Rational Chebyshev tau method for solving Volterra’s population model. (English) Zbl 1038.65149

Parand, K.; Razzaghi, M.

Appl. Math. Comput. 149, No. 3, 893-900 (2004).

Summary: An approximate 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 effect of toxin.
The approach is based on a rational Chebyshev tau method. The Volterra’s population model is first converted to a nonlinear ordinary differential equation. The operational matrices of derivative and product of rational Chebyshev functions are presented. These matrices together with the tau method are then utilized to reduce the solution of the Volterra’s model to the solution of a system of algebraic equations.
Illustrative examples are included to demonstrate the validity and applicability of the technique and a comparison is made with existing results.

MSC:

65R20	Numerical methods for integral equations
45J05	Integro-ordinary differential equations
45G10	Other nonlinear integral equations
92D25	Population dynamics (general)

Keywords:

Rational Chebyshev tau method; Volterra’s population model; nonlinear integro-differential equation; numerical examples

References:

[1]	Wazwaz, A. M., Analytical approximation and Pade approximation for Volterra’s population model, Appl. Math. Comput., 100, 13-25 (1999) · Zbl 0953.92026
[2]	Scudo, F. M., Volterra and theoretical ecology, Theoret. Popul. Biol., 2, 1-23 (1971) · Zbl 0241.92001
[3]	Small, R. D., Population Growth in a Closed System and Mathematical Modelling, Classroom Notes in Applied Mathematics (1989), SIAM: SIAM Philadelphia, PA, pp. 317-320
[4]	TeBeest, K. G., Numerical and analytical solutions of Volterra’s population model, SIAM Rev., 39, 484-493 (1997) · Zbl 0892.92020
[5]	Lanczos, C., Applied Analysis (1956), Englewood Cliffs: Englewood Cliffs Prentice-Hall, NJ · Zbl 0111.12403
[6]	Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral Methods in Fluid Dynamic (1988), Springer: Springer New York · Zbl 0658.76001
[7]	Gottlieb, D.; Hussaini, M.; Orszg, S., Theory and Applications of Spectral Methods in Spectral Methods for Partial Differential Equations (1984), SIAM: SIAM Philadelphia
[8]	Guo, B. Y.; Shen, J.; Wang, Z. Q., Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval, Int. J. Numer. Math. Engng., 53, 65-84 (2002) · Zbl 1001.65129