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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Quadratic Spline Function for the Approximate Solution of an Intermediate Space-Fractional Advection Diffusion Equation

Journal of Applied Nonlinear Dynamics 6(2) (2017) 225--236 | DOI:10.5890/JAND.2017.06.007

E. A. Abdel-Rehi; M. G. Brikaa

Department of Mathematics, Faculty of Science, Suez Canal University, Ismailia, Egypt

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Abstract

The space fractional advection equation is a linear partial pseudodifferential equation with spatial fractional derivatives in space and is used to model transport at the earth surface. This equation arises when velocity variations are heavy tailed. Space fractional diffusion equation mathematically models the solutes that move through fractal media. In this paper, we are interested in finding the approximation solution of an intermediate fractional advection diffusion equation by using the quadratic spline function. The approximation solution is proved to be conditionally stable. Finally, some numerical examples are given based on this method.

References

  1. [1]  Pachepsky, Y., Timlin, D., and Rawls, W. (2003), Generalized Richards’ equation to simulate water transport in unsaturated soils, Journal of hydrology, 272(1-4), 3-13.
  2. [2]  del Castillo Negrete, D., Carreras, B. A., and Lynch, V. E. (2004), Fractional diffusion in Plasma turbulence, Physics of Plasma, 11(8), 3854-3864.
  3. [3]  Scals, E., Gorenflo, R., and Mainardi, F. (2000), Fractional calculus and continuous-time finance, Physica A: Statisstical Mechanics and its Applications, 284(1-4), 376-384.
  4. [4]  Maiardi, F., Raberto, M., Gorenflo, R., and Scalas, E. (2000), Fractional calculus and continuous-time finance II: the waiting-time distribution, Physica A: Statisstical Mechanics and its Applications, 287, 468-481.
  5. [5]  Gorenflo, R., Mainardi, F., Scalas, E., and Raberto, R. (2010), Fractional calculus and continuous finance III: the diffusion limlit in: M Kohlmann and S. Tang, Mathematical Finance, Birkhauser, 171-180.
  6. [6]  Bai, Z. and Lu, H. (2005), Positive solution for boundary value problem of nonlinear fractional differential equation, Journal of Mathemamtical Analysis and Applications, 311, 495-505.
  7. [7]  El-shahed, M.and Nieto, J. (2010), nontrivial solution for a nonlinear multi-point boundary value problem of fractional order, Journal of Computational Mathematcs and Application, 59, 3438-3443.
  8. [8]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential equations, Elsevier.
  9. [9]  Podlubny, J. (1999), Fractional Differential Equations, Academic Press, New York.
  10. [10]  Podlubny, I. (2002), Geometric and physical interpretation of fractional integration and fractional differentiation, Journal of Fractional Calculus, 5(4), 67-386.
  11. [11]  Zhang, S. (2006), Positive solution for boundary value problem of nonlinear frctional differential equations, Electronical Journal of Differential Equation, 2006, 1-2.
  12. [12]  Vladan, D. Djordjevic, Jova Jaric, Ben Fabry, Jeffrey, J. (2003), Fredberg and Dimitrije Stamenovic, Fractional derivatives embody essential features of cell rheological behavior, Annals of Biomedical Engineering, 31(6), 692-699.
  13. [13]  Abdel-Rehim, E. A. (2015), Implicit difference scheme of the space-time fractional advection diffusion equation, Journal of fractional calculus & Applied Analysis, 18(6), 1252-1276.
  14. [14]  Liu, Q., Liu, F., Turner I. W., and Anh, V. (2007), Approximation of the Levy-Feller advection-dispersion process by random walk and finite difference method, Journal of Computational Physics, 222(1), 57-70.
  15. [15]  Huang, Q., Huang, G., and Zhan, H. (2008), A finite element solution for the fractional advection-dispersion equation, Advances in Water Resources, 31(12), 1578-1589.
  16. [16]  Sweilam, N.H., Khader, M.M., and Adel, M. (2014), Chebyshev pseudo-spectral method for solving fractional advection-dispersion equation, Applied Mathematics, 5, 3240-3248.
  17. [17]  Talat S El Danf, (2013), Numerical solution for the linear time and space fractional diffusion equation, Journal of Vibration and Control, 0(0), 1-9.
  18. [18]  Ramdan, M.A. (2005), Spline solution of the first order delay differential equation, Journal of the Egyptian Mathematical Society, 1, 7-18.
  19. [19]  El-Sayed, A.M.A., Behiry, S.H., and Raslan, W.E. (2010), A numerical algorithm for the solution of an intermediate fractional advection dispersion equation, Communications in Nonlinear Science and Numerical Simulation, 15(5), 1253-1258.
  20. [20]  Abdel-Rehim, E.A. (2013), Explicit Approximation Solutions and Proof of Convergence of the Space-Time Fractional Advection Dispersion Equations, Scientific Research: Applied Mathematics, 4, 1427-1440, http://dx.doi.org/10.4236/am.2013.410193 Published Online October 2013 (http://www.scirp.org/journal/am).
  21. [21]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential equations, Elsevier.
  22. [22]  Zahra Waheed, k. and Elkholy, Samah, M. (2012), Quadratic spline solution for boundary value problem of fractional order, Journal of Numerical Algorithms, 59(3), 373-391.

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