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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 203875, 6 pages
http://dx.doi.org/10.1155/2013/203875
Research Article

The Analytical Solution of Some Fractional Ordinary Differential Equations by the Sumudu Transform Method

Hasan Bulut,1 Haci Mehmet Baskonus,2 and Fethi Bin Muhammad Belgacem3

1Department of Mathematics, Firat University, Elazig, Turkey
2Department of Computer Engineering, Tunceli University, Tunceli, Turkey
3Department of Mathematics, Faculty of Basic Education, PAAET, Shamiya, Kuwait

Received 16 March 2013; Accepted 31 July 2013

Academic Editor: Santanu Saha Ray

Copyright © 2013 Hasan Bulut et al. 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. G. K. Watugala, “Sumudu transform: a new integral transform to solve differential equations and control engineering problems,” International Journal of Mathematical Education in Science and Technology, vol. 24, no. 1, pp. 35–43, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. G. K. Watugala, “Sumudu transform—a new integral transform to solve differential equations and control engineering problems,” Mathematical Engineering in Industry, vol. 6, no. 4, pp. 319–329, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. G. K. Watugala, “The Sumudu transform for functions of two variables,” Mathematical Engineering in Industry, vol. 8, no. 4, pp. 293–302, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. S. Weerakoon, “Application of Sumudu transform to partial differential equations,” International Journal of Mathematical Education in Science and Technology, vol. 25, no. 2, pp. 277–283, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. S. Weerakoon, “Complex inversion formula for Sumudu transform,” International Journal of Mathematical Education in Science and Technology, vol. 29, no. 4, pp. 618–621, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. A. B. Deakin, “The “Sumudu transform” and the Laplace transform,” International Journal of Mathematical Education in Science and Technology, vol. 28, p. 159, 1997. View at Google Scholar
  7. S. Weerakoon, “The “Sumudu transform” and the Laplace transform—reply,” International Journal of Mathematical Education in Science and Technology, vol. 28, no. 1, p. 160, 1997. View at Google Scholar
  8. M. A. Asiru, “Sumudu transform and the solution of integral equations of convolution type,” International Journal of Mathematical Education in Science and Technology, vol. 32, no. 6, pp. 906–910, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. A. Aşiru, “Further properties of the Sumudu transform and its applications,” International Journal of Mathematical Education in Science and Technology, vol. 33, no. 3, pp. 441–449, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. F. B. M. Belgacem, A. A. Karaballi, and S. L. Kalla, “Analytical investigations of the Sumudu transform and applications to integral production equations,” Mathematical Problems in Engineering, no. 3-4, pp. 103–118, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  11. F. B. M. Belgacem and A. Karaballi, “Sumudu transform fundamental properties investigations and applications,” Journal of Applied Mathematics and Stochastic Analysis, Article ID 91083, 23 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. F. B. M. Belgacem, “Introducing and analyzing deeper Sumudu properties Sumudu transform fundemantal properties investigations and applications,” Nonlinear Studies Journal, vol. 1, no. 31, pp. 101–114, 2006. View at Google Scholar
  13. F. B. M. Belgacem, “Applications of the Sumudu transform to indefinite periodic parabolic problems,” in Proceedings of the International Conference on Nonlinear Problems and Aerospace Applications (ICNPAA '06), chapter 6, pp. 51–60, 2006.
  14. M. G. M. Hussain and F. B. M. Belgacem, “Transient solutions of Maxwell's equations based on Sumudu transform,” Progress in Electromagnetics Research, vol. 74, pp. 273–289, 2007. View at Google Scholar · View at Scopus
  15. F. B. M. Belgacem, “Sumudu applications to Maxwell's equations,” The Plan Index Enquiry and Retrieval System Online, vol. 5, pp. 355–360, 2009. View at Google Scholar
  16. F. B. M. Belgacem, “Sumudu transform applications to Bessel functions and equations,” Applied Mathematical Sciences, vol. 4, no. 73-76, pp. 3665–3686, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Q. D. Katatbeh and F. B. M. Belgacem, “Applications of the Sumudu transform to fractional differential equations,” Nonlinear Studies, vol. 18, no. 1, pp. 99–112, 2011. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Salinas, A. Jimenez, F. Arteaga, and J. Rodriguez, “Estudio analitico de la transformada de Sumudu y algunas aplicaciones a la teoria de control,” Revista Ingeneria UC, vol. 11, no. 3, pp. 79–86, 2004 (Spanish). View at Google Scholar
  19. M. A. Rana, A. M. Siddiqui, Q. K. Ghori, and R. Qamar, “Application of He's homotopy perturbation method to Sumudu transform,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 8, no. 2, pp. 185–190, 2007. View at Google Scholar · View at Scopus
  20. A. Tamrabet and A. Kadem, “A combined Walsh function and Sumudu transform for solving the two-dimensional neutron transport equation,” International Journal of Mathematical Analysis, vol. 1, no. 9-12, pp. 409–421, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. J. M. Tchuenche and N. S. Mbare, “An application of the double Sumudu transform,” Applied Mathematical Sciences, vol. 1, no. 1–4, pp. 31–39, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J. Zhang, “Sumudu transform based coefficients calculation,” Nonlinear Studies, vol. 15, no. 4, pp. 355–372, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. V. B. L. Chaurasia, R. S. Dubey, and F. B. M. Belgacem, “Fractional radial diffusion equation analytical solution via Hankel and Sumudu transforms,” International Journal of Mathematics in Engineering Science and Aerospace, vol. 3, no. 2, pp. 1–10, 2012. View at Google Scholar
  24. P. Goswami and F. B. M. Belgacem, “Fractional differential equation solutions through a Sumudu rational,” Nonlinear Studies, vol. 19, no. 4, pp. 591–598, 2012. View at Google Scholar · View at MathSciNet
  25. F. B. M. Belgacem, “Introducing and analysing deeper Sumudu properties,” Nonlinear Studies, vol. 13, no. 1, pp. 23–41, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. F. B. M. Belgacem and R. Silambarasan, “Maxwell's equations solutions by means of the natural transform,” International Journal of Mathematics in Engineering, Science And Aeorospace, vol. 3, no. 3, pp. 313–323, 2012. View at Google Scholar
  27. S. K. Alomari, “An estimate of Sumudu transforms for Boehmians,” in Proceedings of the Conference on Mathematical problems in Engineering, Aerospace and Science, Vienna, Austria, 2012.
  28. H. Bulut, H. M. Baskonus, and S. Tuluce, “The solutions of partial differential equations with variable coefficients by sumudu transform method,” AIP Proceeding, vol. 1493, p. 91, 2012. View at Google Scholar
  29. H. Z. Khan and W. A. Khan, “N-transform-properties and applications,” NUST Journal of Engineering Sciences, vol. 1, no. 1, pp. 127–133, 2008. View at Google Scholar
  30. G. Adomian, “A review of the decomposition method and some recent results for nonlinear equations,” Mathematical and Computer Modelling, vol. 13, no. 7, pp. 17–43, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. G. Adomian and R. Rach, “Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations,” Computers & Mathematics with Applications, vol. 19, no. 12, pp. 9–12, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, vol. 60 of Fundamental Theories of Physics, Kluwer Academic Publishers, Boston, Mass, USA, 1994. View at MathSciNet
  33. G. Adomian, “A review of the decomposition method in applied mathematics,” Journal of Mathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. Z.-G. Deng and G.-C. Wu, “Approximate solution of fractional differential equations with uncertainty,” Romanian Journal of Physics, vol. 56, no. 7-8, pp. 868–872, 2011. View at Google Scholar · View at MathSciNet
  35. M. A. Abdou and A. A. Soliman, “Variational iteration method for solving Burger's and coupled Burger's equations,” Journal of Computational and Applied Mathematics, vol. 181, no. 2, pp. 245–251, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  36. Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential equations of fractional order,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 27–34, 2006. View at Google Scholar · View at Scopus
  37. J.-H. He and X.-H. Wu, “Variational iteration method: new development and applications,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 881–894, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  38. N. H. Sweilam, “Fourth order integro-differential equations using variational iteration method,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 1086–1091, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. S. Abbasbandy, “A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 59–63, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. A.-M. Wazwaz, “The variational iteration method for exact solutions of Laplace equation,” Physics Letters A, vol. 363, no. 4, pp. 260–262, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. M. A. Asiru, “Application of the sumudu transform to discrete dynamical systems,” International Journal of Mathematical Education, Science and Technology, vol. 34, no. 6, pp. 944–949, 2003. View at Google Scholar
  42. H. Bulut, H. M. Baskonus, and S. Tuluce, “Homotopy perturbation sumudu transform method for heat equations,” Mathematics in Engineering, Science and Aerospace, vol. 4, no. 1, pp. 49–60, 2013. View at Google Scholar
  43. F. Jarad and K. Taş, “On Sumudu transform method in discrete fractional calculus,” Abstract and Applied Analysis, vol. 2012, Article ID 270106, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. V. G. Gupta, B. Shrama, and A. Kiliçman, “A note on fractional Sumudu transform,” Journal of Applied Mathematics, vol. 2010, Article ID 154189, 9 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. R. C. Mittal and R. Nigam, “Solution of fractional integro-differential equations by adomian decomposition method,” The International Journal of Applied Mathematics and Mechanics, vol. 4, no. 2, pp. 87–94, 2008. View at Google Scholar
  46. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  47. M. Caputo, “Linear model of dissipation whose Q is almost frequency independent-II,” Geophysical Journal of the Royal Astronomical Society, vol. 13, no. 5, pp. 529–539, 1967. View at Google Scholar
  48. F. B. M. Belgacem and R. Silambarasan, “The Sumudu transform analysis by infinite series,” in Proceedings of the AIP Proceeding, Vienna, Austria, 2012.
  49. P. Goswami and F. B. M. Belgacem, “Solving special fractional differential equations by the Sumudu transform,” in Proceedings of the AIP Proceeding, Vienna, Austria, 2012.
  50. Z. M. Odibat and S. Momani, “An algorithm for the numerical solution of differential equations of fractional order,” Journal of Applied Mathematics and Informatics, vol. 26, no. 1-2, pp. 15–127.
  51. N. J. Ford and A. C. Simpson, “The numerical solution of fractional differential equations: speed versus accuracy,” Numerical Analysis Report no. 385, A Report in Association with Chester College, 2003. View at Google Scholar