Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 871635, 6 pages
http://dx.doi.org/10.1155/2015/871635
Research Article

Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

1Department of Mathematics, Faculty of Science, Assiut University, New Valley Branch, El-Kharja 72511, Egypt
2Department of Mathematics, Faculty of Science, Northern Border University, Arar 91431, Saudi Arabia
3Department of Mathematics, Faculty of Science, Qassim University, Buraydah 51452, Saudi Arabia

Received 27 May 2014; Revised 25 August 2014; Accepted 26 August 2014

Academic Editor: Zhong-Ke Gao

Copyright © 2015 Emad A.-B. Abdel-Salam and Zeid I. A. Al-Muhiameed. 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, Elsevier, Amsterdam, Netherlands, 2006. View at MathSciNet
  2. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, New Jersey, NJ, USA, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  3. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  4. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  5. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.
  6. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet
  7. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  8. M. Cui, “Compact finite difference method for the fractional diffusion equation,” Journal of Computational Physics, vol. 228, no. 20, pp. 7792–7804, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. X. Zhao and Z. Z. Sun, “Compact Crank-Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium,” Journal of Scientific Computing, 2014. View at Publisher · View at Google Scholar
  10. Q. Huang, G. Huang, and H. Zhan, “A finite element solution for the fractional advection–dispersion equation,” Advances in Water Resources, vol. 31, no. 12, pp. 1578–1589, 2008. View at Publisher · View at Google Scholar
  11. A. M. A. El-Sayed and M. Gaber, “The Adomian decomposition method for solving partial differential equations of fractal order in finite domains,” Physics Letters A, vol. 359, no. 3, pp. 175–182, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  12. A. M. A. El-Sayed, S. H. Behiry, and W. E. Raslan, “Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1759–1765, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  13. Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194–199, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  14. J. H. He, “Variational iteration method for delay differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 2, no. 4, pp. 235–236, 1997. View at Google Scholar
  15. G. 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
  16. S. Guo and L. Mei, “The fractional variational iteration method using He's polynomials,” Physics Letters A, vol. 375, no. 3, pp. 309–313, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  17. J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  18. J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  19. Q. Xu and J. S. Hesthaven, “Stable multi-domain spectral penalty methods for fractional partial differential equations,” Journal of Computational Physics, vol. 257, pp. 241–258, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  20. M. Zayernouri and G. E. Karniadakis, “Exponentially accurate spectral and spectral element methods for fractional ODEs,” Journal of Computational Physics, vol. 257, pp. 460–480, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  21. Q. Xu and J. S. Hesthaven, “Discontinuous Galerkin method for fractional convection-diffusion equations,” SIAM Journal on Numerical Analysis, vol. 52, no. 1, pp. 405–423, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  22. G. Pang, W. Chen, and Z. Fu, “Space-fractional advection-dispersion equations by the Kansa method,” Journal of Computational Physics, 2014. View at Publisher · View at Google Scholar
  23. S. Zhang and H. Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, vol. 375, no. 7, pp. 1069–1073, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  24. S. Guo, L. Mei, Y. Li, and Y. Sun, “The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics,” Physics Letters A, vol. 376, no. 4, pp. 407–411, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  25. E. A. Abdel-Salam and E. A. Yousif, “Solution of nonlinear space-time fractional differential equations using the fractional Riccati expansion method,” Mathematical Problems in Engineering, vol. 2013, Article ID 846283, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  26. G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  27. G. Jumarie, “New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations,” Mathematical and Computer Modelling, vol. 44, no. 3-4, pp. 231–254, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. G. Jumarie, “Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-LIOuville derivative,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1659–1664, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  29. R. Almeida and D. F. M. Torres, “Fractional variational calculus for nondifferentiable functions,” Computers Mathematics with Applications, vol. 61, pp. 3097–3104, 2011. View at Publisher · View at Google Scholar
  30. K. M. Kolwankar, “Decomposition of Lebesgue-Cantor devil’s staircase,” Fractals, vol. 12, no. 4, pp. 375–380, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  31. G. Jumarie, “An approach to differential geometry of fractional order via modified Riemann-Liouville derivative,” Acta Mathematica Sinica, vol. 28, no. 9, pp. 1741–1768, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  32. P. G. Drazin and R. S. Johnson, Solitons : An Introduction, Cambridge University Press, Cambridge, UK, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  33. A. H. Khater, O. H. El-Kakaawy, and D. K. Callebaut, “Bäcklund transformations and exact solutions for Alfvén solitons in a relativistic electron–positron plasma,” Physica Scripta, vol. 58, no. 6, p. 545, 1998. View at Publisher · View at Google Scholar
  34. T. Kakutani and N. Yamasaki, “Solitary waves on a two-layer fluid,” Journal of the Physical Society of Japan, vol. 45, no. 2, pp. 674–679, 1978. View at Publisher · View at Google Scholar · View at Scopus
  35. E. A.-B. Abdel-Salam, “Quasi-periodic, periodic waves, and soliton solutions for the combined KdV-mKdV equation,” Zeitschrift fur Naturforschung A, vol. 64, no. 9-10, pp. 639–645, 2009. View at Google Scholar
  36. M. I. Nouh and A. S. Saad, “A new analytical solution to the relativistic polytropic fluid spheres,” International Review of Physics, vol. 7, no. 1, pp. 16–21, 2013. View at Google Scholar