Numerical Solution for Poisson Fractional Equation Via Finite Differences Theta-method
-
2981
Downloads
-
3997
Views
Authors
Mohammad Aslefallah
- Department of Mathematics, Imam Khomeini International University, Qazvin, Iran.
Davood Rostamy
- Department of Mathematics, Imam Khomeini International University, Qazvin, Iran.
Abstract
In this paper we examine \(\theta\)-method for solving fractional Possion differential equations for\((0\leq\theta\leq 1)\). Consistency, stability and convergence analysis of the method is discussed. At the end, numerical examples have been presented. The obtained results reveal that the proposed technique is very effective, convenient and quite accurate to such considered problems.
Share and Cite
ISRP Style
Mohammad Aslefallah, Davood Rostamy, Numerical Solution for Poisson Fractional Equation Via Finite Differences Theta-method, Journal of Mathematics and Computer Science, 12 (2014), no. 2, 132-142
AMA Style
Aslefallah Mohammad, Rostamy Davood, Numerical Solution for Poisson Fractional Equation Via Finite Differences Theta-method. J Math Comput SCI-JM. (2014); 12(2):132-142
Chicago/Turabian Style
Aslefallah, Mohammad, Rostamy, Davood. "Numerical Solution for Poisson Fractional Equation Via Finite Differences Theta-method." Journal of Mathematics and Computer Science, 12, no. 2 (2014): 132-142
Keywords
- Fractional PDE (FPDE)
- Finite differences \(\theta\)-method
- Riemann-Liouville derivative
- Shifted Grunwald formula.
MSC
References
-
[1]
M. Aslefallah, D. Rostamy, A Numerical Scheme For Solving Space-Fractional Equation By Finite Differences Theta-Method, Int. J. of Adv. in Aply. Math. and Mech, 1(4) (2014), 1-9.
-
[2]
D. A. Benson, M. M. Meerschaert, J. Revielle, Fractional calculus in hydrologic modeling: A numerical perspective, Advances in Water Resources , 51 (2013), 479-497.
-
[3]
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(5) (2013), 427-435.
-
[4]
R. Gorenflo, F. Mainardi, E. Scalas, M. Raberto, Fractional calculus and continuous-time finance, III., The diffusion limit, Mathematical finance (Konstanz, 2000), Trends in Mathematics, (2001), 171-180.
-
[5]
R. Hilfer , Applications of Fractional Calculus in Physics , World Scientific, Singapore (2000)
-
[6]
M. Alipour, D. Rostamy, Solving nonlinear fractional differential equations by Bernstein polynomials operational matrices, The J. of Math. And computer sci. (TJMS), 5(3) (2012), 185-196.
-
[7]
B. Lundstrom, M. Higgs, W. Spain, A. Fairhall, Fractional differentiation by neocortical pyramidal neurons, Nature Neuroscience, 11 (2008), 1335-1342.
-
[8]
M. M. Meerschaert, C. Tadjeran, Finite Difference Approximations for two-sided space-fractional partial differential equations, Applied Numerical Mathematics, 56 (2006), 80-90.
-
[9]
M. M. Meerschaert, C. Tadjeran, H. P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, Journal of Computational Physics, 213 (2006), 205-213.
-
[10]
M. M. Meerschaert, D. A. Benson, B. Baeumer, Multidimensional advection and fractional dispersion, Physical Review E , 59 (1999), 5026-5028.
-
[11]
E. Salehpoor, H. Jafari, Variational iteration method: A tool for solving partial differential equations, The J. of Math. And Computer Sci. (TJMS), 2(2) (2011), 388-393.
-
[12]
I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999)
-
[13]
J. P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains, Journal of Computational and Applied Mathematics , 193(1) (2006), 243-268.
-
[14]
Y. A. Rossikhin, M. V. Shitikova, Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems, Acta Mechanica , 120 (1997), 109-125.
-
[15]
L. Sabatelli, S. Keating, J. Dudley, P. Richmond, Waiting time distributions in financial markets , European Physical Journal B , 27 (2002), 273-275.
-
[16]
S. Samko, A. Kibas, O. Marichev, Fractional Integrals and derivatives: Theory and Applications, Gordon and Breach, London (1993. )
-
[17]
R. Schumer, D. A. Benson, M. M. Meerschaert, S. W. Wheatcraft, Eulerian derivation of the fractional advection-dispersion equation, Journal of Contaminant Hydrology , 48 (2001), 69-88.
-
[18]
R. Schumer, D. A. Benson, M. M. Meerschaert, B. Baeumer, Multiscaling fractional advection-dispersion equations and their solutions, Water Resources Research , 39 (2003), 1022-1032.
-
[19]
G. D. Smith, Numerical Solution of Partial Differential Equations : Finite Difference Methods, Oxford University Press, (1978)
-
[20]
M. S. Tavazoei, M. Haeri, Describing function based methods for predicting chaos in a class of fractional order differential equations, Nonlinear Dynamics, 57 (3) (2009), 363--373.
-
[21]
S. B. Yuste, L. Acedo, K. Lindenberg, Reaction front in an A+B→C reaction-subdiffusion process, Physical Review E , 69(3) (2004), 036-126.
-
[22]
Y. W. Zhang , Formulationand solution to time-fractional Sharma-Tassoo-Olever equation via variational methods, Int. J. of Appl. Math and Mech, 9 (18) (2013), 15-27.