Numerical approach for solving time fractional diffusion equation
DOI:
https://doi.org/10.11121/ijocta.01.2017.00492Abstract
In this article one of the fractional partial differential equations was solved by finite difference scheme based on five point and three point central space method with discretization in time. We use between the Caputo and the Riemann-Liouville derivative definition and the Grünwald-Letnikov operator for the fractional calculus. The stability analysis of this scheme is examined by using von-Neumann method. A comparison between exact solutions and numerical solutions is made. Some figures and tables are included.
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References
Ciesielski, M., Leszczynski J. (2003). Numerical simulations of anomalous diffusion, Computer Methods in Mechanics, Gliwice, Poland, June 3-6.
Luenberger, D.G., (1973). Introduction to Linear and Nonlinear Programming. Addison-Wesley, California
Berkowitz, B., Scher, H., (1997). Anomalous transport in random fracture networks, Physical Review Letters, 79, 4038-4041.
Klemm, A., Müller, H.P., Kimmich R., (1997). NMR-microscopy of pore-space backbones in rock, Phys. Rev. E, 55,4413.
Crank, J., (1989). The Mathematics of Diffusion – 2nd ed., Oxford University Press.
Palade, L.I., Attane, P., Huilgol, R.R., Mena, B., (1999). Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models, International Journal of Engineering Science., 37, 315-329.
Amblard, F., Maggs, A.C., Yurke, B.,.Pargellis, A.N., Leibler, S., (1996). Subdiffusion and anomalous local viscoelasticity in actin networks, Physical Review Letters, 77, 4470-4473.
Baskonus, H.M., Bulut, H., (2015). On the Numerical Solutions of Some Fractional Ordinary Differential Equations by Fractional Adams-Bashforth-Moulton Method, Open Mathematics, 13(1), 547–556.
Baskonus, H.M., Mekkaoui, T., Hammouch, Z. and Bulut, H., (2015). Active Control of a Chaotic Fractional Order Economic System, Entropy, 17(8), 5771-5783.
Baskonus, H.M., Bulut, H., (2016).Regarding on the prototype solutions for the nonlinear fractional-order biological population model, AIP Conference Proceedings 1738, 290004.
Baskonus, H.M., Hammouch, Z., Mekkaoui, T. and Bulut, H., (2016).Chaos in the fractional order logistic delay system: Circuit realization and synchronization, AIP Conference Proceedings, 1738, 290005.
Gencoglu, M. T., Baskonus, H. M. and Bulut, H.,(2017).Numerical simulations to the nonlinear model of interpersonal Relationships with time fractional derivative, AIP Conference Proceedings, 1798, 1-9 (020103).
Li C., Zeng F., (2015). Numerical Methods for Fractional Calculus, Taylor & Francis Group.
Nakagawa, J., Sakamoto K., Yamamoto M., (2010). Overwiev to mathematical analysis for fractional diffusion equation-new mathematical aspects motivated by industrial collaboration, Journal of Math-for-Industry, 2, 99-108.
Narahari Achar B., Hanneken J., (2004). Fractional radial diffusion in a cylinder, Journal of Molecular Liquids, 114, 147-151.
Langlands, T., (2006). Fractional Dynamics, Physica A 367, 136.
Diethelm, K., Ford, J., Freed, A., Luchko, Y., (2005). Algorithms for the fractional calculus: a selection of numerical methods, Computer Methods in Applied Mechanics and Engineering, 194(6), 743-773.
He, J.H., (1998). Approximate analytical solution for seepage flow with fractional derivatives in porous media, Computer Methods in Applied Mechanics and Engineering, 167, 57-68.
Kurulay, M., Bayram, M., (2009). Approximate analytical solution for the fractional modified KdV by differential transform method, Communications in Nonlinear Science and Numerical Simulation, 15(7), 1777-178.
Reutskiy, S. Yu., A. (2017). New semi-analytical collocation method for solving multi-term fractional partial differential equations with time variable coefficients, Applied Mathematical Modelling, 45, 238–254.
Smith, G. D., (1985). Numerical Solution of Partial Difference Equations, Finite Difference Methods, Oxford University.
Yuste, S. B., Acedo, L., (2005). An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations, Society for Industrial and Applied Mathematics, 42(5), 1862-1874.
Murillo, J. Q., Yuste, S. B., (2011). An Explicit Numerical Method for the Fractional Cable Equation, Hindawi Publishing Corporation International Journal of Differential Equations.
Lubich, C.,(1986).Discretized fractional calculus, SIAM Journal on Mathematical Analysis, 17(3),704-719.
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