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

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.

Mathematics Subject Classification: Primary: 34A34, 35A05, 35K57; Secondary: 65L05, 65M06, 93C10.

 Citation:

• 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 $\alpha=0.25$ $\alpha=0.50$ $\alpha=0.75$ $\alpha=1.25$ $\alpha=1.50$ $\alpha=1.75$ FDM 9.2570e-06 1.8864e-05 2.8615e-05 4.8107e-05 5.7776e-05 6.7399e-05 0.1674s 0.1682s 0.1693s 0.1718s 0.1673s 0.1685s FSM 2.7055e-09 6.2174e-09 1.0710e-08 2.4231e-08 3.4382e-08 4.7695e-08 0.1664s 0.1663s 0.1665s 0.1677s 0.1663s 0.1659s

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-06 0.21 4.57E-05 0.27 1.33E-04 0.27 2.26E-04 0.27 $200$ 7.17E-06 0.27 3.58E-05 0.27 1.04E-04 0.27 1.77E-04 0.27 $300$ 2.86E-08 0.26 1.43E-05 0.28 4.14E-05 0.22 7.06E-08 0.26 $400$ 1.34E-06 0.26 6.71E-06 0.27 1.93E-05 0.27 3.29E-05 0.27 IMEX3PC $100$ 4.49E-06 0.26 2.43E-05 0.27 7.30E-05 0.27 1.24E-04 0.26 $200$ 3.51E-06 0.27 1.90E-05 0.27 5.72E-05 0.27 9.75E-05 0.27 $300$ 1.39E-06 0.27 7.54E-06 0.28 2.29E-05 0.29 3.90E-05 0.28 $400$ 6.43E-07 0.27 3.48E-06 0.27 1.07E-05 0.27 1.83E-05 0.28 ETD3RK $100$ 1.87E-07 0.26 1.01E-06 0.27 3.04E-06 0.26 5.18E-06 0.26 $200$ 1.46E-07 0.27 7.93E-07 0.28 2.38E-06 0.27 4.06E-06 0.27 $300$ 5.79E-08 0.27 3.14E-07 0.29 9.54E-07 0.27 1.62E-06 0.28 $400$ 2.68E-08 0.27 1.45E-07 0.28 4.48E-07 0.27 7.64E-07 0.27

Figures(6)

Tables(2)