June  2019, 12(3): 567-590. doi: 10.3934/dcdss.2019037

High-order solvers for space-fractional differential equations with Riesz derivative

Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa

* Corresponding author: mkowolax@yahoo.com (K.M. Owolabi)

Received  January 2017 Revised  September 2017 Published  September 2018

Fund Project: The research contained in this report is supported by South African National Research Foundation.

This paper proposes the computational approach for fractional-in-space reaction-diffusion equation, which is obtained by replacing the space second-order derivative in classical reaction-diffusion equation with the Riesz fractional derivative of order $ α $ in $ (0, 2] $. The proposed numerical scheme for space fractional reaction-diffusion equations is based on the finite difference and Fourier spectral approximation methods. The paper utilizes a range of higher-order time stepping solvers which exhibit third-order accuracy in the time domain and spectral accuracy in the spatial domain to solve some fractional-in-space reaction-diffusion equations. The numerical experiment shows that the third-order ETD3RK scheme outshines its third-order counterparts, taking into account the computational time and accuracy. Applicability of the proposed methods is further tested with a higher dimensional system. Numerical simulation results show that pattern formation process in the classical sense is the same as in fractional scenarios.

Citation: Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037
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show all references

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A. H. BhrawyM. A. Zaky and R. A. Van Gorder, A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation, Numerical Algorithms, 71 (2016), 151-180.  doi: 10.1007/s11075-015-9990-9.  Google Scholar

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[21]

N. F. Britton, Reaction-diffusion Equations and their Applications to Biology, Academic Press, London, 1986.  Google Scholar

[22]

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[24]

M. Caputo and M. Fabrizio, Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2 (2016), 1-11.   Google Scholar

[25]

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[26]

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[27]

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Figure 1.  Stability regions of (a) ETD3RK, (b) IMEX3PC with choice $(\mu, \psi, \eta) = (1, 0, 0)$
Figure 2.  Convergence results of different schemes for one-dimensional problem (1) at (a) $t = 0.1$ and (b) $t = 2.0$ for $\alpha = 1.45$, $d = 8$. Simulation runs for $N = 200$
Figure 3.  Solution of the fractional chemical system (42) in two-dimensions for subdiffusive (upper-row) and supperdiffusive (lower-row) scenarios. The parameters used are: $D = 0.39, d = 4, \varpi = 0.79, \beta = -0.91, \tau_2 = 0.278$ and $\tau_3 = 0.1$ at $t = 2$ for $N = 200$
Figure 4.  Superdiffusive distribution of chemical system (42) mitotic patterns in two dimensions at some instances of $\alpha$ with initial conditions: $u_0 = 1-\exp(-10(x-0.5)^2+(y-0.5)^2), \;\;v_0 = \exp(-10(x-0.5)^2+2(y-0.5)^2)$. Other parameters are given in Figure 3 caption
Figure 5.  Three dimensional results of system (42) showing the species evolution at subdiffusive ($\alpha = 0.35$) and superdiffusive ($\alpha = 1.91$) cases for $\tau_3 = 0.21$, $N = 50$ and final time $t = 5$. Other parameters are given in Figure 3 caption
Figure 6.  Three dimensional results for system (42) at different instances of fractional power $\alpha$, with $\tau_3 = 0.26$ and final time $t = 5$. The first and second columns correspond to subdiffusive and superdiffusive cases. Other parameters are given in Figure 3 caption
Table 1.  The maximum norm error and timing results for solving equation (1) in one-dimensional space with the exact solution and source term (40) using the FDM and FSM in conjunction with the IMEX3RK scheme at some instances of fractional power $\alpha$ in sub- and supper-diffusive scenarios for $t = 1$, $d = 0.5$ and $N = 200$
Method$\alpha=0.25$$\alpha=0.50$$\alpha=0.75$$\alpha=1.25$$\alpha=1.50$$\alpha=1.75$
FDM9.2570e-061.8864e-052.8615e-054.8107e-055.7776e-056.7399e-05
0.1674s0.1682s0.1693s0.1718s0.1673s0.1685s
FSM2.7055e-096.2174e-091.0710e-082.4231e-083.4382e-084.7695e-08
0.1664s0.1663s0.1665s0.1677s0.1663s0.1659s
Method$\alpha=0.25$$\alpha=0.50$$\alpha=0.75$$\alpha=1.25$$\alpha=1.50$$\alpha=1.75$
FDM9.2570e-061.8864e-052.8615e-054.8107e-055.7776e-056.7399e-05
0.1674s0.1682s0.1693s0.1718s0.1673s0.1685s
FSM2.7055e-096.2174e-091.0710e-082.4231e-083.4382e-084.7695e-08
0.1664s0.1663s0.1665s0.1677s0.1663s0.1659s
Table 2.  The maximum norm errors for two dimensional problem (1) with exact solution and local source term (41) obtained with different scheme at some instances of fractional power $\alpha$ and $N$ at final time $t = 1.5$ and $d = 10$
Method$N$$0<\alpha<1$ $1<\alpha< 2$
$\alpha=0.15$CPU(s)$\alpha=0.63$CPU(s)$\alpha=1.37$CPU(s)$\alpha=1.89$CPU(s)
IMEX3RK$100$9.15E-060.214.57E-050.271.33E-040.272.26E-040.27
$200$7.17E-060.273.58E-050.271.04E-040.271.77E-040.27
$300$2.86E-080.261.43E-050.284.14E-050.227.06E-080.26
$400$1.34E-060.266.71E-060.271.93E-050.273.29E-050.27
IMEX3PC$100$4.49E-060.262.43E-050.277.30E-050.271.24E-040.26
$200$3.51E-060.271.90E-050.275.72E-050.279.75E-050.27
$300$1.39E-060.277.54E-060.282.29E-050.293.90E-050.28
$400$6.43E-070.273.48E-060.271.07E-050.271.83E-050.28
ETD3RK$100$1.87E-070.261.01E-060.273.04E-060.265.18E-060.26
$200$1.46E-070.277.93E-070.282.38E-060.274.06E-060.27
$300$5.79E-080.273.14E-070.299.54E-070.271.62E-060.28
$400$2.68E-080.271.45E-070.284.48E-070.277.64E-070.27
Method$N$$0<\alpha<1$ $1<\alpha< 2$
$\alpha=0.15$CPU(s)$\alpha=0.63$CPU(s)$\alpha=1.37$CPU(s)$\alpha=1.89$CPU(s)
IMEX3RK$100$9.15E-060.214.57E-050.271.33E-040.272.26E-040.27
$200$7.17E-060.273.58E-050.271.04E-040.271.77E-040.27
$300$2.86E-080.261.43E-050.284.14E-050.227.06E-080.26
$400$1.34E-060.266.71E-060.271.93E-050.273.29E-050.27
IMEX3PC$100$4.49E-060.262.43E-050.277.30E-050.271.24E-040.26
$200$3.51E-060.271.90E-050.275.72E-050.279.75E-050.27
$300$1.39E-060.277.54E-060.282.29E-050.293.90E-050.28
$400$6.43E-070.273.48E-060.271.07E-050.271.83E-050.28
ETD3RK$100$1.87E-070.261.01E-060.273.04E-060.265.18E-060.26
$200$1.46E-070.277.93E-070.282.38E-060.274.06E-060.27
$300$5.79E-080.273.14E-070.299.54E-070.271.62E-060.28
$400$2.68E-080.271.45E-070.284.48E-070.277.64E-070.27
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