Fast \(O(N)\) hybrid Laplace transform-finite difference method in solving 2D time fractional diffusion equation
Volume 23, Issue 2, pp 110--123
http://dx.doi.org/10.22436/jmcs.023.02.04
Publication Date: October 15, 2020
Submission Date: May 12, 2020
Revision Date: August 15, 2020
Accteptance Date: September 15, 2020
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Authors
Fouad Mohammad Salama
- School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia.
Norhashidah Hj. Mohd Ali
- School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia.
Nur Nadiah Abd Hamid
- School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia.
Abstract
It is time-memory consuming when numerically solving time fractional partial differential equations, as it requires \(O(N^2)\) computational cost and \(O(MN)\) memory complexity with finite difference methods, where, \(N\) and \(M\) are the total number of time steps and spatial grid points, respectively. To surmount this issue, we develop an efficient hybrid method with \(O(N)\) computational cost and \(O(M)\) memory complexity in solving two-dimensional time fractional diffusion equation. The presented method is based on the Laplace transform method and a finite difference scheme. The stability and convergence of the proposed method are analyzed rigorously by the means of the Fourier method. A comparative study drawn from numerical experiments shows that the hybrid method is accurate and reduces the computational cost, memory requirement as well as the CPU time effectively compared to a standard finite difference scheme.
Share and Cite
ISRP Style
Fouad Mohammad Salama, Norhashidah Hj. Mohd Ali, Nur Nadiah Abd Hamid, Fast \(O(N)\) hybrid Laplace transform-finite difference method in solving 2D time fractional diffusion equation, Journal of Mathematics and Computer Science, 23 (2021), no. 2, 110--123
AMA Style
Salama Fouad Mohammad, Ali Norhashidah Hj. Mohd, Hamid Nur Nadiah Abd, Fast \(O(N)\) hybrid Laplace transform-finite difference method in solving 2D time fractional diffusion equation. J Math Comput SCI-JM. (2021); 23(2):110--123
Chicago/Turabian Style
Salama, Fouad Mohammad, Ali, Norhashidah Hj. Mohd, Hamid, Nur Nadiah Abd. "Fast \(O(N)\) hybrid Laplace transform-finite difference method in solving 2D time fractional diffusion equation." Journal of Mathematics and Computer Science, 23, no. 2 (2021): 110--123
Keywords
- Caputo fractional derivative
- fractional diffusion equation
- Laplace transform
- finite difference scheme
- stability and convergence analyses
MSC
References
-
[1]
A. Atangana, Fractional discretization: the African's tortoise walk, Chaos Solitons Fractals, 130 (2020), 24 pages
-
[2]
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 Conference Proceedings, Johor Bahru, Malaysia), 1750 (2016), 7 pages
-
[3]
P. M. Basha, V. Shanthi, A robust second order numerical method for a weakly coupled system of singularly perturbed reaction-diffusion problem with discontinuous source term, Int. J. Comput. Sci. Math., 11 (2020), 63--80
-
[4]
D. A. Benson, S. W. Wheatcraft, M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), 1403--1412
-
[5]
M. Bishehniasar, S. Salahshour, A. Ahmadian, F. Ismail, D. Baleanu, An accurate approximate-analytical technique for solving time-fractional partial differential equations, Complexity, 2017 (2017), 12 pages
-
[6]
A. Carpinteri, F. Mainardi, Fractals and fractional calculus in continuum mechanics, Springer, New York (2014)
-
[7]
J.-F. Cheng, Y.-M. Chu, Solution to the linear fractional differential equation using adomian decomposition method, Math. Probl. Eng., 2011 (2011), 14 pages
-
[8]
M. Dehghan, J. Manafian, A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differential Equations, 26 (2010), 448--479
-
[9]
K. Diethelm, An efficient parallel algorithm for the numerical solution of fractional differential equations, Fract. Calc. Appl. Anal., 14 (2011), 475--490
-
[10]
R. Du, W. Cao, Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model., 34 (2010), 2998--3007
-
[11]
B. Ghanbari, A. Atangana, An efficient numerical approach for fractional diffusion partial differential equations, Alex. Eng. J., 59 (2020), 2171--2180
-
[12]
C. Y. Gong, W. M. Bao, G. J. Tang, Y. W. Jiang, J. Liu, Computational challenge of fractional differential equations and the potential solutions: a survey, Math. Probl. Eng., 2015 (2015), 13 pages
-
[13]
C. Gong, W. N. Bao, G. J. Tang, B. Yang, J. Liu, An efficient parallel solution for caputo fractional reaction–diffusion equation, J. Supercomput., 68 (2014), 1521--1537
-
[14]
S. D. Jiang, J. Zhang, Q. Zhang, Z. M. Zhang, Fast evaluation of the caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650--678
-
[15]
I. Karatay, N. Kale, S. R. Bayramoglu, A new difference scheme for time fractional heat equations based on the crank-nicholson method, Fract. Calc. Appl. Anal., 16 (2013), 892--910
-
[16]
S. Karimi Vanani, A. Aminataei, Tau approximate solution of fractional partial differential equations, Comput. Math. Appl., 62 (2011), 1075--1083
-
[17]
D. Kumar, J. Singh, D. Baleanu, S. Rathore, Analysis of a fractional model of the ambartsumian equation, Eur. Phys. J. Plus, 133 (2018), 13 pages
-
[18]
Q. Liu, Y. T. Gu, P. Zhuang, F. Liu, Y. F. Nie, An implicit RBF meshless approach for time fractional diffusion equations, Comput. Mech., 48 (2011), 1--12
-
[19]
R. L. Magin, Fractional calculus in bioengineering, Begell House, Redding (2006)
-
[20]
F. Mainardi, Fractional calculus and waves in linear viscoelasticity, Imperial College Press, London (2010)
-
[21]
S. T. Mohyud-Din, T. Akram, M. Abbas, A. I. Ismail, N. H. M. Ali, A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection-diffusion equation, Adv. Difference Equ., 2018 (2018), 17 pages
-
[22]
M. Moshrefi-Torbati, J. K. Hammond, Physical and geometrical interpretation of fractional operators, J. Franklin Inst., 335 (1998), 1077--1086
-
[23]
K. M. Owolabi, A. Atangana, Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations, Chaos Solitons Fractals, 111 (2018), 119--127
-
[24]
H.-K. Pang, H.-W. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), 693--703
-
[25]
I. Podlubny, Fractional differential equations, Academic press, San Diego (1999)
-
[26]
M. Ranjbar, H. Ghafouri, A. Khani, Application of cubic $B$-spline quasi-interpolation for solving time-fractional partial differential equation, Comput. Methods Differ. Equ., 2020 (2020), 1--13
-
[27]
J. C. Ren, Z.-Z. Sun, W. Z. Dai, New approximations for solving the caputo-type fractional partial differential equations, Appl. Math. Model., 40 (2016), 2625--2636
-
[28]
F. M. Salama, N. H. M Ali, Fast O(N) Hybrid method for the solution of two dimensional time fractional cable equation, Compusoft, 8 (2019), 3453--3461
-
[29]
F. M. Salama, N. H. M. Ali, Computationally efficient hybrid method for the numerical solution of the 2D time fractional advection-diffusion equation, Int. J. Math. Eng. Manag. Sci., 5 (2020), 432--446
-
[30]
F. M. Salama, N. H. M. Ali, N. N. Abd Hamid, Efficient hybrid group iterative methods in the solution of two-dimensional time fractional cable equation, Adv. Difference Equ., 2020 (2020), 20 pages
-
[31]
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
-
[32]
P. J. Torvik, R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294--298
-
[33]
S. Vong, P. Lyu, X. Chen, S.-L. Lei, High order finite difference method for time-space fractional differential equations with caputo and riemann-liouville derivatives, Numer. Algorithms, 72 (2016), 195--210
-
[34]
Y. F. Xu, Z. M. He, The short memory principle for solving abel differential equation of fractional order, Comput. Math. Appl, 62 (2011), 4796--4805
-
[35]
Y.-N. Zhang, Z.-Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230 (2011), 8713--8728