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June  2019, 12(3): 543-566. doi: 10.3934/dcdss.2019036

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

1. 

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

2. 

Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria

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

Received  May 2017 Revised  October 2017 Published  September 2018

Fund Project: 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.

Citation: Kolade M. Owolabi. Numerical analysis and pattern formation process for space-fractional superdiffusive systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 543-566. doi: 10.3934/dcdss.2019036
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References:
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B. S. T. Alkahtani, Chua's circuit model with Atangana-Baleanu derivative with fractional order, Chaos, Solitons and Fractals, 89 (2016), 547-551. Google Scholar

[2]

U. M. AscherS. J. Ruth and B. T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM Journal on Numerical Analysis, 32 (1995), 797-823. doi: 10.1137/0732037. Google Scholar

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[4]

A. Atangana, On the stability and convergence of the time-fractional variable order telegraph equation, Journal of Computational Physics, 293 (2015), 104-114. doi: 10.1016/j.jcp.2014.12.043. Google Scholar

[5]

A. Atangana, On the new fractional derivative and application to Fisher's reaction-diffusion, Applied Mathematics and Computation, 273 (2016), 948-956. doi: 10.1016/j.amc.2015.10.021. Google Scholar

[6]

A. Atangana and B. S. T. Alkahtani, New model of groundwater owing within a confine aquifer: Application of Caputo-Fabrizio derivative, Arabian Journal of Geosciences, 9 (2016), 1-6. Google Scholar

[7]

A. Atangana and R. T. Alqahtani, Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Advances in Difference Equations, 2016 (2016), 1-13. doi: 10.1186/s13662-016-0871-x. Google Scholar

[8]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. doi: 10.2298/TSCI160111018A. Google Scholar

[9]

A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solitons and Fractals, 89 (2016), 447-454. doi: 10.1016/j.chaos.2016.02.012. Google Scholar

[10]

T. Bakkyaraj and R. Sahadevan, Invariant analysis of nonlinear fractional ordinary differential equations with Riemann-Liouville fractional derivative, Nonlinear Dynamics, 80 (2015), 447-455. doi: 10.1007/s11071-014-1881-4. Google Scholar

[11]

D. BaleanuR. Caponetto and J. T. Machado, Challenges in fractional dynamics and control theory, Journal of Vibration and Control, 22 (2016), 2151-2152. doi: 10.1177/1077546315609262. Google Scholar

[12]

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

[13]

A. H. Bhrawy and M. A. Abdelkawy, A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations, Journal of Computational Physics, 294 (2015), 462-483. doi: 10.1016/j.jcp.2015.03.063. Google Scholar

[14]

A. H. Bhrawy, A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations, Numerical Algorithms, 73 (2016), 91-113. doi: 10.1007/s11075-015-0087-2. Google Scholar

[15]

A. Bueno-OrovioD. Kay and K. Burrage, Fourier spectral methods for fractional-in-space reaction-diffusion equations, BIT Numerical mathematics, 54 (2014), 937-954. doi: 10.1007/s10543-014-0484-2. Google Scholar

[16]

M. Caputo, Linear models of dissipation whose $ \mathcal{Q} $ is almost frequency independent: Part Ⅱ, J. R. Astr. Soc., 13 (1967), 529-539: Reprinted in: Fractional Calculus and Applied Analysis, 11 (2008), 4-14. Google Scholar

[17]

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. doi: 10.18576/pfda/020101. Google Scholar

[18]

F. ChenQ. Xu and J. S. Hesthaven, A multi-domain spectral method for time-fractional differential equations, Journal of Computational Physics, 293 (2015), 157-172. doi: 10.1016/j.jcp.2014.10.016. Google Scholar

[19]

W. ChenL. Ye and H. Sun, Fractional diffusion equations by Kansa method, Computers and Mathematics with Applications, 59 (2010), 1614-1620. doi: 10.1016/j.camwa.2009.08.004. Google Scholar

[20]

A. Coronel-EscamillaJ. F. Gómez-AguilarM. G. López-LópezV. M. Alvarado-Martínez and G. V. Guerrero-Ramírez, Triple pendulum model involving fractional derivatives with different kernels, Chaos, Solitons and Fractals, 91 (2016), 248-261. doi: 10.1016/j.chaos.2016.06.007. Google Scholar

[21]

S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics, 176 (2002), 430-455. doi: 10.1006/jcph.2002.6995. Google Scholar

[22]

X. Li Ding and Y. Lin-Jiang, Analytical solutions for the multi-term time-space fractional advection-diffusion equations with mixed boundary conditions, Nonlinear Analysis: Real World Applications, 14 (2013), 1026-1033. doi: 10.1016/j.nonrwa.2012.08.014. Google Scholar

[23]

E. H. Doha, A. H. Bhrawy and S. S. Ezz-Eldien, An efficient Legendre spectral tau matrix formulation for solving fractional sub-diffusion and reaction sub-diffusion equations, Journal of Computational and Nonlinear Dynamics, 10 (2015), 021019.Google Scholar

[24]

J. F. Gómez-AguilarT. Córdova-FragaJ. E. Escalante-MartínezC. Calderón-Ramón and R. F. Escobar-Jiménez, Electrical circuits described by a fractional derivative with regular kernel, Rev. Mex. Fis, 62 (2016), 144-154. Google Scholar

[25]

J. F. Gómez-AguilarM. G. López-LópezV. M. Alvarado-MartínezJ. Reyes-Reyes and M. Adam-Medina, Modeling diffusive transport with a fractional derivative without singular kernel, Physica A: Statistical Mechanics and its Applications, 447 (2016), 467-481. doi: 10.1016/j.physa.2015.12.066. Google Scholar

[26]

J. F. Gómez-AguilarL. TorresH. Yépez-MartínezD. BaleanuJ. M. Reyes and I. O. Sosa, Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel, Advances in Difference Equations, 2016 (2016), 1-13. doi: 10.1186/s13662-016-0908-1. Google Scholar

[27]

M. H. HeydariM. R. Hooshmandasl and F. Mohammadi, Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions, Applied Mathematics and Computation, 234 (2014), 267-276. doi: 10.1016/j.amc.2014.02.047. Google Scholar

[28]

C. IngoT. R. BarrickA. G. Webb and I. Ronen, Accurate Padé global approximations for the Mittag-Leffler function, its inverse, and its partial derivatives to efficiently compute convergent power series, International Journal of Applied and Computational Mathematics, 3 (2017), 347-362. doi: 10.1007/s40819-016-0158-7. Google Scholar

[29]

Y. JiaoL.-L. Wang and C. Huang, Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis, Journal of Computational Physics, 305 (2016), 1-28. doi: 10.1016/j.jcp.2015.10.029. Google Scholar

[30]

N. A. KhanN. U. KhanA. Ara and M. Jamil, Approximate analytical solutions of fractional reaction-diffusion equations, Journal of King Saud University-Science, 24 (2012), 111-118. doi: 10.1016/j.jksus.2010.07.021. Google Scholar

[31]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Netherlands, 2006. Google Scholar

[32]

Q. LiuF. LiuY. GuP. ZhuangJ. Chen and I. Turner, A meshless method based on Point Interpolation Method (PIM) for the space fractional diffusion equation, Applied Mathematics and Computation, 256 (2015), 930-938. doi: 10.1016/j.amc.2015.01.092. Google Scholar

[33]

J. MaJ. Liu and Z. Zhou, Convergence analysis of moving finite element methods for space fractional differential equations, Journal of Computational and Applied Mathematics, 255 (2014), 661-670. doi: 10.1016/j.cam.2013.06.021. Google Scholar

[34]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, Journal of Computational and Applied Mathematics, 172 (2004), 65-77. doi: 10.1016/j.cam.2004.01.033. Google Scholar

[35]

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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$$t=1.5$
1.35$64$5.2561E-056.7664E-051.7077E-075.8514E-073.7106E-072.8463E-07
$128$1.8173E-052.3564E-051.0202E-073.4986E-072.2171E-071.7003E-07
$256$2.8351E-063.8046E-064.6865E-081.6608E-071.0564E-078.1003E-08
1.55$64$5.2503E-057.7713E-052.3292E-076.3852E-074.6629E-073.8821E-07
$128$1.8137E-052.7075E-051.4398E-073.9483E-072.8825E-072.3996E-07
$256$2.8174E-064.3804E-066.8250E-081.9398E-071.4159E-071.1786E-07
1.75$64$5.1624E-058.7724E-052.7025E-075.9352E-074.9862E-074.5041E-07
$128$1.7833E-053.0563E-051.7156E-073.7683E-073.1654E-072.8593E-07
$256$2.7699E-064.9451E-068.6179E-081.8931E-071.5901E-071.4363E-07
$\alpha$$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-056.7664E-051.7077E-075.8514E-073.7106E-072.8463E-07
$128$1.8173E-052.3564E-051.0202E-073.4986E-072.2171E-071.7003E-07
$256$2.8351E-063.8046E-064.6865E-081.6608E-071.0564E-078.1003E-08
1.55$64$5.2503E-057.7713E-052.3292E-076.3852E-074.6629E-073.8821E-07
$128$1.8137E-052.7075E-051.4398E-073.9483E-072.8825E-072.3996E-07
$256$2.8174E-064.3804E-066.8250E-081.9398E-071.4159E-071.1786E-07
1.75$64$5.1624E-058.7724E-052.7025E-075.9352E-074.9862E-074.5041E-07
$128$1.7833E-053.0563E-051.7156E-073.7683E-073.1654E-072.8593E-07
$256$2.7699E-064.9451E-068.6179E-081.8931E-071.5901E-071.4363E-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\|$$T(s)$
1.15$100$1.44E-070.343.40E-050.211.02E-070.206.13E-080.23
$200$6.65E-080.176.38E-060.194.69E-080.192.81E-080.20
$300$4.26E-080.172.05E-060.173.00E-080.211.80E-080.17
$400$3.14E-080.178.96E-070.172.21E-080.171.33E-080.17
1.45$100$3.30E-070.244.30E-050.172.64E-070.171.58E-070.18
$200$1.63E-070.178.09E-060.171.30E-070.177.82E-080.17
$300$1.06E-070.172.61E-060.178.54E-080.175.12E-080.18
$400$7.96E-080.171.15E-060.176.36E-080.173.81E-080.17
1.81$100$3.96E-070.225.36E-050.183.67E-070.182.20E-070.61
$200$2.06E-070.171.01E-050.211.90E-070.171.14E-070.17
$300$1.37E-070.173.27E-060.171.27E-070.177.63E-080.17
$400$1.02E-070.171.44E-060.219.53E-080.175.71E-080.17
$N$ $(r, s)=(1, 0)$ $(r, s)=(1, -1)$
$\|u^c-u^e\|_{\infty}$$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-070.343.40E-050.211.02E-070.206.13E-080.23
$200$6.65E-080.176.38E-060.194.69E-080.192.81E-080.20
$300$4.26E-080.172.05E-060.173.00E-080.211.80E-080.17
$400$3.14E-080.178.96E-070.172.21E-080.171.33E-080.17
1.45$100$3.30E-070.244.30E-050.172.64E-070.171.58E-070.18
$200$1.63E-070.178.09E-060.171.30E-070.177.82E-080.17
$300$1.06E-070.172.61E-060.178.54E-080.175.12E-080.18
$400$7.96E-080.171.15E-060.176.36E-080.173.81E-080.17
1.81$100$3.96E-070.225.36E-050.183.67E-070.182.20E-070.61
$200$2.06E-070.171.01E-050.211.90E-070.171.14E-070.17
$300$1.37E-070.173.27E-060.171.27E-070.177.63E-080.17
$400$1.02E-070.171.44E-060.219.53E-080.175.71E-080.17
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