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Marzban, H. R.; Hoseini, S. M.; Razzaghi, M.

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

Math. Methods Appl. Sci. 32, No. 2, 127-134 (2009).
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.

Cited in 29 Documents

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
[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.