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Positivity and boundedness preserving schemes for the fractional reaction-diffusion equation. (English) Zbl 1292.65095
Summary: We design a semi-implicit scheme for the scalar time fractional reaction-diffusion equation. We theoretically prove that the numerical scheme is stable without the restriction on the ratio of the time and space stepsizes, and numerically show that the convergence orders are 1 in time and 2 in space. As a concrete model, the subdiffusive predator-prey system is discussed in detail. First, we prove that the analytical solution to the system is positive and bounded. Then, we use the provided numerical scheme to solve the subdiffusive predator-prey system, and theoretically prove and numerically verify that the numerical scheme preserves the positivity and boundedness.

MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35R11 Fractional partial differential equations 35K57 Reaction-diffusion equations 35Q92 PDEs in connection with biology, chemistry and other natural sciences 92D25 Population dynamics (general)
FODE
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References:
 [1] Agrawal, O P, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynam, 29, 145-155, (2002) · Zbl 1009.65085 [2] Aly, S; Kim, I; Sheen, D, Turing instability for a ratio-dependent predator-prey model with diffusion, Appl Math Comp, 217, 7265-7281, (2011) · Zbl 1211.92054 [3] Bartumeus, F; Alonso, D; Catalan, J, Self organized spatial structures in a ratio dependent predator-prey model, Physica A, 295, 53-57, (2001) · Zbl 0978.35016 [4] Cavani, M; Farkas, M, Bifurcation in a predator-prey model with memory and diffusion II: Turing bifurcation, Acta Math Hungar, 63, 375-393, (1994) · Zbl 0809.92018 [5] Chen, C M; Liu, F; Turner, I; etal., A Fourier method for the fractional diffusion equation describing sub-diffusion, J Comput Phys, 227, 886-897, (2007) · Zbl 1165.65053 [6] Cui, M R, Compact finite difference method for the fractional diffusion equation, J Comput Phys, 228, 7792-7804, (2009) · Zbl 1179.65107 [7] Deng, W H, Numerical algorithm for the time fractional Fokker-Planck equation, J Comput Phys, 227, 1510-1522, (2007) · Zbl 1388.35095 [8] Gao, G H; Sun, Z Z, A compact finite difference scheme for the fractional sub-diffusion equations, J Comput Phys, 230, 586-595, (2011) · Zbl 1211.65112 [9] Gorenflo, R; Mainardi, F; Moretti, D; etal., Time fractional diffusion: a discrete random walk approach, Nonlinear Dynam, 29, 129-143, (2002) · Zbl 1009.82016 [10] Lin, Y; Xu, C, Finite difference/spectral approximations for the time-fractional diffusion equation, J Comput Phys, 225, 1533-1552, (2007) · Zbl 1126.65121 [11] Ma, J T; Jiang, Y J, Moving collocation methods for time fractional differential equations and simulation of blowup, Sci China Math, 54, 611-622, (2011) · Zbl 1217.34010 [12] Mainardi, F, The fundamental solutions for the fractional diffusion-wave equation, Appl Math Lett, 9, 23-28, (1996) · Zbl 0879.35036 [13] Metzler, R; Klafter, J, The random walks guide to anomalous diffusion: A fractional dynamics approach, Phys Rep, 339, 1-77, (2000) · Zbl 0984.82032 [14] Schneider, W R; Wyss, W, Fractional diffusion and wave equations, J Math Phys, 30, 134-144, (1989) · Zbl 0692.45004 [15] Pang, P Y H; Wang, M X, Qualitative analysis of a ratio-dependent predator prey system with diffusion, Proc R Soc Edinburgh Ser A, 133, 919-942, (2003) · Zbl 1059.92056 [16] Podlubny I. Fractional Differential Equations. San Diego: Academic Press, 1999 · Zbl 0924.34008 [17] Wang, M, Stationary patterns for a prey predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D, 196, 172-192, (2004) · Zbl 1081.35025 [18] Wyss, W, The fractional diffusion equation, J Math Phys, 27, 2782-2785, (1986) · Zbl 0632.35031 [19] Yun, A; Jeong, D; Kim, J, An efficient and accurate numerical scheme for Turing instability on a predator-prey model, Int J Bifur Chaos, 22, 125-139, (2012) · Zbl 1270.65053 [20] Yuste, S B; Acedo, L, An explicit finite difference method and a new von-Neumann type stability analysis for fractional diffusion equations, SIAM J Numer Anal, 42, 1862-1874, (2005) · Zbl 1119.65379 [21] Zhang, Y N; Sun, Z Z, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J Comput Phys, 230, 8713-8728, (2011) · Zbl 1242.65174
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