 2008. · Zbl 1227.34017 · doi:10.1016/j.physleta.2008.11.009 |

[18] | S. M. Moghadas, M. E. Alexander, and B. D. Corbett, “A non-standard numerical scheme for a generalized Gause-type predator-prey model,” Physica D, vol. 188, no. 1-2, pp. 134-151, 2004. · Zbl 1043.92040 · doi:10.1016/S0167-2789(03)00285-9 |

[19] | B. Batiha, M. S. M. Noorani, I. Hashim, and E. S. Ismail, “The multistage variational iteration method for a class of nonlinear system of ODEs,” Physica Scripta, vol. 76, no. 4, pp. 388-392, 2007. · Zbl 1132.34008 · doi:10.1088/0031-8949/76/4/018 |

[20] | B. Batiha, M. S. M. Noorani, and I. Hashim, “Variational iteration method for solving multispecies Lotka-Volterra equations,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 903-909, 2007. · Zbl 1141.65370 · doi:10.1016/j.camwa.2006.12.058 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.