Numerical study of time-fractional hyperbolic partial differential equations
-
2543
Downloads
-
4122
Views
Authors
Saima Arshed
- Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan.
Abstract
In this study, a numerical scheme is developed for the solution of time-fractional hyperbolic partial differential equation.
In the proposed scheme, cubic B-spline collocation is used for space discretization and time discretization is obtained by using
central difference formula. Caputo fractional derivative is used for time-fractional derivative. The stability and convergence
of the developed scheme, have also been proved. The numerical examples support the theoretical results.
Share and Cite
ISRP Style
Saima Arshed, Numerical study of time-fractional hyperbolic partial differential equations, Journal of Mathematics and Computer Science, 17 (2017), no. 1, 53-65
AMA Style
Arshed Saima, Numerical study of time-fractional hyperbolic partial differential equations. J Math Comput SCI-JM. (2017); 17(1):53-65
Chicago/Turabian Style
Arshed, Saima. "Numerical study of time-fractional hyperbolic partial differential equations." Journal of Mathematics and Computer Science, 17, no. 1 (2017): 53-65
Keywords
- Time-fractional hyperbolic equation
- cubic B-spline
- collocation method
- convergence analysis
- stability analysis.
MSC
References
-
[1]
A. Atangana, Convergence and stability analysis of a novel iteration method for fractional biological population equation, Neural Comput. Appl., 25 (2014), 1021–1030.
-
[2]
A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 293 (2015), 104–114.
-
[3]
A. Atangana, On the stability of iteration methods for special solution of time-fractional generalized nonlinear ZK-BBM equation, J. Vib. Control, 22 (2016), 1769–1776.
-
[4]
A. Atangana, T. Tuluce Demiray, H. Bulut, Modelling the nonlinear wave motion within the scope of the fractional calculus, Abstr. Appl. Anal., 2014 (2014), 7 pages.
-
[5]
H. M. Baskonus, F. B. M. Belgacem, H. Bulut, Solutions of nonlinear fractional differential equations systems through the implementation of the variational ieration method, Fractional Dynamics Book edited by Carlo Cattani, De Gruyter Open, Chap. 19. (2015)
-
[6]
H. M. Baskonus, H. Bulut, On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method, Open Math., 13 (2015), 547–556.
-
[7]
H. M. Baskonus, H. Bulut, On the complex analytical solutions for the fractional nonlinear double sinh-Poisson equation, 1st International Symposium on Computational Mathematics and Engineering Sciences, Errichidia, Morocco (2016)
-
[8]
H. M. Baskonus, T. Mekkaoui, Z. Hammouch, H. Bulut, Active control of a chaotic fractional order economic system, Entropy, 17 (2015), 5771–5783.
-
[9]
H. Bulut, H. M. Baskonus, F. B. M. Belgacem, The analytical solution of some fractional ordinary differential equations by the Sumudu transform method, Abstr. Appl. Anal., 2013 (2013 ), 6 pages.
-
[10]
H. Bulut, H. M. Baskonus, Y. Pandir, The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation, Abstr. Appl. Anal., 2013 (2013 ), 8 pages.
-
[11]
H. Bulut, F. B. M. Belgacem, H. M. Baskonus, Some new analytical solutions for the nonlinear time-fractional KdVBurgers- Kuramoto equation, Adv. Math. Stat. Sci., 2 (2015), 118–129.
-
[12]
H. Bulut, Y. Pandir, S. T. Demiray, Exact solutions of time-fractional KdV equations by using generalized Kudryashov method, Int. J. Model. Optim., 4 (2014), 315–320.
-
[13]
J. Chen, F. Liu, V. Anh, S. Shen, Q. Liu, C. Liao, The analytical solution and numerical solution of the fractional diffusion-wave equation with damping, Appl. Math. Comput., 219 (2012), 1737–1748.
-
[14]
H.-F. Ding, C.-P. Li, Numerical algorithms for the fractional diffusion-wave equation with reaction term, Abstr. Appl. Anal., 2013 (2013), 15 pages.
-
[15]
W. Li, X. Da, Finite central difference/finite element approximations for parabolic integro-differential equations, Computing, 90 (2010), 89–111.
-
[16]
C.-P. Li, Z.-G. Zhao, Y.-Q. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62 (2011), 855–875.
-
[17]
Y. Lin, C.-J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552.
-
[18]
F. Liu, V. V. Anh, I. Turner, P. Zhuang, Time fractional advection-dispersion equation, J. Appl. Math. Comput., 13 (2003), 233–245.
-
[19]
F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput., 191 (2007), 12–20.
-
[20]
V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, H. R. Hicks, Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys., 192 (2003), 406–421.
-
[21]
M. M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65–77.
-
[22]
S. S. Siddiqi, S. Arshed, Numerical Solution of Convection-Diffusion Integro-Differential Equations with a Weakly Singular Kernel, J. Basic. Appl. Sci. Res., 3 (2013), 106–120.
-
[23]
E. Sousa, How to approximate the fractional derivative of order \(1 < \alpha\leq 2\), Internat. J. Bifur. Chaos Appl. Sci. Engrg., 22 (2012), 13 pages.
-
[24]
V. K. Srivastava, M. K. Awasthi, M. Tamsir, RDTM solution of Caputo time fractional-order hyperbolic telegraph equation, AIP Adv., 3 (2013), 11 pages.
-
[25]
N. H. Sweilam, T. A. Rahman Assiri, Numerical simulations for the space-time variable order nonlinear fractional wave equation, J. Appl. Math., 2013 (2013 ), 7 pages.
-
[26]
C. Tadjeran, M. M. Meerschaert, H. P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213 (2006), 205–213.
-
[27]
S. Tuluce Demiray, Y. Pandir, H. Bulut, Generalized Kudryashov method for time-fractional differential equations, Abstr. Appl. Anal., 2014 (2014 ), 13 pages.
-
[28]
L.-L. Wei, H. Dai, D.-L. Zhang, Z.-Y. Si, Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation, Calcolo, 51 (2014), 175–192.
-
[29]
Y. Zhang, A finite difference method for fractional partial differential equation, Appl. Math. Comput., 215 (2009), 524– 529.
-
[30]
H.-X. Zhang, X. Han, Quasi-wavelet method for time-dependent fractional partial differential equation, Int. J. Comput. Math., 90 (2013), 2491–2507.