A MODIFIED ACCELERATED OVERRELAXATION (MAOR)
SCHEME FOR SOLVING THE SHIFTED GRÜNWALD
ESTIMATION FRACTIONAL POISSON EQUATION
Nik Amir Syafiq1, Mohamed Othman1,2,
Norazak Senu1,3, Fudziah Ismail1,3 1 Laboratory of Computational Sciences and
Mathematical Physics
Institute for Mathematical Research
Universiti Putra Malaysia
43400, Selangor, MALAYSIA
2 Department of Communication Technology
and Network
Universiti Putra Malaysia
43400, Selangor, MALAYSIA
3 Department of Mathematics
Faculty of Science
Universiti Putra Malaysia
43400, Selangor, MALAYSIA
This research studies the Modified Accelerated Overrelaxation (MAOR) scheme on fractional Poisson equation. The equation is discretized using the fractional finite difference method with the shifted Grünwald estimate. The superiority of this scheme is shown through comparison with previous well known relaxation schemes. This research also presents the convergence analysis for this scheme. Then the numerical result is used to compare and discuss all the schemes.
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References
[1] O.P. Agrawal, Solution for a fractional diffusion-wave equation defined in a
bounded domain, mechanical engineering and energy processes, Nonlinear
Dynam., 19 (2002), 145-155.
[2] O.P. Agrawal, O. Defterli, D. Baleanu, Fractional optimal control problems
with several state and control variables, J. Vib. Control, 16, No 13 (2010),
1967-1976.
[3] A. Ali, N.H.M. Ali, Explicit group iterative methods for the solution of twodimensional time-fractional telegraph equation, AIP Conf. Proc., 2138
(2019), 030006.
[4] A. Ali, N.H.M. Ali, Explicit group iterative methods in the solution of
two dimensional time-fractional diffusion-waves equation, An Intern. J. of
Advanced Computer Technology, 7, No 11 (2018), 2931-2938.
[5] A. Ali, N.H.M. Ali, On skewed grid point iterative method for solving 2D
hyperbolic telegraph fractional differential equation, Advances in Difference Equations, 303 (2019).
[6] R.L. Bagley, P.J. Torvik, On the appearance of the fractional derivative in
the behavior of real materials, J. Appl. Mech., 51 (1984), 294-298.
[7] R.L. Bagley, P.J. Torvik, Theoretical basis for the application of fractional
calculus to viscoelasticity, J. Rheol., 27, No 3 (2013), 201-210.
[8] Baharuddin, A. Sunarto, J. Dalle, QSAOR iterative method for the solution of time-fractional diffusion equation, J. of Engineering and Applied
Sciences, 12, No 12 (2017), 3220-3224.
[9] A.T. Balasim, N.H.M. Ali, Group iterative methods for the solution of
two-dimensional time-fractional diffusion equation, AIP Conf. Proc., 1750
(2016), 030003.
[10] A.T. Balasim, N.H.M. Ali, New group iterative schemes in the numerical
solution of the two-dimensional time fractional advection-diffusion equation, Cogent Mathematics, 4 (2017), 1412241.
[11] D.A. Benson, S.W. Wheatcraft, Application of a fractional advectiondispersion equation, Water Resour. Res., 36, No 6 (2000), 1403-1412.
[12] A. Borhanifar, S. Valizadeh, A fractional finite difference method for solving the fractional poisson equation based on the shifted grünwald estimate,
Walailak J Sci & Tech, 10, No 5 (2013), 427-435.
[13] J.V.L. Chew, J. Sulaiman, Implicit solution of 1D nonlinear porous
medium equation using the four-point newton-EGMSOR iterative method,
J. of Appl. Math. and Comput. Mech., 15, No 2 (2016), 11-21.
[14] A.A. Dahalan, A. Saudi, J. Sulaiman, Autonomous navigation on modified AOR iterative method in static indoor environment, J. of Physics:
Conference Series, 1366 (2019), 012020.
[15] J.H. Eng, A. Saudi, J. Sulaiman, Application of SOR iteration for poisson
image blending, Intern. Conf. on High Performance Compilation, Computing and Communications (2017), 60-64.
[16] A. Hadjidimos, Accelerated overrelaxation method, Mathematics of Computation, 32 (1978), 149-157.
[17] A. Hadjidimos, A. Psimarni, A.K. Yeyios, On the convergence of the modified accelerated overrelaxation (MAOR) method, Computer Science Technical Reports (1989), Paper 790.
[18] R. Herrmann, Fractional Calculus: An Introduction for Physicists, World
Scientific, Hackensack (2011).
[19] A. Jajarmi, D. Baleanu, Suboptimal control of fractional-order dynamic
systems with delay argument, J. Vib. Control, 24, No 12 (2018), 24302446.
[20] D.R. Kincaid, D.M. Young, The modified successive over relaxation
method with fixed parameters, Mathematics of Computation, 26 (1972),
705-717.
[21] D. Kumar, J. Singh, D. Baleanu, S. Rathore, Analysis of a fractional model
of the Ambartsumian equation, Eur. Phys. J. Plus, 133 (2018), 259.
[22] R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Danbury (2006).
[23] F. Mainardi, Fractals and Fractional Calculus Continuum Mechanics,
Springer, Bologna (1997).
[24] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity:
An Introduction to Mathematical Models, Imperial College Press, London
(2010).
[25] K. Miller, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York (1993).
[26] F.A. Muhiddin, J. Sulaiman, A. Sunarto, Four-point EGSOR iteration for
the Grünwald implicit finite difference solution of one-dimensional timefractional parabolic equations, J. of Physics: Conference Series, 1366
(2019), 012086.
[27] J. Munkhammar, Riemann-Liouville Fractional Derivatives and the
Taylor-Riemann Series, Department of Mathematics Uppsala University
(2004).
[28] M. Othman, A.R. Abdullah, An Efficient Four Points Modified Explicit
Group Poisson Solver, Intern. J. of Computer Mathematics, 76, No 2
(2000), 203-217.
[29] M. Othman, A.R. Abdullah, D.J. Evans, A Parallel Four Points Modified
Explicit Group Algorithm on Shared Memory Multiprocessors, Intern. J.
of Parallel, Emergent and Distributed Systems, 19, No 1 (2004), 1-9.
[30] I. Podlubny, Fractional Differential Equations, Academic Press, New York
(1999).
[31] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives:
Theory and Applications, Gordon and Breach, London (1993).
[32] J. Singh, D. Kumar, D. Baleanu, S. Rathore, An efficient numerical algorithm for the fractional Drinfeld-Sokolov-Wilson equation, Appl. Math.
Comput., 335 (2018), 12-24.
[33] A. Sunarto, J. Sulaiman, Solving space-fractional diffusion equations by
using HSSOR method, Far East J. of Appl. Mathematics, 97, No 4 (2017),
159-169.
[34] A. Sunarto, J. Sulaiman, A. Saudi, Application of the full-sweep AOR iteration concept for space-fractional diffusion equation, J. of Physics: Conference Series, 710 (2016), 012019.
[35] A. Sunarto, J. Sulaiman, A. Saudi, Caputo’s implicit solution of spacefractional diffusion equations by QSSOR iteration, Advanced Science Letters, 24, No 3 (2018), 1297-1931.
[36] A. Sunarto, J. Sulaiman, A. Saudi, Caputo’s implicit solution of timefractional diffusion equation using half-sweep AOR iteration, Global J. of
Pure and Appl. Math., 12, No 4 (2016), 3469-3479.
[37] A. Sunarto, J. Sulaiman, A. Saudi, Full-sweep SOR iterative method to
solve space-fractional diffusion equations, Australian J. of Basic and Appl.
Sciences, 8, No 24 (2014), 153-158.
[38] A. Sunarto, J. Sulaiman, A. Saudi, Implicit finite difference solution for
time-fractional diffusion equations using AOR method, J. of Physics: Conference Series, 495 (2014), 012032.
[39] N.A. Syafiq, M. Othman, N. Senu, F. Ismail, An experimental study of
the modified accelerated overrelaxation (MAOR) scheme on stationary
helmholtz equation, J. of Physics: Conference Series, 1366 (2019), 012093.
[40] D.M. Young, Iterative methods for solving partial dfferential equations of
elliptic type, Trans. Amer. Math. Soc., 76 (1954), 92-111.
[41] D.M. Young, Iterative Solution of Large Linear Systems, Academic Press,
New York (1971).
[42] J. Zhou, Y. Pu, K. Liao, Fractional Order Calculus Principle and Its Application in Latest Single Analysis and Processing, Beijing Science Press,
Beijing (2010).
[43] P. Zhuang, F. Liu, Finite difference approximation for two-dimensional
time fractional diffusion equation, J. of Algorithms & Computational Technology, 1, No 1 (2007), 1-15.