Composite spectral functions for solving Volterra’s population model. (English) Zbl 1127.92033

Ramezani, M.; Razzaghi, M.; Dehghan, M.

Chaos Solitons Fractals 34, No. 2, 588-593 (2007).

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 toxins. The approach is based upon composite spectral function approximations. The properties of composite spectral functions consisting of few terms of orthogonal functions are presented and are utilized to reduce the solution of the Volterra model to the solution of a system of algebraic equations. The method is easy to implement and yields very accurate result.

MSC:

92D25	Population dynamics (general)
45J05	Integro-ordinary differential equations
34K07	Theoretical approximation of solutions to functional-differential equations

References:

[1]	Scudo, F.M., Volterra and theoretical ecology, Theor populat biol, 2, 1-23, (1971) · Zbl 0241.92001
[2]	TeBeest, K.G., Numerical and analytical solutions of volterra’s population model, SIAM rev, 39, 484-493, (1997) · Zbl 0892.92020
[3]	Al-Khaled, K., Numerical approximations for population growth models, Appl math comput, 160, 865-873, (2005) · Zbl 1062.65142
[4]	Small, R.D., (), 317-320
[5]	Wazwaz, A.M., Analytical approximations and pade approximant for volterra’s population model, Appl math comput, 100, 13, (1999) · Zbl 0953.92026
[6]	Chen, C.F.; Hasio, C.H., A Walsh series direct method for solving variational problems, J franklin inst, 300, 265-280, (1975) · Zbl 0339.49017
[7]	Hwang, C.; Shih, Y.P., Solutions of stiff differential equations via generalized block pulse functions optimal control of delay systems via block-pulse functions, J chem eng, 27, 81-86, (1983)
[8]	Hwang, C.; Shih, Y.P., Laguerre series direct method for variational problems, J optim theory appl, 39, 143-149, (1983) · Zbl 0481.49005
[9]	Chang, R.Y.; Wang, M.L., Shifted Legendre function approximation of differential equations; application to crystallization processes, Comput chem eng, 8, 117-125, (1984)
[10]	Horng, I.R.; Chou, J.H., Shifted Chebyshev direct method for solving variational problems, Int J syst sci, 16, 855-861, (1985) · Zbl 0568.49019
[11]	Razzaghi, M.; Razzaghi, M., Fourier series direct method for variational problems, Int J control, 48, 887-895, (1988) · Zbl 0651.49012
[12]	Razzaghi, M.; Ordokhani, Y., An application of rationalized Haar functions for variational problems, Appl math comput, 122, 353-364, (2001) · Zbl 1020.49026
[13]	Ricot, J.J.; Acha, E., Analysis of linear time-varying systems via Hartley series, Int J syst sci, 29, 541-549, (1998)
[14]	Rao, G.P.; Palanisamy, K.R.; Srinivasan, T., Extension of computation beyond the limit of normal interval in Walsh series analysis of dynamical systems, IEEE trans automat control, 25, 317-319, (1980) · Zbl 0442.93021
[15]	Lancaster, P., Theory of matrices, (1969), Academic Press New York · Zbl 0186.05301
[16]	Balachandran, K.; Murugesan, K., Analysis of nonlinear singular systems via STWS method, Int J comput math, 36, 9-12, (1990) · Zbl 0725.65073
[17]	Razzaghi, M.; Sepehrian, B., Single-term Walsh series direct method for the solution of nonlinear problems in the calculus of variations, J vib control, 10, 1071-1081, (2004) · Zbl 1092.49022
[18]	Sepehrian, B.; Razzaghi, M., Single-term Walsh series method for the Volterra integro-differential equations, Eng anal bound elem, 28, 1315-1319, (2004) · Zbl 1081.65551
[19]	Datta, K.B.; Mohan, B.M., Orthogonal functions in systems and control, (1995), World Scientific Publishing Co. · Zbl 0819.93036
[20]	Sannuti, P., Analysis and synthesis of dynamic systems via block-pulse functions, Proc IEE, 124, 569-571, (1977)
[21]	Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamic, (1986), Prentice-Hall Englewood Cliffs (NJ)