Article Contents
Article Contents

# Numerical analysis and pattern formation process for space-fractional superdiffusive systems

The research contained in this report is supported by South African National Research Foundation

• In this paper, we consider the numerical solution of fractional-in-space reaction-diffusion equation, which is obtained from the classical reaction-diffusion equation by replacing the second-order spatial derivative with a fractional derivative of order $α∈(1, 2]$. We adopt a class of second-order approximations, based on the weighted and shifted Grünwald difference operators in Riemann-Liouville sense to numerically simulate two multicomponent systems with fractional-order in higher dimensions. The efficiency and accuracy of the numerical schemes are justified by reporting the norm infinity and norm relative errors as well as their convergence. The complexity of the dynamics in the equation is theoretically discussed by conducting its local and global stability analysis and Numerical experiments are performed to back-up the theoretical claims.

Mathematics Subject Classification: Primary: 65L05, 65L06, 93C10; Secondary: 34A34, 49M25, 65M70.

 Citation:

• Figure 1.  Convergence for one-dimensional example with $(r, s) = (1, 0)$, $\eta = 1.15$, $t = 2$ at some instances of fractional power index $\alpha$

Figure 2.  Convergence for two-dimensional example (29) with $(r, s) = (1, 0)$, $\eta = 1.1$, $t = 2$ at some instances of fractional power index $\alpha$

Figure 3.  Coexistence of the species. Panels (a-c) show the periodic behaviour of the three species as a function of time. Panel (d) depicts their limit cycle (species attractor) obtained at $t = 2000$. Parameter values: $a = 1, b = 0.1, c = 0.5$ with initial data $u_0 = v_0 = w_0 = 0.5$

Figure 4.  Two-dimensional evolution of fractional system (31) at different values of $\alpha$ for $a = 0.035$, $b = 0.065$, $\delta_1 = 2.05e-5$ and $\delta_2 = 1.0e-5$ with time step of $1.0$ on physical domain size $[0, L]\times [0, L], L = 2$. Simulation runs for $N = 200$ and final time $t = 5000$

Figure 5.  The 2D simulation results fractional reaction-diffusion system (32) showing spatiotemporal oscillations of the species at different instances of $\alpha$. Parameters: $\alpha = 1.35 (first-column), \alpha = 1.95 (second-column), $$L = 5, a = 0.9; b = 0.1, c = 1.5,$$ \delta_1 = 2, \delta_2 = 0.05, \delta_3 = 0.1$ and $t = 200$. Simulation runs for $N = 200$

Figure 6.  Distribution of two-component system (31) in 3D at $\alpha = 1.15$ (top-row), $\alpha = 1.55$ (middle-row) and $\alpha = 1.75$ (bottom-row). The Figures was captured in a $[128\times 128\times 128]$ Fourier modes with dimension $[0, L]^3, L = 1$. Other parameters are given in Figure 4

Figure 7.  The 3D simulation results showing different evolution of multicomponent fractional reaction-diffusion system (32) at various instances of $\alpha$. Parameters: $p = 1, a = 1; b = 0.1, c = 1.5, \delta_1 = 2, \delta_2 = 0.05, \delta_3 = 0.1$ and $t = 5$. Simulation runs for $N = 100$

Table 1.  The norm infinity and norm relative of errors for one dimensional problem (18) obtained at some instances of fractional power $\alpha$ and final time $t$, approximated with the Crank-Nicolson weighted and shifted Grünwald difference scheme with $\kappa = \hbar$ and $(r, s) = (1, 0)$

 $\alpha$ $N$ $\|u^c-u^e\|_{\infty}$ $\|u^c-u^e\|$ $t=0.5$ $t=1.0$ $t=1.5$ $t=0.5$ $t=1.0$ $t=1.5$ 1.35 $64$ 5.2561E-05 6.7664E-05 1.7077E-07 5.8514E-07 3.7106E-07 2.8463E-07 $128$ 1.8173E-05 2.3564E-05 1.0202E-07 3.4986E-07 2.2171E-07 1.7003E-07 $256$ 2.8351E-06 3.8046E-06 4.6865E-08 1.6608E-07 1.0564E-07 8.1003E-08 1.55 $64$ 5.2503E-05 7.7713E-05 2.3292E-07 6.3852E-07 4.6629E-07 3.8821E-07 $128$ 1.8137E-05 2.7075E-05 1.4398E-07 3.9483E-07 2.8825E-07 2.3996E-07 $256$ 2.8174E-06 4.3804E-06 6.8250E-08 1.9398E-07 1.4159E-07 1.1786E-07 1.75 $64$ 5.1624E-05 8.7724E-05 2.7025E-07 5.9352E-07 4.9862E-07 4.5041E-07 $128$ 1.7833E-05 3.0563E-05 1.7156E-07 3.7683E-07 3.1654E-07 2.8593E-07 $256$ 2.7699E-06 4.9451E-06 8.6179E-08 1.8931E-07 1.5901E-07 1.4363E-07

Table 2.  The norm infinity and norm relative of errors for one dimensional problem (29) obtained at some instances of fractional power $\alpha$ at final time $t = 1.0$ approximated with the Crank-Nicolson weighted and shifted Grünwald difference scheme for $\eta = 1.8$, $\kappa = \hbar$

 $N$ $(r, s)=(1, 0)$ $(r, s)=(1, -1)$ $\|u^c-u^e\|_{\infty}$ $T(s)$ $\|u^c-u^e\|$ $T(s)$ $\|u^c-u^e\|_{\infty}$ $T(s)$ $\|u^c-u^e\|$ $T(s)$ 1.15 $100$ 1.44E-07 0.34 3.40E-05 0.21 1.02E-07 0.20 6.13E-08 0.23 $200$ 6.65E-08 0.17 6.38E-06 0.19 4.69E-08 0.19 2.81E-08 0.20 $300$ 4.26E-08 0.17 2.05E-06 0.17 3.00E-08 0.21 1.80E-08 0.17 $400$ 3.14E-08 0.17 8.96E-07 0.17 2.21E-08 0.17 1.33E-08 0.17 1.45 $100$ 3.30E-07 0.24 4.30E-05 0.17 2.64E-07 0.17 1.58E-07 0.18 $200$ 1.63E-07 0.17 8.09E-06 0.17 1.30E-07 0.17 7.82E-08 0.17 $300$ 1.06E-07 0.17 2.61E-06 0.17 8.54E-08 0.17 5.12E-08 0.18 $400$ 7.96E-08 0.17 1.15E-06 0.17 6.36E-08 0.17 3.81E-08 0.17 1.81 $100$ 3.96E-07 0.22 5.36E-05 0.18 3.67E-07 0.18 2.20E-07 0.61 $200$ 2.06E-07 0.17 1.01E-05 0.21 1.90E-07 0.17 1.14E-07 0.17 $300$ 1.37E-07 0.17 3.27E-06 0.17 1.27E-07 0.17 7.63E-08 0.17 $400$ 1.02E-07 0.17 1.44E-06 0.21 9.53E-08 0.17 5.71E-08 0.17
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