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Mathematical Problems in Engineering
Volume 2013, Article ID 391901, 7 pages
http://dx.doi.org/10.1155/2013/391901
Research Article

On Approximate Solutions for Fractional Logistic Differential Equation

1Department of Mathematics and Statistics, College of Science, Al-Imam Mohammed Ibn Saud Islamic University (IMSIU), P.O. Box 65892, Riyadh 11566, Saudi Arabia
2Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

Received 6 March 2013; Revised 1 April 2013; Accepted 2 April 2013

Academic Editor: Guo-Cheng Wu

Copyright © 2013 M. M. Khader and Mohammed M. Babatin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  2. O. P. Agrawal, “Formulation of Euler-Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 368–379, 2002. View at Publisher · View at Google Scholar · View at Scopus
  3. R. Almeida and D. F. M. Torres, “Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1490–1500, 2011. View at Publisher · View at Google Scholar · View at Scopus
  4. M. M. Khader and A. S. Hendy, “The approximate and exact solutions of the fractional-order delay differential equations using Legendre pseudospectral method,” International Journal of Pure and Applied Mathematics, vol. 74, no. 3, pp. 287–297, 2012. View at Google Scholar
  5. M. M. Khader, T. S. EL Danaf, and A. S. Hendy, “A computational matrix method for solving systems of high order fractional differential equations,” Applied Mathematical Modelling, vol. 37, pp. 4035–4050, 2013. View at Google Scholar
  6. F. Mainardi, “The fundamental solutions for the fractional diffusion-wave equation,” Applied Mathematics Letters, vol. 9, no. 6, pp. 23–28, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus and Applied Analysis, vol. 4, no. 2, pp. 153–192, 2001. View at Google Scholar · View at MathSciNet
  8. N. H. Sweilam, M. M. Khader, and A. M. S. Mahdy, “Numerical studies for fractional-order logistic differential equation with two different delays,” Journal of Applied Mathematics, vol. 2012, Article ID 764894, 14 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  9. J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998. View at Google Scholar · View at Scopus
  10. T. Odzijewicz, A. B. Malinowska, and D. F. M. Torres, “Fractional variational calculus with classical and combined Caputo derivatives,” Nonlinear Analysis. Theory, Methods and Applications, vol. 75, no. 3, pp. 1507–1515, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  11. G. C. Wu and D. Baleanu, “Variational iteration method for fractional calculus-a universal approach by Laplace transform,” Advances in Difference Equations, vol. 2013, article 18, 2013. View at Publisher · View at Google Scholar
  12. G.-C. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,” Physics Letters A, vol. 374, no. 25, pp. 2506–2509, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  13. Q. Wang, “Homotopy perturbation method for fractional KdV equation,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1795–1802, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  14. V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 508–518, 2005. View at Publisher · View at Google Scholar · View at Scopus
  15. J. S. Duan, R. Rach, D. Buleanu, and A. M. Wazwaz, “A review of the Adomian decomposition method and its applications to fractional differential equations,” Communications in Fractional Calculus, vol. 3, pp. 73–99, 2012. View at Google Scholar
  16. M. M. Khader, “On the numerical solutions for the fractional diffusion equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 6, pp. 2535–2542, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. N. H. Sweilam and M. M. Khader, “A Chebyshev pseudo-spectral method for solving fractional-order integro-differential equations,” The ANZIAM Journal, vol. 51, no. 4, pp. 464–475, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  18. J. M. Cushing, An Introduction to Structured Population Dynamics, vol. 71, SIAM, Philadelphia, Pa, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  19. H. Pastijn, “Chaotic growth with the logistic model of P.-F. Verhulst, understanding complex systems,” in The Logistic Map and the Route to Chaos, pp. 3–11, Springer, Berlin, Germany, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  20. K. T. Alligood, T. D. Sauer, and J. A. Yorke, An Introduction to Dynamical Systems, Springer, New York, NY, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  21. M. Ausloos, The Logistic Map and the Route to Chaos: From the Beginnings to Modern Applications, Springer, Berlin, Germany, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  22. Y. Suansook and K. Paithoonwattanakij, “Dynamic of logistic model at fractional order,” in Proceedings of the IEEE International Symposium on Industrial Electronics (IEEE ISIE '09), pp. 718–723, July 2009. View at Publisher · View at Google Scholar · View at Scopus
  23. A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “On the fractional-order logistic equation,” Applied Mathematics Letters, vol. 20, no. 7, pp. 817–823, 2007. View at Publisher · View at Google Scholar · View at Scopus
  24. A. M. A. El-Sayed, F. M. Gaafar, and H. H. G. Hashem, “On the maximal and minimal solutions of arbitrary-orders nonlinear functional integral and differential equations,” Mathematical Sciences Research Journal, vol. 8, no. 11, pp. 336–348, 2004. View at Google Scholar · View at MathSciNet
  25. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, vol. 3, World Scientific Publishing, Singapore, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  26. Y. Chen, B. M. Vinagre, and I. Podlubny, “Continued fraction expansion approaches to discretizing fractional order derivatives-an expository review,” Nonlinear Dynamics, vol. 38, no. 1–4, pp. 155–170, 2004. View at Publisher · View at Google Scholar · View at Scopus
  27. R. A. El-Nabulsi, “Universal fractional Euler-Lagrange equation from a generalized fractional derivate operator,” Central European Journal of Physics, vol. 9, no. 1, pp. 250–256, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. R. A. El-Nabulsi, “A fractional action-like variational approach of some classical, quantum and geometrical dynamics,” International Journal of Applied Mathematics, vol. 17, no. 3, pp. 299–317, 2005. View at Google Scholar · View at MathSciNet
  29. C.-l. Xu and B.-Y. Guo, “Laguerre pseudospectral method for nonlinear partial differential equations,” Journal of Computational Mathematics, vol. 20, no. 4, pp. 413–428, 2002. View at Google Scholar · View at MathSciNet
  30. R. Askey and G. Gasper, “Convolution structures for Laguerre polynomials,” Journal d'Analyse Mathématique, vol. 31, no. 1, pp. 48–68, 1977. View at Publisher · View at Google Scholar · View at Scopus
  31. F. Talay Akyildiz, “Laguerre spectral approximation of Stokes' first problem for third-grade fluid,” Nonlinear Analysis: Real World Applications, vol. 10, no. 2, pp. 1029–1041, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. I. K. Khabibrakhmanov and D. Summers, “The use of generalized Laguerre polynomials in spectral methods for nonlinear differential equations,” Computers and Mathematics with Applications, vol. 36, no. 2, pp. 65–70, 1998. View at Google Scholar · View at Scopus
  33. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, USA, 1964.
  34. Z. Lewandowski and J. Szynal, “An upper bound for the Laguerre polynomials,” Journal of Computational and Applied Mathematics, vol. 99, no. 1-2, pp. 529–533, 1998. View at Google Scholar · View at Scopus
  35. M. Michalska and J. Szynal, “A new bound for the Laguerre polynomials,” Journal of Computational and Applied Mathematics, vol. 133, no. 1-2, pp. 489–493, 2001. View at Publisher · View at Google Scholar · View at Scopus