Approximate Solution of a Model Describing Biological Species Living Together by Taylor Collocation Method

Abstract

In this paper, a numerical method is presented to obtain approximate solutions for the system of nonlinear delay integrodifferential equations derived from considering biological species living together. This method is essentially based on the truncatedTaylor series and its matrix representations with collocation points. Also, to illustrate the pertinent features of the method examples arepresented and results are compared to the Adomian decomposition method, the variational iteration method, pseudospectral Legendremethod. All numerical computations have been performed on the computer algebraic system Maple 15

Keywords

• System of nonlinear integro-differential equations,Taylor polynomials and series,Biological species,Collocation points

Approximate solution of a model describing biological species living together by Taylor collocation method

Öz

References

1. Kot, M., Elements of Mathematical Ecology, Cambridge University Press, (2001).
2. Pougaza, D. B., The Lotka integral equation as a stable population model, Postgraduate Essay, African Institute for Mathematical Sciences (AIMS), (2007).
3. Kopeikin I.D., V.P. Shishkin, Integral form of the general solution of equations of steady-state thermoelasticity, Journal of Appl. Math. Mech. (PMM U.S.S.R.), 48(1), (1984), 117-119.
4. Lotka, A. J., On an integral equation in population analysis, Ann. Math. Stat., 10, (1939), 144-161.
5. Bloom, F., Asymptotic bounds for solutions to a system of damped integro-differential of electromagnetic theory, J Math Anal Appl., 73, (1980), 524-542.
6. Abdou, M.A., Fredholm-Volterra integral equation of the first kind and contact problem, Appl. Math. and Comput., 125,(2002), 177-1
7. Baker, C.T.H., A perspective on the numerical treatment of Volterra equation, J Comput Appl Math.,125, (2000), 217-249.
8. Linz, P., Analytical and numerical methods for Volterra equations, Philadelphia (PA): SIAM, (1985).

9. Volterra, V., Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Memorie del R. Comitato talassografico italiano, (1927).
10. Jerri, A.J., Introduction to integral equations with applications, John Wiley and Sons Inc., (1999).
11. Babolian, E., Biazar, J., Solving the problem of biological species living together by Adomian decomposition method, Appl. Math. and Comput., 129, (2002), 339-343.
12. Parand, K., Razzaghi M., Rational Chebyshev tau method for solving Volterra’s population model, Appl. Math. and Comput., 149, (2004), 893-900.
13. Shakeri, F., Dehghan M., Solution of a model describing biological species living together using the variational iteration method, Mathematical and Computer Modelling, 48, (2008), 685-699.
14. Yousefi, S. A., Numerical Solution of a Model Describing Biological Species Living Together by Using Legendre Multiwavelet Method, International Journal of Nonlinear Science, 11, (2011), 109-113.
15. Sezer, M., Taylor polynomial solutions of Volterra integral equations, Int. J. Math. Educ. Sci. Technol., 25(5), (1994), 625-633.
16. Aky¨uz A., Sezer M., Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients, Appl. Math. Comput., 144, (2003), 237-247.
17. G¨ulsu M, Sezer M, Taylor collocation method for solution of systems of high-order linear Fredholm-Volterra integro-differential equations, Int. J. Comput. Math., 83(4), (2006), 429-448.
18. Kurt N, Sezer M, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, Journal of Franklin Institude, 345(8), (2008), 839-850.
19. Yalc¸ınbas¸ S., Sezer M., Sorkun H., Legendre polynomial solutions of high-order linear Fredholm integro-differential equations, Appl. Math. Comput., 210, (2009), 334-349.
20. B¨ulb¨ul B., G¨ulsu M., Sezer M., A new Taylor collocation method for non-linear Fredholm-Volterra integro-differential equations, Journal of Numerical Methods Partial Differential Equations, 26, (2010), 1006-1020.
21. G¨ulsu M., G¨urb¨uz B., ¨Ozt¨urk Y., Sezer M., Laguerre polynomial approach for solving linear delay difference equations, Appl. Math. Comput., 217, (2011), 6765-6776.
22. Yalc¸ınbas¸ S., Aynıg¨ul M., Sezer M., A collocation method using Hermite polynomials for approximate solution of pantograph equations, Journal of the Franklin Institute, 348, (2011), 1128-1139.
23. Is¸ık O.R., Sezer M., G¨uney Z., Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel, Appl. Math. Comput., 217(16), (2011), 7009-7020.
24. Y¨uzbas¸´y S¸., S¸ahin N., Sezer M., A collocation approach for solving modeling the pollution of a system of lakes, Mathematical and Computer Modelling, 55(3-4), (2012), 330-341.
25. G¨ulsu M., ¨Ozt¨urk Y., Numerical approach for the solution of hypersingular integro-differential equations, Appl. Math. Comput., 230, (2014), 701-710.