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

• Autorzy: Pimenov V. G. , Hendy A. S.

Identyfikatory

DOI

10.1515/amcs-2017-0033

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

fractional diffusion equation; difference scheme; convergence analysis

PL

równanie dyfuzji ułamkowe; schemat różnicowy; analiza zbieżności

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2017
• Tom: Vol. 27, no. 3
• Strony: 477--488
• Opis fizyczny: Bibliogr. 35 poz., tab.

Twórcy

autor

Pimenov V. G.

• Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 Kovalevskoy St., Yekaterinburg 620000, Russia; Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002, Russia

autor

Hendy A. S.

• Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., Yekaterinburg 620002, Russia; Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt

Bibliografia

• [1] Alikhanov, A.A. (2015). A new difference scheme for the time fractional diffusion equation, Journal of Computational Physics 280: 424–438.
• [2] Bagley, R.L. and Torvik, P.J. (1983). A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology 27(201): 201–210.
• [3] Balachandran, K. and Kokila, J. (2012). On the controllability of fractional dynamical systems, International Journal of Applied Mathematics and Computer Science 22(3): 523–531, DOI: 10.2478/v10006-012-0039-0.
• [4] Batzel, J.J. and Kappel, F. (2011). Time delay in physiological systems: Analyzing and modeling its impact, Mathematical Biosciences 234(2): 61–74.
• [5] Bellen, A. and Zennaro, M. (2003). Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford.
• [6] Benson, D., Schumer, R., Meerschaert, M.M. and Wheatcraft, S.W. (2001). Fractional dispersion, Levy motion, and the made tracer tests, Transport in Porous Media 42(1–2): 211–240.
• [7] Chen, F. and Zhou, Y. (2011). Attractivity of fractional functional differential equations, Computers and Mathematics with Applications 62(3): 1359–1369.
• [8] 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 Biology 46: 425–444.
• [9] Ferreira, J.A. (2008). Energy estimates for delay diffusion-reaction equations, Computational and Applied Mathematics 26(4): 536–553.
• [10] 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 Computation 275: 238–254.
• [11] Hatano, Y. and Hatano, N. (1998). Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resources Research 34(5): 1027–1033.
• [12] Höfling, F. and Franosch, T. (2013). Anomalous transport in the crowded world of biological cells, Reports on Progress in Physics 76(4): 46602.
• [13] Holte, J.M. (2009). Discrete Gronwall lemma and applications, MAA North Central Section Meeting at UND, Grand Forks, ND, USA, p. 8, http://homepages.gac.edu/˜holte/publications/GronwallLemma.pdf.
• [14] 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 Simulation 96: 95–103.
• [15] 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 Analysis 16(4): 893–910.
• [16] Kruse, R. and Scheutzow, M. (2016). A discrete stochastic Gronwall lemma, Mathematics and Computers in Simulation, DOI: 10.1016/j.matcom.2016.07.002.
• [17] Lakshmikantham, V. (2008). Theory of fractional functional differential equations, Nonlinear Analysis: Theory, Methods and Applications 69(10): 3337–3343.
• [18] 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 Computation 256: 83–93.
• [19] Liu, P.-P. (2015). Periodic solutions in an epidemic model with diffusion and delay, Applied Mathematics and Computation 265: 275–291.
• [20] Meerschaert, M.M. and Tadjeran, C. (2004). Finite difference approximations for fractional advection-dispersion flow equations, Computational and Applied Mathematics 172(1): 65–77.
• [21] Miller, K.S. and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Miller, New York, NY.
• [22] 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.
• [23] 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 Mathematics 318: 433–443.
• [24] Raberto, M., Scalas, E. and Mainardi, F. (2002). Waiting-times returns in high frequency financial data: An empirical study, Physica A 314(1–4): 749–755.
• [25] Ren, J. and Sun, Z.Z. (2015). Maximum norm error analysis of difference schemes for fractional diffusion equations, Applied Mathematics and Computation 256: 299–314.
• [26] Rihan, F.A. (2009). Computational methods for delay parabolic and time-fractional partial differential equations, Numerical Methods for Partial Differential Equations 26(6): 1557–1571.
• [27] Samarskii, A.A. and Andreev, V.B. (1976). Finite Difference Methods for Elliptic Equations, Nauka, Moscow, (in Russian).
• [28] Scalas, E., Gorenflo, R. and Mainardi, F. (2000). Fractional calculus and continuous-time finance, Physica A 284(1–4): 376–384.
• [29] Schneider, W. and Wyss, W. (1989). Fractional diffusion and wave equations, Journal of Mathematical Physics 30(134): 134–144.
• [30] Sikora, B. (2016). Controllability criteria for time-delay fractional systems with a retarded state, International Journal of Applied Mathematics and Computer Science 26(3): 521–531, DOI: 10.1515/amcs-2016-0036.
• [31] 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 Mathematics 318: 580–590.
• [32] 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 Algorithms 7: 79–97.
• [33] Wyss, W. (1986). The fractional diffusion equation, Journal of Mathematical Physics 27: 2782–2785.
• [34] 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 Simulation 82(9): 1572–1585.
• [35] Zhang, Z.B. and Sun, Z.Z. (2013). A linearized compact difference scheme for a class of nonlinear delay partial differential equations, Applied Mathematical Modelling 37(3): 742–752.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).