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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 270106, 16 pages
http://dx.doi.org/10.1155/2012/270106
Research Article

On Sumudu Transform Method in Discrete Fractional Calculus

Department of Mathematics and Computer Science, Cankaya University, Eskisehir Yolu 29 km, 06810 Ankara, Turkey

Received 16 April 2012; Revised 4 June 2012; Accepted 5 June 2012

Academic Editor: Paul Eloe

Copyright © 2012 Fahd Jarad and Kenan Taş. 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. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
  2. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives—Theory and Applications, Gordon and Breach Science Publishers, Linghorne, Pa, USA, 1993.
  3. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  4. R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Redding, Conn, USA, 2006.
  5. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar
  6. N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives,” Rheologica Acta, vol. 45, pp. 765–771, 2006.
  7. K. S. Miller and B. Ross, “Fractional difference calculus,” in Proceedings of the Univalent Functions, Fractional Calculus, and Their Applications, pp. 139–152, Nihon University, 1989.
  8. F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165–176, 2007.
  9. F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981–989, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. F. M. Atıcı and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1–12, 2009. View at Zentralblatt MATH
  11. T. Abdeljawad and D. Baleanu, “Fractional differences and integration by parts,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 574–582, 2011. View at Zentralblatt MATH
  12. T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602–1611, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. 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
  14. 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 Zentralblatt MATH
  15. 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
  16. 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
  17. 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
  18. F. B. M. Belgacem and A. A. Karballi, “Sumudu transform fundemantal properties investigations and applications,” Journal of Applied Mathematics and Stochastic Analysis, vol. 2006, Article ID 91083, 23 pages, 2006. View at Publisher · View at Google Scholar
  19. A. Kılıçman and H. Eltayeb, “On the applications of Laplace and Sumudu transforms,” Journal of the Franklin Institute, vol. 347, no. 5, pp. 848–862, 2010. View at Publisher · View at Google Scholar
  20. F. B. M. Belgacem, “Introducing and analysing deeper Sumudu properties,” Nonlinear Studies, vol. 13, no. 1, pp. 23–41, 2006. View at Zentralblatt MATH
  21. F. Jarad and K. Tas, “Application of Sumudu and double Sumudu transforms to Caputo-Fractional dierential equations,” Journal of Computational Analysis and Applications, vol. 14, no. 3, pp. 475–483, 2012.
  22. 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 Zentralblatt MATH
  23. M. Bohner and G. Sh. Guseinov, “The h-Laplace and q-Laplace transforms,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 75–92, 2010. View at Publisher · View at Google Scholar
  24. S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990. View at Zentralblatt MATH
  25. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001. View at Publisher · View at Google Scholar
  26. M. Bohner and A. Peterson, Advances in Dynamic equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. View at Publisher · View at Google Scholar
  27. M. T. Holm, The theory of discrete fractional calculus: development and application [Ph.D. thesis], 2011.
  28. F. Jarad, K. Bayram, T. Abdeljawad, and D. Baleanu, “On the discrete sumudu transform,” Romanian Reports in Physics. In press.