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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 620529, 7 pages
http://dx.doi.org/10.1155/2014/620529
Research Article

Local Fractional Sumudu Transform with Application to IVPs on Cantor Sets

1Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4
2Department of Physics, Urmia Branch, Islamic Azad University, P.O. Box 969, Orumiyeh, Iran
3Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
4Institute of Space Sciences, Magurele, 077125 Bucharest, Romania
5Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey
6Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Received 13 February 2014; Accepted 10 May 2014; Published 26 May 2014

Academic Editor: Jordan Hristov

Copyright © 2014 H. M. Srivastava 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. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Company, San Francisco, Calif, USA, 1982. View at MathSciNet
  2. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, New York, NY, USA, 1990. View at MathSciNet
  3. K. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, Chichester, UK, 1997. View at MathSciNet
  4. G. A. Edgar, Integral, Probability, and Fractal Measures, Springer, New York, NY, USA, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  5. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at MathSciNet
  6. K. S. Miller and B. Ross, An Introduction to the Fractional Integrals and Derivatives Theory and Applications, John Wiley and Sons, New York, NY, USA, 1993.
  7. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. View at MathSciNet
  8. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  9. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  10. 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 B.V., Amsterdam, The Netherlands, 2006. View at MathSciNet
  11. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  12. V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Background and Theory, Nonlinear Physical Science, Springer, Berlin, Germany, 2013. View at MathSciNet
  13. 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 · View at MathSciNet
  14. W. Chen, H. Sun, X. Zhang, and D. Korošak, “Anomalous diffusion modeling by fractal and fractional derivatives,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1754–1758, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,” Physical Review Letters, vol. 80, no. 2, pp. 214–217, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Parvate and A. D. Gangal, “Calculus on fractal subsets of real line. I. Formulation,” Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society, vol. 17, no. 1, pp. 53–81, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. A. Parvate, S. Satin, and A. D. Gangal, “Calculus on fractal curves in 𝐑n,” Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society, vol. 19, no. 1, pp. 15–27, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. K. Golmankhaneh, A. K. Golmankhaneh, and D. Baleanu, “About Maxwell's equations on fractal subsets of R3,” Central European Journal of Physics, vol. 11, no. 6, pp. 863–867, 2013. View at Google Scholar
  19. A. K. Golmankhaneh, V. Fazlollahi, and D. Baleanu, “Newtonian mechanics on fractals subset of real-line,” Romanian Reports in Physics, vol. 65, no. 1, pp. 84–93, 2013. View at Google Scholar · View at Scopus
  20. X.-J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
  21. X.-J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic, Hong Kong, 2011.
  22. X.-J. Yang, D. Baleanu, and J. A. Tenreiro Machado, “Systems of Navier-Stokes equations on Cantor sets,” Mathematical Problems in Engineering, vol. 2013, Article ID 769724, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  23. Y. Zhao, D. Baleanu, C. Cattani, D.-F. Cheng, and X.-J. Yang, “Maxwell's equations on Cantor sets: a local fractional approach,” Advances in High Energy Physics, vol. 2013, Article ID 686371, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  24. X.-J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of the Heisenberg uncertainty principle within local fractional Fourier analysis,” Boundary Value Problems, vol. 2013, 131, 2013. View at Publisher · View at Google Scholar
  25. X.-J. Ma, H. M. Srivastava, D. Baleanu, and X.-J. Yang, “A new Neumann series method for solving a family of local fractional Fredholm and Volterra integral equations,” Mathematical Problems in Engineering, vol. 2013, Article ID 325121, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  26. D. Baleanu, J. A. Tenreiro Machado, C. Cattani, M. C. Baleanu, and X.-J. Yang, “Local fractional variational iteration and decomposition methods for wave equation on Cantor sets within local fractional operators,” Abstract and Applied Analysis, vol. 2014, Article ID 535048, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  27. J.-H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. C. F. Liu, S. S. Kong, and S. J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,” Thermal Science, vol. 17, no. 3, pp. 715–721, 2013. View at Google Scholar
  29. C.-G. Zhao, A.-M. Yang, H. Jafari, and A. Haghbin, “The Yang-Laplace transform for solving the IVPs with local fractional derivative,” Abstract and Applied Analysis, vol. 2014, Article ID 386459, 5 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  30. S.-Q. Wang, Y.-J. Yang, and H. K. Jassim, “Local fractional function decomposition method for solving inhomogeneous wave equations with local fractional derivative,” Abstract and Applied Analysis, vol. 2014, Article ID 176395, 7 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  31. 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
  32. 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
  33. M. A. B. Deakin, “The Sumudu transform and the Laplace transform,” International Journal of Mathematical Education in Science and Technology, vol. 28, no. 1, pp. 159–160, 1997. View at Google Scholar
  34. 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
  35. 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
  36. H. Eltayeb and A. Kılıçman, “A note on the Sumudu transforms and differential equations,” Applied Mathematical Sciences, vol. 4, no. 21–24, pp. 1089–1098, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. 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