| Title:
|
A new application of the homotopy analysis method in solving the fractional Volterra's population system (English) |
| Author:
|
Ghasemi, Mehdi |
| Author:
|
Fardi, Mojtaba |
| Author:
|
Ghaziani, Reza Khoshsiar |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
59 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
319-330 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Volterra's population system of fractional order |
| Keyword:
|
Caputo's fractional derivative |
| Keyword:
|
bi-parametric homotopy method |
| Keyword:
|
convergence region |
| MSC:
|
26A33 |
| MSC:
|
34K37 |
| MSC:
|
92B25 |
| idZBL:
|
Zbl 06362229 |
| idMR:
|
MR3232633 |
| DOI:
|
10.1007/s10492-014-0057-3 |
| . |
| Date available:
|
2014-05-20T07:39:14Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143775 |
| . |
| Reference:
|
[1] Al-Khaled, K.: Numerical approximations for population growth models.Appl. Math. Comput. 160 (2005), 865-873. Zbl 1062.65142, MR 2113123, 10.1016/j.amc.2003.12.005 |
| Reference:
|
[2] Caputo, M.: Linear models of dissipation whose $Q$ is almost frequency independent II.Geophys. J. R. Astron. Soc. 13 (1967), 529-539. 10.1111/j.1365-246X.1967.tb02303.x |
| Reference:
|
[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. Zbl 1119.65352, MR 2105648, 10.1016/j.cma.2004.06.006 |
| Reference:
|
[4] He, J.: Approximate analytical solution for seepage flow with fractional derivatives in porous media.Comput. Methods Appl. Mech. Eng. 167 (1998), 57-68. Zbl 0942.76077, MR 1665221, 10.1016/S0045-7825(98)00108-X |
| Reference:
|
[5] He, J.: Nonlinear oscillation with fractional derivative and its applications.International Conference on Vibrating Engineering, Dalian, China, 1998 288-291. |
| Reference:
|
[6] He, J.: Some applications of nonlinear fractional differential equations and their approximations.Bull. Sci. Technol. 15 (1999), 86-90. |
| Reference:
|
[7] Liao, S.: Boundary element method for general nonlinear differential operators.Eng. Anal. Bound. Elem. 20 (1997), 91-99. 10.1016/S0955-7997(97)00043-X |
| Reference:
|
[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. |
| Reference:
|
[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). Zbl 0924.34008, MR 1658022 |
| Reference:
|
[10] Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation.Fract. Calc. Appl. Anal. 5 (2002), 367-386. Zbl 1042.26003, MR 1967839 |
| Reference:
|
[11] Scudo, F. M.: Vito Volterra and theoretical ecology.Theor. Population Biology 2 (1971), 1-23. Zbl 0241.92001, MR 0408866, 10.1016/0040-5809(71)90002-5 |
| Reference:
|
[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). |
| Reference:
|
[13] TeBeest, K. G.: Numerical and analytical solutions of Volterra's population model.SIAM Rev. 39 (1997), 484-493. Zbl 0892.92020, MR 1469945, 10.1137/S0036144595294850 |
| Reference:
|
[14] Wazwaz, A.-M.: Analytical approximations and Padé approximants for Volterra's population model.Appl. Math. Comput. 100 (1999), 13-25. Zbl 0953.92026, MR 1665900, 10.1016/S0096-3003(98)00018-6 |
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