Previous |  Up |  Next


Volterra's population system of fractional order; Caputo's fractional derivative; bi-parametric homotopy method; convergence region
This paper considers a Volterra's population system of fractional order and describes a bi-parametric homotopy analysis method for solving this system. The homotopy method offers a possibility to increase the convergence region of the series solution. Two examples are presented to illustrate the convergence and accuracy of the method to the solution. Further, we define the averaged residual error to show that the obtained results have reasonable accuracy.
[1] Al-Khaled, K.: Numerical approximations for population growth models. Appl. Math. Comput. 160 (2005), 865-873. DOI 10.1016/j.amc.2003.12.005 | MR 2113123 | Zbl 1062.65142
[2] Caputo, M.: Linear models of dissipation whose $Q$ is almost frequency independent II. Geophys. J. R. Astron. Soc. 13 (1967), 529-539. DOI 10.1111/j.1365-246X.1967.tb02303.x
[3] Diethelm, K., Ford, N. J., Freed, A. D., Luchko, Y.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194 (2005), 743-773. DOI 10.1016/j.cma.2004.06.006 | MR 2105648 | Zbl 1119.65352
[4] He, J.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167 (1998), 57-68. DOI 10.1016/S0045-7825(98)00108-X | MR 1665221 | Zbl 0942.76077
[5] He, J.: Nonlinear oscillation with fractional derivative and its applications. International Conference on Vibrating Engineering, Dalian, China, 1998 288-291.
[6] He, J.: Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15 (1999), 86-90.
[7] Liao, S.: Boundary element method for general nonlinear differential operators. Eng. Anal. Bound. Elem. 20 (1997), 91-99. DOI 10.1016/S0955-7997(97)00043-X
[8] Luchko, Y., Gorenflo, R.: The initial value problem for some fractional differential equations with the Caputo derivatives. Fachbereich Mathematik und Informatik, Freie Universität Berlin (1998), Preprint A-98-08.
[9] Podlubny, I.: Fractional Differential Equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Mathematics in Science and Engineering 198 Academic Press, San Diego (1999). MR 1658022 | Zbl 0924.34008
[10] Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5 (2002), 367-386. MR 1967839 | Zbl 1042.26003
[11] Scudo, F. M.: Vito Volterra and theoretical ecology. Theor. Population Biology 2 (1971), 1-23. DOI 10.1016/0040-5809(71)90002-5 | MR 0408866 | Zbl 0241.92001
[12] Small, R. D.: Population growth in a closed system. Mathematical Modelling: Classroom Notes in Applied Mathematics M. S. Klamkin Society for Industrial and Applied Mathematics Philadelphia (1989).
[13] TeBeest, K. G.: Numerical and analytical solutions of Volterra's population model. SIAM Rev. 39 (1997), 484-493. DOI 10.1137/S0036144595294850 | MR 1469945 | Zbl 0892.92020
[14] Wazwaz, A.-M.: Analytical approximations and Padé approximants for Volterra's population model. Appl. Math. Comput. 100 (1999), 13-25. DOI 10.1016/S0096-3003(98)00018-6 | MR 1665900 | Zbl 0953.92026
Partner of
EuDML logo