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**Solution of Volterra’s population model via block-pulse functions and Lagrange-interpolating polynomials.**
*(English)*
Zbl 1156.65106

Summary: A numerical method for solving Volterra’s population model for population growth of a species in a closed system is proposed. Volterra’s model is a nonlinear integro-differential equation where the integral term represents the effects of toxin. The approach is based on hybrid function approximations. The properties of hybrid functions that consist of block-pulse and Lagrange-interpolating polynomials are presented. The associated operational matrices of integration and product are then utilized to reduce the solution of Volterra’s model to the solution of a system of algebraic equations. The method is easy to implement and computationally very attractive. Applications are demonstrated through an illustrative example.

### MSC:

65R20 | Numerical methods for integral equations |

45J05 | Integro-ordinary differential equations |

45G10 | Other nonlinear integral equations |

92D25 | Population dynamics (general) |

### Keywords:

Volterra’s population model; hybrid function; block-pulse functions; Lagrange-interpolating polynomials; nonlinear integro-differential equation; orthogonal functions; numerical examples
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\textit{H. R. Marzban} et al., Math. Methods Appl. Sci. 32, No. 2, 127--134 (2009; Zbl 1156.65106)

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### References:

[1] | Chen, A Walsh series direct method for solving variational problems, Journal of the Franklin Institute 300 pp 265– (1975) · Zbl 0339.49017 |

[2] | Hwang, Optimal control of delay systems via block-pulse functions, Journal of Chemical Engineering 27 pp 81– (1983) |

[3] | Chang, Shifted Legendre function approximation of differential equations; application to crystallization processes, Computers and Chemical Engineering 8 pp 117– (1984) |

[4] | Horng, Shifted Chebyshev direct method for solving variational problems, International Journal of Systems Science 16 pp 855– (1985) · Zbl 0568.49019 |

[5] | Razzaghi, Fourier series direct method for variational problems, International Journal of Control 48 pp 887– (1988) · Zbl 0651.49012 |

[6] | Razzaghi, An application of rationalized Haar functions for variational problems, Applied Mathematics and Computation 122 pp 353– (2001) · Zbl 1020.49026 |

[7] | Ricot, Analysis of linear time-varying systems via Hartley series, International Journal of Systems Science 29 pp 541– (1998) |

[8] | Scudo, Volterra and theoretical ecology, Theoretical Population Biology 2 pp 1– (1971) · Zbl 0241.92001 |

[9] | TeBeest, Numerical and analytical solutions of Volterra’s population model, SIAM Review 39 pp 484– (1997) · Zbl 0892.92020 |

[10] | Small, Population Growth in a Closed System and Mathematical Modelling pp 317– (1989) |

[11] | Wazwaz, Analytical approximations and Pade approximant for Volterra’s population model, Applied Mathematics and Computation 100 pp 13– (1999) |

[12] | Al-Khaled, Numerical approximations for population growth models, Applied Mathematics and Computation 160 pp 865– (2005) · Zbl 1062.65142 · doi:10.1016/j.amc.2003.12.005 |

[13] | Davis, Methods of Numerical Integration (1984) · Zbl 0537.65020 |

[14] | Marzban, Solution of time-varying delay systems by hybrid functions, Mathematics and Computers in Simulation 64 pp 597– (2004) · Zbl 1039.65053 |

[15] | Ramezani, Composite spectral functions for solving Volterra’s population model, Chaos, Solitons and Fractals 34 pp 588– (2007) · Zbl 1127.92033 |

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