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

Kolade M. Owolabi; Abdon Atangana

doi:10.3934/dcdss.2019037

Article Contents

2019, Volume 12, Issue 3: 567-590. Doi: 10.3934/dcdss.2019037

This issue Previous Article Numerical analysis and pattern formation process for space-fractional superdiffusive systems Next Article Efficient numerical method for a model arising in biological stoichiometry of tumour dynamics

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

Kolade M. Owolabi^, and
Abdon Atangana

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

Received: December 31, 2016

Revised: August 31, 2017

Published: October 2018

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

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Abstract

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.

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

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Figure 1. Stability regions of (a) ETD3RK, (b) IMEX3PC with choice $(\mu, \psi, \eta) = (1, 0, 0)$

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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 $N = 200$

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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 $N = 200$

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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

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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

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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

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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
FDM	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
FSM	0.1664s	0.1663s	0.1665s	0.1677s	0.1663s	0.1659s

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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	$N$	$0<\alpha<1$				$1<\alpha< 2$
Method	$N$	$\alpha=0.15$	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

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