Table of Contents Author Guidelines Submit a Manuscript
Table of Contents Alerts

To receive news and publication updates for Abstract and Applied Analysis, enter your email address in the box below.

Abstract and Applied Analysis
Volume 2016, Article ID 2913539, 5 pages
http://dx.doi.org/10.1155/2016/2913539
Research Article

The Approximate Solutions of Three-Dimensional Diffusion and Wave Equations within Local Fractional Derivative Operator

Hassan Kamil Jassim

Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq

Received 5 June 2016; Accepted 24 August 2016

Academic Editor: Zhenhua Guo

Copyright © 2016 Hassan Kamil Jassim. 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. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.
  2. B. Q. Li, Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer, Computational Fluid and Solid Mechanics, Springer, London, UK, 2006. View at MathSciNet
  3. H. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics, Prentice Hall, Upper Saddle River, NJ, USA, 2007.
  4. X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic, Hong Kong, 2011.
  5. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Scientific, New York, NY, USA, 2012.
  6. V. Christianto and B. Rahul, “A derivation of proca equations on cantor sets: a local fractional approach,” Bulletin of Mathematical Sciences & Applications, vol. 10, pp. 48–56, 2014. View at Publisher · View at Google Scholar
  7. H.-Y. Liu, J.-H. He, and Z.-B. Li, “Fractional calculus for nanoscale flow and heat transfer,” International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24, no. 6, pp. 1227–1250, 2014. View at Publisher · View at Google Scholar · View at Scopus
  8. Y.-J. Hao, H. M. Srivastava, H. Jafari, and X.-J. Yang, “Helmholtz and diffusion equations associated with local fractional derivative operators involving the Cantorian and Cantor-type cylindrical coordinates,” Advances in Mathematical Physics, vol. 2013, Article ID 754248, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  9. A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, NY, USA, 1997.
  10. A. M. Spasic and M. P. Lazarevic, “Electroviscoelasticity of liquid/liquid interfaces: fractional-order model,” Journal of Colloid and Interface Science, vol. 282, no. 1, pp. 223–230, 2005. View at Publisher · View at Google Scholar · View at Scopus
  11. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999. View at MathSciNet
  12. X. J. Yang, D. Baleanu, and W. P. Zhong, “Approximation solutions for diffusion equation on Cantor time-space,” Proceeding of the Romanian Academy A, vol. 14, no. 2, pp. 127–133, 2013. View at Google Scholar
  13. Z.-P. Fan, H. K. Jassim, R. K. Raina, and X.-J. Yang, “Adomian decomposition method for three-dimensional diffusion model in fractal heat transfer involving local fractional derivatives,” Thermal Science, vol. 19, supplement 1, pp. S137–S141, 2015. View at Publisher · View at Google Scholar · View at Scopus
  14. H. Jafari and H. K. Jassim, “Local fractional adomian decomposition method for solving two dimensional heat conduction equations within local fractional operators,” Journal of Advance in Mathematics, vol. 9, no. 4, pp. 2574–2582, 2014. View at Google Scholar
  15. D. Baleanu, J. A. T. 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 · View at Scopus
  16. X.-J. Yang, D. Baleanu, Y. Khan, and S. T. Mohyud-Din, “Local fractional variational iteration method for diffusion and wave equations on Cantor sets,” Romanian Journal of Physics, vol. 59, no. 1-2, pp. 36–48, 2014. View at Google Scholar · View at MathSciNet · View at Scopus
  17. S. Xu, X. Ling, Y. Zhao, and H. K. Jassim, “A novel schedule for solving the two-dimensional diffusion problem in fractal heat transfer,” Thermal Science, vol. 19, supplement 1, pp. S99–S103, 2015. View at Publisher · View at Google Scholar · View at Scopus
  18. A.-M. Yang, X.-J. Yang, and Z.-B. Li, “Local fractional series expansion method for solving wave and diffusion equations on Cantor sets,” Abstract and Applied Analysis, vol. 2013, Article ID 351057, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Y. Cao, W. G. Ma, and L. C. Ma, “Local fractional functional method for solving diffusion equations on Cantor sets,” Abstract and Applied Analysis, vol. 2014, Article ID 803693, 6 pages, 2014. View at Publisher · View at Google Scholar
  20. 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 · View at Scopus
  21. H. K. Jassim, “Local fractional Laplace decomposition method for nonhomogeneous heat equations arising in fractal heat flow with local fractional derivative,” International Journal of Advances in Applied Mathematics and Mechanics, vol. 2, no. 4, pp. 1–7, 2015. View at Google Scholar
  22. Y. Zhang, C. Cattani, and X.-J. Yang, “Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains,” Entropy, vol. 17, no. 10, pp. 6753–6764, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. X.-J. Yang, D. Baleanu, and H. M. Srivastava, “Local fractional similarity solution for the diffusion equation defined on Cantor sets,” Applied Mathematics Letters, vol. 47, pp. 54–60, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. X.-J. Yang, J. A. T. Machado, and H. M. Srivastava, “A new numerical technique for solving the local fractional diffusion equation: two-dimensional extended differential transform approach,” Applied Mathematics and Computation, vol. 274, pp. 143–151, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. H. Jafari, H. K. Jassim, F. Tchier, and D. Baleanu, “On the approximate solutions of local fractional differential equations with local fractional operator,” Entropy, vol. 18, no. 150, pp. 1–12, 2016. View at Publisher · View at Google Scholar