EXPLICIT GROUP ITERATIVE METHODS IN THE SOLUTION OF TWO DIMENSIONAL TIME-FRACTIONAL DIFFUSION-WAVES EQUATION
AbstractIn this paper, we present the preliminary study of the formulation of fractional explicit group (FEG) and fractional explicit de-coupled group (FEDG) iterative methods in solving the two dimensional second order diffusion wave equation of fractional order. Both FEG and FEDG iterative methods are derived from the fractional standard and fractional rotated five points Crank-Nicolson discretizations respectively. Their computational complexity is presented and numerical experiments are conducted to demonstrate the efficiency and adeptness of the newly developed explicit group formulations in terms of CPU timings and total number of operations. AMS Subject Classification: 65N14
O. P. Agrawal, O. Defterli, D. Baleanu, Fractional optimal control problems with several state and control variables, Journal of Vibration and Control, 16, No. 13, (2010), 1967-1976.
N. H. M. Ali, L. M. Kew, New explicit group iterative methods in the solution of two dimensional hyperbolic equations, Journal of Computational Physics, 231, (2012), 6953-6968.
N. H. M. Ali, K. F. Ng, Modified explicit de-coupled group method in the solution of 2-D elliptic PDEs, Proceedings 12th WSEAS International Conference on Applied Mathematics, Dec. 29-31, (2007), Cairo Egypt, 12-17.
R. Azim, Y. Farzane, An extension of the singular boundary method for solving two-dimensional time fractional diffusion equations, Engineering Analysis with Boundary Elements, 83, (2017), 167-179.
R. L. Bagley, P. J. Torvik, Theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology, 27, No. 3, (2013), 201-210.
R. L. Bagley, P. J. Torvik, On the appearance of the fractional derivative in the behavior of real materials, Journal of Applied Mechanics, 51, (1984), 294-298.
A. T. Balasim, N. H. M. Ali, A rotated Crank-Nicolson iterative method for the solution of two-dimensional time-fractional diffusion equation, Indian Journal of Science and Technology, 8, No. 32, (2015), 1-7.
A. T. Balasim, N. H. M. Ali, The solution of 2-D time-fractional diffusion equation by the fractional modified explicit group iterative method, International Conference on Mathematics Engineering and Industrial Applications, AIP Conf. Proc. 1775, (2016), 030014-1030014-8.doi: 10.1063/1.4965134
A. T. Balasim, N. H. M. Ali, Group iterative methods for the solution of two-dimensional time-fractional diffusion equation, Advances in Industrial and Applied Mathematics, AIP Conf. Proc.1750, (2016), 030003-1030003-7. doi: 10.1063/1.4954539.
D. A. Benson, S. W. Wheatcraft, Application of a fractional advection-dispersion equation, Water Resources Research, 36, No. 6, (2000), 1403-1412.
D. J. Evans, M. S. Sahimi, The alternative group explicit (AGE) iterative method to solve parabolic equations 1 and 2 dimensional problems, International Journal Computer of Mathematics, 24, (1988), 31 l-341.
D. J. Evans, L. Changjun, The AGE iterative method and its parallel implementations, Iterative Methods in Linear Algebra, IMACS, (1992), 243-250.
D. J. Evans, W. S. Yousif, Explicit group iterative methods for solving elliptic partial di_erential equations in 3-space dimensions, International journal of Computer Mathematics, 18 No. 3, (1986), 323-340.
D. J. Evans, Group explicit iterative methods for solving large linear systems, International Journal of Computer Mathematics 17, (1985), 81-108.
R. Herrmann, Fractional calculus: An introduction for physicists, World Scientific, Hackensack, USA, (2011).
K. P. Hong, W. S. Hai, Fourth order finite difference schemes for time-space fractional sub-diffusion equations, Computers and Mathematics with Applications, 71, (2016), 1287-1302.
L. M. Kew, N. H. M. Ali, New explicit group iterative methods in the solution of three dimensional hyperbolic telegraph equations, Journal of Computational Physics, 294, (2015), 382-404.
M. M. Khader, M. H. Adel, Numerical solutions of fractional wave equations using an efficient class of FDM based on the Hermite formula, Advances in Difference Equations, 34, (2016), 1-10.
R. L. Magin, Fractional calculus in bioengineering, BegellHouse Publisher, Inc. Connecticut, USA, (2006).
F. Mainardi, Fractals and Fractional Calculus Continuum Mechanics, Springer Verlag, USA, 291-348, (1997).
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, An Introduction to Mathematical Models, Imperial College Press, London, (2010).
M. Moshre_, J. K. Hammond, Physical and geometrical interpretations of fractional operators, Journal of the Franklin Institute, 335, No. 6, (1998), 1077-1086.
M. Othman, A. R. Abdullah, An efficient four points modified explicit group poisson solver, International journal of Computer Mathematics, 76, No. 2, (2000), 203-217.
Z. G. Shi, Y. M. Zhao, F. Liu, Y. F. Tang, F. L. Wanga, Y.H. Shi, High accuracy analysis of an H1-Galerkin mixed finite element method for two-dimensional time fractional diffusion equations, Computers and Mathematics with Applications, 74, (2017), 1903-1914.
G. Shimin, M. Liquan, L. Ying, An efficient Galerkin spectral method for two-dimensional fractional nonlinear reaction diffusion wave equation, Computers and Mathematics with Applications, 74, (2017), 2449-2465.
R. H. Vahid, S. Elyas, C. Wen, Local radial point interpolation (MLRPI) method for solving time fractional diffusion-wave equation with damping, Journal of Computational Physics, 312, (2016), 307-332.
Lincy, G., Ponnavaikko, M., & Lenin Anselm, W. A. (2018). Economic analysis of distributed generation with photovoltaic systems. International Journal of Pharmaceutical Research, 10(3), 528â€“536.
N. Z. Ya, Z. S. Zhi, Z. Xuan, Comapct alternating direction implicit scheme for the two-dimensional fractional diffusion wave equation, SIAM Journal on Numerical Analysis, 50, No. 3, (2012), 1535-1555.
J. Zhou, Y. Pu, K. Liao, Fractional order calculus principle and its application in latest single analysis and processing, Beijing Science Press, (2010), 1-33.
The submitter hereby warrants that the Work (collectively, the “Materials”) is original and that he/she is the author of the Materials. To the extent the Materials incorporate text passages, figures, data or other material from the works of others, the undersigned has obtained any necessary permissions. Where necessary, the undersigned has obtained all third party permissions and consents to grant the license above and has all copies of such permissions and consents.
The submitter represents that he/she has the power and authority to make and execute this assignment. The submitter agrees to indemnify and hold harmless the COMPUSOFT from any damage or expense that may arise in the event of a breach of any of the warranties set forth above. For authenticity, validity and originality of the research paper the author/authors will be totally responsible.