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Numerical approximations for population growth model by rational Chebyshev and Hermite functions collocation approach: a comparison. (English) Zbl 1204.65159

Summary: This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF) collocation approach to solve Volterra’s model for population growth of a species within a closed system. This model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. This approach is based on orthogonal functions, which will be defined. The collocation method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare these methods with some other numerical results and show that the present approach is applicable for solving nonlinear integro-differential equations.

MSC:

65R20 Numerical methods for integral equations
92D25 Population dynamics (general)
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
45D05 Volterra integral equations
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References:

[1] Coulaud, Laguerre spectral approximation of elliptic problems in exterior domains, Computer Methods in Applied Mechanics and Engineering 80 (1-3) pp 451– (1990) · Zbl 0734.73090
[2] Funaro, Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Mathematics of Computation 57 pp 597– (1991) · Zbl 0764.35007
[3] Funaro, Computational aspects of pseudospectral Laguerre approximations, Applied Numerical Mathematics 6 (6) pp 447– (1990) · Zbl 0708.65072
[4] Guo, Error estimation of Hermite spectral method for nonlinear partial differential equations, Mathematics of Computation 68 (227) pp 1067– (1999) · Zbl 0918.65069
[5] Guo, Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval, Numerische Mathematik 86 (4) pp 635– (2000) · Zbl 0969.65094
[6] Maday, Reappraisal of Laguerre type spectral methods, La Recherche Aerospatiale 6 pp 13– (1985) · Zbl 0604.42026
[7] Shen, Stable and efficient spectral methods in unbounded domains using Laguerre functions, SIAM Journal on Numerical Analysis 38 (4) pp 1113– (2000) · Zbl 0979.65105
[8] Siyyam, Laguerre Tau methods for solving higher-order ordinary differential equations, Journal of Computational Analysis and Applications 3 (2) pp 173– (2001) · Zbl 1024.65049
[9] Iranzo, Some spectral approximations for differential equations in unbounded domains, Computer Methods in Applied Mechanics and Engineering 98 (1) pp 105– (1992) · Zbl 0762.76081
[10] Guo, Gegenbauer approximation and its applications to differential equations on the whole line, Journal of Mathematical Analysis and Applications 226 (1) pp 180– (1998) · Zbl 0913.41020
[11] Guo, Jacobi spectral approximation and its applications to differential equations on the half line, Journal of Computational Mathematics 18 pp 95– (2000)
[12] Guo, Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, Journal of Mathematical Analysis and Applications 243 (2) pp 373– (2000) · Zbl 0951.41006
[13] Boyd, Chebyshev and Fourier Spectrals Method (2001)
[14] Christov, A complete orthogonal system of functions in L2(-, ) space, SIAM Journal on Applied Mathematics 42 pp 1337– (1982)
[15] Boyd, Spectral methods using rational basis functions on an infinite interval, Journal of Computational Physics 69 (1) pp 112– (1987) · Zbl 0615.65090
[16] Boyd, Orthogonal rational functions on a semi-infinite interval, Journal of Computational Physics 70 (1) pp 63– (1987) · Zbl 0614.42013
[17] Guo, A rational approximation and its applications to differential equations on the half line, Journal of Scientific Computing 15 (2) pp 117– (2000)
[18] Boyd, Pseudospectral methods on a semi-infinite interval with application to the Hydrogen atom: a comparison of the mapped Fourier-sine method with Laguerre series and rational Chebyshev expansions, Journal of Computational Physics 188 (1) pp 56– (2003) · Zbl 1028.65086
[19] Parand, Rational Chebyshev Tau method for solving Volterra’s population model, Applied Mathematics and Computation 149 (3) pp 893– (2004) · Zbl 1038.65149
[20] Parand, Rational Chebyshev Tau method for solving higher-order ordinary differential equations, International Journal of Computer Mathematics 81 (1) pp 73– (2004) · Zbl 1047.65052
[21] Parand, Rational Legendre approximation for solving some physical problems on semi-infinite intervals, Physica Scripta 69 pp 353– (2004) · Zbl 1063.65146
[22] Parand, Rational Chebyshev pseudospectral approach for solving Thomas-Fermi equation, Physics Letters A 373 pp 210– (2009) · Zbl 1227.49050
[23] Parand, Rational scaled generalized Laguerre function collocation method for solving the Blasius equation, Journal of Computational and Applied Mathematics 233 (4) pp 980– (2009) · Zbl 1259.65127
[24] Parand, Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane-Emden type, Journal of Computational Physics 228 (23) pp 8830– (2009) · Zbl 1177.65100
[25] Parand, An approximational algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method, Computer Physics Communications (2010) · Zbl 1216.65098
[26] Scudo, Vito Volterra and theoretical ecology, Theoretical Population Biology 2 (1) pp 1– (1971) · Zbl 0241.92001
[27] Small, Population growth in a closed system, SIAM Review 25 (1) pp 93– (1983) · Zbl 0502.92012
[28] TeBeest, Numerical and analytical solutions of Volterra’s population model, SIAM Review 39 (3) pp 484– (1997) · Zbl 0892.92020
[29] Al-Khaled, Numerical approximations for population growth models, Applied Mathematics and Computation 160 (3) pp 865– (2005) · Zbl 1062.65142
[30] Wazwaz, Analytical approximations and Padé approximants for Volterra’s population model, Applied Mathematics and Computation 100 (1) pp 13– (1999) · Zbl 0953.92026
[31] Parand, Solving Volterra’s population model using new second derivative multistep methods, American Journal of Applied Sciences 5 (8) pp 1019– (2008)
[32] Ramezani, Composite spectral functions for solving Volterras population model, Chaos, Solitons and Fractals 34 (2) pp 588– (2007)
[33] Marzban, Solution of Volterras population model via block-pulse functions and Lagrange-interpolating polynomials, Mathematical Methods in the Applied Sciences 32 pp 127– (2009) · Zbl 1156.65106
[34] Momani, Numerical approximations and Padé approximants for a fractional population growth model, Applied Mathematical Modelling 31 pp 1907– (2007) · Zbl 1167.45300
[35] Xu, Analytical approximations for a population growth model with fractional order, Communications in Nonlinear Science and Numerical Simulation 14 pp 1978– (2009)
[36] Guo, Chebyshev rational spectral and pseudospectral methods on a semi-infinite interval, International Journal for Numerical Methods in Engineering 53 (1) pp 65– (2002) · Zbl 1001.65129
[37] Boyd, The optimzation of convergence for Chebyshev polynomial methods in an unbounded domain, Journal of Computational Physics 45 pp 43– (1982)
[38] Shen, Some recent advances on spectral methods for unbounded domains, Communications in Computational Physics 5 (2-4) pp 195– (2009) · Zbl 1364.65265
[39] Shen, High Order Numerical Methods and Algorithms (2005)
[40] Shen, Spectral Methods Algorithms, Analyses and Applications (2010)
[41] Guo, Spectral and pseudospectral approximations using Hermite functions: application to the Dirac equation, Advances in Computational Mathematics 19 (1-3) pp 35– (2003) · Zbl 1032.33004
[42] Liu, The numerical computation of connecting orbits in dynamical systems: a rational spectral approach, Journal of Computational Physics 111 (2) pp 373– (1994) · Zbl 0797.65055
[43] Tang, The Hermite spectral method for Gaussian-type functions, SIAM Journal on Scientific Computing 14 (3) pp 594– (1993) · Zbl 0782.65110
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