A numerical method based on explicit finite difference for solving fractional hyperbolic PDE’s

  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract

    In this paper, a new numerical scheme based on explicit finite difference approximation for solving fractional hyperbolic partial differential equations (FHPDE’s) is formulated. Numerical studies for the model problems are presented to confirm the accuracy and the effectiveness of the proposed method. The obtained results of proposed system are compared with exact solutions and the original system to show the efficient of the new method.

  • Keywords

    Fractional Hyperbolic Partial Differential Equations; Preconditioned Explicit Finite Difference Method.

  • References

      [1] K. Nishimoto, Fractional Calculus: Integrations and Differentiations of Arbitrary Order, Descartes Press Company Koriyama Japan, 1983.

      [2] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys.Rep. 339, 1(2000) 1-77. https://doi.org/10.1016/S0370-1573(00)00070-3.

      [3] G. Zaslavsky, Chaos, fractional kinetics and anomalous transport. Phys. Rep. 371, 6, (2002) 461- 580.

      [4] Magin, RL Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32, 1(2004)1- 104.

      [5] H. Jafari, V. D. Gejji, Solving linear and nonlinear fractional diffusion and wave equations by adomian decomposition. Appl. Math. and Comput., 180(2006) 488-497. https://doi.org/10.1016/j.amc.2005.12.031.

      [6] N. H. Sweilam, A. M. Nagy, Numerical solution of fractional wave equation using Crank-Nicolson method. World Appl. Sci. J., 13(2011) 71-75.

      [7] N. H. Sweilam, M. M. Khader , A. M. Nagy, Numerical solution of two-sided space fractional wave equation using finite difference method.Journal of Computational and Applied Mathematics, 235, 8(2011) 2832-2841. https://doi.org/10.1016/j.cam.2010.12.002.

      [8] A. R. Abdullah, the Four Point Explicit Decoupled Group (EDG) Method: A Fast Poisson Solver. International Journal of Comp. Math. 38 (1991) 61-70. https://doi.org/10.1080/00207169108803958.

      [9] M. Othman, A. R. Abdullah, An Efficient Four Points Modified Explicit Group Poisson Solver, International Journal of Computer Mathematics, 76 (2000) 203-217.

      [10] A. M. Saeed and N. H. M. Ali, Preconditioned Modified Explicit Decoupled Group Method In The Solution Of Elliptic PDEs, Applied Mathematical Sciences.4 ,24 (2010) 1165-1181.

      [11] A. M. Saeed, N. H. M. Ali, On the Convergence of the Preconditioned Group Rotated Iterative Methods In The Solution of Elliptic PDEs, Applied Mathematics &Information Sciences, 5, 1, (2011) 65-73.

      [12] M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional Advection-dispersion flow equations. J. Comput. Appl. Math. 172, 1, (2004) 65-77.

      [13] M. Meerschaert, C. Tadjeran, finite difference approximation for two –sided space fractional partial differential equations, Applied Numerical Mathematics, 56 (2006)80-90. https://doi.org/10.1016/j.apnum.2005.02.008.

      [14] K. Nishimoto, Fractional Calculus: Integrations and Differentiations of Arbitrary Order, Descartes Press Company Koriyama Japan, 1983.

      [15] G. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press, 1978.

      [16] L. Sidiqi, some finite difference method for solving fractional differential equations, M.Sc. Thesis, college of Science, Al-Nahrain University, 2007.




Article ID: 6887
DOI: 10.14419/ijamr.v5i4.6887

Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.