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International Journal of Latest Research in Science and Technology

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EL-GENDI NODAL GALERKIN METHOD FOR SOLVING LINEAR AND NONLINEAR PARTIAL FRACTIONAL SPACE EQUATIONS

Research Paper Open Access

International Journal of Latest Research in Science and Technology Vol.2 Issue 6, pp 10-17,Year 2013

EL-GENDI NODAL GALERKIN METHOD FOR SOLVING LINEAR AND NONLINEAR PARTIAL FRACTIONAL SPACE EQUATIONS

M. El-Kady, Salah M. El-Sayed, Heba. E. Salem

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Received : 21 November 2013; Accepted : 15 December 2013 ; Published : 31 December 2013

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Abstract

In this paper, an efficient numerical technique is presented to solve the partial fractional space equations with variable coefficients on a finite domain. This technique based on nodal Galerkin method. The fractional derivatives are described in the Caputo sense. Also, a fully discrete scheme is given for a type of nonlinear space-fractional anomalous advection-diffusion equation. In this paper, the problems can be reduced to a set of linear algebraic equations by using the Chebyshev nodal Galerkin method. The existence and uniqueness of the solution for the linear semi discrete weak form solutions are proved. And the stability analysis for the linear semi and fully discrete schemes are discussed. Numerical solutions obtained by this method are in excellent agreement and efficient to use with those obtained by previous work in the literature.

Key Words   
Shifted Chebyshev polynomial; Nodal Galerkin method; Fractional diffusion equation

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References
  1. Boyd, J.P., “Chebyshev and Fourier spectral methods,” Dover, Mineola, 2001.
  2. Choi, H. W., Chung, S. K., Lee, Y. J., “Numerical solutions for spacefractional dispersion equations with nonlinear source terms,” Bull. Korean Math. Soc. 47, 1225-1234, 2010.
  3. Dalir, M., Bashour, M., “Applications of fractional calculus, ” Appl. Math. Sci. 4, 1021–1032, 2010.
  4. Diethelm, K., “The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type,” Springer, 2004.
  5. Elbarbary, M.E., El-Sayed, M., “Higher order pseudospectral differentiation matrices,” Applied Numerical Mathematics. 55, 425–438, 2005.
  6. Ervin, V. J., Heuer, N., Roop, J. P., “Numerical approximation of a time dependent, nonlinear, spacefractional diffusion equation,” SIAM Journal on Numerical Analysis. 45 (2), 572–591, 2008.
  7. Ervin, V.J., Roop, J.P., “Variational formulation for the stationary fractional advection dispersion equation,” Numer. Meth. Part. D. E. 22(3), 558-576, 2006.
  8. Henry, B. I., M. Langlands, T. A., Wearne, S. L., “Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations, Physical Review E, article 031116, 74 (3), 2006.
  9. Hesthaven, J.S., Gottlieb, S., Gottlieb, D., “Spectral Methods for Time-Dependent Problems,” The Cambridge monographs on applied and computational mathematics, 2007.
  10. Khader, M.M., Sweilam, N.H., Mahdy, A.S., “An efficient numerical method for solving the fractional diffusion equation,” Journal of Applied Mathematics & Bioinformatics. 1, 1-12, 2011.
  11. Khater, A.H., Temsah, R.S., Hassan, M.M., “A Chebyshev spectral collocation method for solving Burgers’-type equations,” J. of computational and applied mathematics. 222, 333–350, 2008.
  12. Khader, M.M., “On the numerical solutions for the fractional diffusion equation,” Commun. Nonlinear Sci. Numer. Simul. 16, 2535–2542, 2011.
  13. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., “Theory and Applications of Fractional Differential Equations,” Elsevier, San Diego, 2006.
  14. Li, C. P., Zhao, Z. G., Chen, Y. Q., “Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion,” Computers & Mathematics with Applications. 62, 855–875, 2011.
  15. Li, X.J., Xu, C.J., “A space-time spectral method for the time fractional differential equation,” SIAM J. Numer. Anal. 47(3), 2108-2131, 2009.
  16. Magin, R. L., Abdullah, O., Baleanu, D., Zhou, X. J., “Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation,” Journal of Magnetic Resonance. 190 (2), 255–270, 2008.
  17. Meerschaert, M. M., Benson, D. A., Baeumer, B., “Operator Levy motion and multiscaling anomalous diffusion,” Physical Review E. article 021112, 63 (2I), 2001.
  18. Ortega, J. M., Rheinboldt, W. C., “Iterative Solution of Nonlinear Equations in Several Variables,” Academic Press, New York-London, 1970.
  19. Podlubny, I., “Fractional Differential Equations,” vol. 198, Academic Press, San Diego, Calif, USA, 1999.
  20. Ray, S.S., “Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method,” Commun. Nonlinear Sci. Numer. Simul. 14, 1295–1306, 2009.
  21. Saadatmandi, A., Dehghan, M., “A tau approach for solution of the space fractional diffusion equation,” Computers and Mathematics with Applications. 62 1135–1142, 2011.
  22. Shen, J., Tang, T., “Spectral and High-Order Methods with Applications,” Science Press of China, 2006.
  23. Zheng, Y., Zhao, Z., “A fully discrete Galerkin method for a nonlinear Space-fractional diffusion Equation,” Mathematical Problems in Engineering. Article 171620, 20 pages, 2011.
To cite this article

M. El-Kady, Salah M. El-Sayed, Heba. E. Salem , " El-gendi Nodal Galerkin Method For Solving Linear And Nonlinear Partial Fractional Space Equations ", International Journal of Latest Research in Science and Technology . Vol. 2, Issue 6, pp 10-17 , 2013

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