A numerical solution for a class of time fractional diffusion equations with delay

Vladimir G. Pimenov; Ahmed S. Hendy

doi:10.1515/amcs-2017-0033

$A numerical solution for a class of time fractional diffusion equations with delay Cover$

.blurhash-client-img { display: none !important; }

A numerical solution for a class of time fractional diffusion equations with delay

International Journal of Applied Mathematics and Computer Science

Volume 27 (2017): Issue 3 (September 2017)

By: Vladimir G. Pimenov and Ahmed S. Hendy

Open Access

|Sep 2017

Abstract

This paper describes a numerical scheme for a class of fractional diffusion equations with fixed time delay. The study focuses on the uniqueness, convergence and stability of the resulting numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence order O(τ^2−α+ h⁴) in L_∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results.

References

Alikhanov, A.A. (2015). A new difference scheme for the time fractional diffusion equation, Journal of Computational Physics280: 424–438.10.1016/j.jcp.2014.09.031
Search in Google Scholar Back to article
Bagley, R.L. and Torvik, P.J. (1983). A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology27(201): 201–210.10.1122/1.549724
Open DOI Search in Google Scholar Back to article
Balachandran, K. and Kokila, J. (2012). On the controllability of fractional dynamical systems, International Journal of Applied Mathematics and Computer Science22(3): 523–531, DOI: 10.2478/v10006-012-0039-0.10.2478/v10006-012-0039-0
Open DOI Search in Google Scholar Back to article
Batzel, J.J. and Kappel, F. (2011). Time delay in physiological systems: Analyzing and modeling its impact, Mathematical Biosciences234(2): 61–74.10.1016/j.mbs.2011.08.00621945380
Search in Google Scholar Back to article
Bellen, A. and Zennaro, M. (2003). Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford.10.1093/acprof:oso/9780198506546.001.0001
Search in Google Scholar Back to article
Benson, D., Schumer, R., Meerschaert, M.M. and Wheatcraft, S.W. (2001). Fractional dispersion, Levy motion, and the made tracer tests, Transport in Porous Media42(1–2): 211–240.10.1023/A:1006733002131
Open DOI Search in Google Scholar Back to article
Chen, F. and Zhou, Y. (2011). Attractivity of fractional functional differential equations, Computers and Mathematics with Applications62(3): 1359–1369.10.1016/j.camwa.2011.03.062
Search in Google Scholar Back to article
Culshaw, R. V., Ruan, S. and Webb, G. (2003). A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay, Mathematical Biology46: 425–444.10.1007/s00285-002-0191-512750834
Search in Google Scholar Back to article
Ferreira, J.A. (2008). Energy estimates for delay diffusion-reaction equations, Computational and Applied Mathematics26(4): 536–553.
Search in Google Scholar Back to article
Hao, Z., Fan, K., Cao, W. and Sun, Z. (2016). A finite difference scheme for semilinear space-fractional diffusion equations with time delay, Applied Mathematics and Computation275: 238–254.10.1016/j.amc.2015.11.071
Search in Google Scholar Back to article
Hatano, Y. and Hatano, N. (1998). Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resources Research34(5): 1027–1033.10.1029/98WR00214
Open DOI Search in Google Scholar Back to article
Höfling, F. and Franosch, T. (2013). Anomalous transport in the crowded world of biological cells, Reports on Progress in Physics76(4): 46602.10.1088/0034-4885/76/4/04660223481518
Search in Google Scholar Back to article
Holte, J.M. (2009). Discrete Gronwall lemma and applications, MAA North Central SectionMeeting at UND, Grand Forks, ND, USA, p. 8, http://homepages.gac.edu/~holte/publications/GronwallLemma.pdf.
Search in Google Scholar Back to article
Jackiewicz, Z., Liu, H., Li, B. and Kuang, Y. (2014). Numerical simulations of traveling wave solutions in a drift paradox inspired diffusive delay population model, Mathematics and Computers in Simulation96: 95–103.10.1016/j.matcom.2012.06.004
Open DOI Search in Google Scholar Back to article
Karatay, I., Kale, N. and Bayramoglu, S.R. (2013). A new difference scheme for time fractional heat equations based on Crank–Nicholson method, Fractional Calculus and Applied Analysis16(4): 893–910.10.2478/s13540-013-0055-2
Search in Google Scholar Back to article
Kruse, R. and Scheutzow, M. (2016). A discrete stochastic Gronwall lemma, Mathematics and Computers in Simulation, DOI: 10.1016/j.matcom.2016.07.002.10.1016/j.matcom.2016.07.002
Open DOI Search in Google Scholar Back to article
Lakshmikantham, V. (2008). Theory of fractional functional differential equations, Nonlinear Analysis: Theory, Methods and Applications69(10): 3337–3343.10.1016/j.na.2007.09.025
Search in Google Scholar Back to article
Lekomtsev, A. and Pimenov, V. (2015). Convergence of the scheme with weights for the numerical solution of a heat conduction equation with delay for the case of variable coefficient of heat conductivity, Applied Mathematics and Computation256: 83–93.10.1016/j.amc.2014.12.149
Search in Google Scholar Back to article
Liu, P.-P. (2015). Periodic solutions in an epidemic model with diffusion and delay, Applied Mathematics and Computation265: 275–291.10.1016/j.amc.2015.05.028
Search in Google Scholar Back to article
Meerschaert, M.M. and Tadjeran, C. (2004). Finite difference approximations for fractional advection-dispersion flow equations, Computational and Applied Mathematics172(1): 65–77.10.1016/j.cam.2004.01.033
Search in Google Scholar Back to article
Miller, K.S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Miller, New York, NY.
Search in Google Scholar Back to article
Pimenov, V.G. and Hendy, A.S. (2015). Numerical studies for fractional functional differential equations with delay based on BDF-type shifted Chebyshev polynomials, Abstract and Applied Analysis, 2015(2015), Article ID 510875, DOI: 10.1155/2015/510875.10.1155/2015/510875
Open DOI Search in Google Scholar Back to article
Pimenov, V.G., Hendy, A.S. and De Staelen, R.H. (2017). On a class of non-linear delay distributed order fractional diffusion equations, Journal of Computational and Applied Mathematics318: 433–443.10.1016/j.cam.2016.02.039
Search in Google Scholar Back to article
Raberto, M., Scalas, E. and Mainardi, F. (2002). Waiting-times returns in high frequency financial data: An empirical study, Physica A314(1–4): 749–755.10.1016/S0378-4371(02)01048-8
Search in Google Scholar Back to article
Ren, J. and Sun, Z.Z. (2015). Maximum norm error analysis of difference schemes for fractional diffusion equations, Applied Mathematics and Computation256: 299–314.10.1016/j.amc.2014.12.151
Search in Google Scholar Back to article
Rihan, F.A. (2009). Computational methods for delay parabolic and time-fractional partial differential equations, Numerical Methods for Partial Differential Equations26(6): 1557–1571.10.1002/num.20504
Search in Google Scholar Back to article
Samarskii, A.A. and Andreev, V.B. (1976). Finite Difference Methods for Elliptic Equations, Nauka, Moscow, (in Russian).
Search in Google Scholar Back to article
Scalas, E., Gorenflo, R. and Mainardi, F. (2000). Fractional calculus and continuous-time finance, Physica A284(1–4): 376–384.10.1016/S0378-4371(00)00255-7
Search in Google Scholar Back to article
Schneider, W. and Wyss, W. (1989). Fractional diffusion and wave equations, Journal of Mathematical Physics30(134): 134–144.10.1063/1.528578
Open DOI Search in Google Scholar Back to article
Sikora, B. (2016). Controllability criteria for time-delay fractional systems with a retarded state, International Journal of Applied Mathematics and Computer Science26(3): 521–531, DOI: 10.1515/amcs-2016-0036.10.1515/amcs-2016-0036
Open DOI Search in Google Scholar Back to article
Solodushkin, S.I., Yumanova, I.F. and De Staelen, R.H. (2017). A difference scheme for multidimensional transfer equations with time delay, Journal of Computational and Applied Mathematics318: 580–590.10.1016/j.cam.2015.12.011
Search in Google Scholar Back to article
Tumwiine, J., Luckhaus, S., Mugisha, J.Y.T. and Luboobi, L.S. (2008). An age-structured mathematical model for the within host dynamics of malaria and the immune system, Journal of Mathematical Modelling and Algorithms7: 79–97.10.1007/s10852-007-9075-4
Search in Google Scholar Back to article
Wyss, W. (1986). The fractional diffusion equation, Journal of Mathematical Physics27: 2782–2785.10.1063/1.527251
Open DOI Search in Google Scholar Back to article
Yan, Y. and Kou, C. (2012). Stability analysis of a fractional differential model of HIV infection of CD4+ T-cells with time delay, Mathematics and Computers in Simulation82(9): 1572–1585.10.1016/j.matcom.2012.01.004
Open DOI Search in Google Scholar Back to article
Zhang, Z.B. and Sun, Z.Z. (2013). A linearized compact difference scheme for a class of nonlinear delay partial differential equations, Applied Mathematical Modelling37(3): 742–752.10.1016/j.apm.2012.02.036
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.1515/amcs-2017-0033 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X

Journal RSS Feed

Language: English

Page range: 477 - 488

Submitted on: Nov 29, 2016

Accepted on: May 4, 2017

Published on: Sep 23, 2017

Published by: University of Zielona Góra

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

fractional diffusion equation with delay,

Related subjects:

© 2017 Vladimir G. Pimenov, Ahmed S. Hendy, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 27 (2017): Issue 3 (September 2017)