 Research
 Open Access
Mathematical analysis and numerical simulation of twocomponent system with nonintegerorder derivative in high dimensions
 Kolade M Owolabi^{1}Email author and
 Abdon Atangana^{1}
https://doi.org/10.1186/s136620171286z
© The Author(s) 2017
 Received: 6 April 2017
 Accepted: 19 July 2017
 Published: 3 August 2017
Abstract
In this paper, we propose efficient and reliable numerical methods to solve two notable nonintegerorder partial differential equations. The proposed algorithm adapts the Fourier spectral method in space, coupled with the exponential integrator scheme in time. As an advantage over existing methods, our method yields a full diagonal representation of the noninteger fractional operator, with better accuracy over a finite difference scheme. We realize in this work that evolution equations formulated in the form of fractionalinspace reactiondiffusion systems can result in some amazing examples of pattern formation. Numerical experiments are performed in two and three space dimensions to justify the theoretical results. Simulation results revealed that pattern formation in a fractional medium is practically the same as in classical reactiondiffusion scenarios.
Keywords
 Fourier spectral method
 exponential integrator
 fractional reactiondiffusion
 nonlinear PDEs
 numerical simulations
 Turing instability
MSC
 34A34
 35A05
 35K57
 65L05
 65M06
 93C10
1 Introduction
Systems with noninteger order are commonly referred to as fractional differential equations. They are systems containing fractional integrals or fractional derivatives, which have received a lot of attention across disciplines such as biology, chemistry and physics. More importantly, they are mostly used in dynamical systems with chaotic and spatiotemporal dynamical behavior, quasichaotic dynamical systems, the dynamics of porous media or complex material and random walks with memory. The concepts of fractional differential equations, with fractional integral equations and fractional partial differential equations, have gained a wider application in diverse fields of applied science and engineering.
In some years back, the interest of some researchers was devoted to research on the equations involving the fractional differential equations applied to mechanics, physics, and other disciplines. For instance, the timefractional reactiondiffusion equations have been studied by Podlubny [1], Podlubny et al. [2], Gorenflo et al. [3], Guo et al. [4], GómexAguilar et al. [5–8], Zhou [9], Ilic et al. [10] and Kilbas et al. [11] and the references therein, to mention a few. The interested reader is referred to the monographs and literature on fractional calculus.
However, recent years have been characterized by fast growing applications of fractional calculus to various scientific and engineering fields pertaining to anomalous diffusion, signal processing and control, constitutive modelling in viscoelasticity, image processing, fluid mechanics and findings on soft matter behavior, to mention a few. Unlike the integerorder ordinary or partial differential equations, fractional calculus is capable of providing a more detail, simple and accurate description of complex dynamical, mechanical, chemical and physical processes that feature historical dependence and space nonlocality, which has induced the occurrences of a series of fractional differential equations.
However, the mathematical theory and the efficient numerical algorithms of fractionalorder differential equations require further study. Most analytical solutions obtained for fractional differential problems are given in terms of special functions, which make numerical evaluation difficult and almost impossible. Until now, finite difference schemes and series approximation techniques such as the variational iteration method and the Adomian decomposition method remain the dominant numerical methods for the solution of fractional reactiondiffusion equations.
More importantly, little is now known about the systematic analyses on the issue of stability of numerical methods regarding fractional calculus, together with the solution techniques for highdimensional fractional reactiondiffusion equations, most especially for nonlinear equations. The present paper introduces the Fourier spectral method as a better alternative to finite difference methods for solving fractionalinspace reactiondiffusion systems in one and high dimensions.
The remainder part of this paper is broken into sections. In Section 2, we introduce the general twocomponents partial differential equations formulated in both classical and fractional reactiondiffusion systems. We provide conditions for the emergence of Turing instability in the two scenarios. Section 3 deals with the basics of fractional derivatives and various methods of numerical approximations. Numerical experiments in high dimensions are presented with some notable examples taken from the literature in Section 4. Section 5 concludes the paper.
2 Model equation
In what follows, we shall examine some existing background theorems and definitions that are well established for the general twocomponent reactiondiffusion system (2.1) subject to zeroflux boundary conditions \(\nu\binom{u}{v}=0\), on \(\partial\Omega\times[0,T)\) with the initial function \(u=u_{0} v=v_{0}\) on \(\Omega\times\{t=0\}\), on smooth and bounded domain Ω, satisfying the conditions: (i) existence and uniqueness, (ii) existence for all times t, (iii) continuously dependency on the initial functions, (iv) for nonnegative initial data, the solution is nonnegative, and (v) the solution is bounded for all given bounded initial data.
Definition 2.1
Sectorial operator [15]
Theorem 2.2
Proof
The reader is referred to Theorem 1.3.4 in [15]. □
Lemma 2.3
Proof
Theorem 2.4
 (c1):

\(D>0\);
 (c2):

\(u_{0}\ge0\) and \(v_{0}\ge0\) are continuous on Ω̄, \(u_{0},v_{0}\in C^{0}_{L^{\infty}}(\Omega)\);
 (c3):

\(\mathcal{F}\) and \(\mathcal{G}\) are said to be continuously differentiable from \(\mathbb{R}^{2}_{+}\rightarrow\mathbb {R}\) with \(\mathcal{F}(0,y,t)\ge0\) and \(\mathcal{G}(x,0,t)\ge0\) for all \(x,y,t\ge0\), a situation applicable when \((x,y)\mapsto\mathcal {F}(x,y)\) and \((x,y)\mapsto\mathcal{G}(x,y)\) are differentiable for \(_{\Omega}^{\inf}\underline{u}\le x\le {}_{\Omega}^{\sup}\bar{u}, _{\Omega}^{\inf} \underline{v}\le x\le{}_{\Omega}^{\sup}\bar{v}\) by the mean value theorem;
 (c4):

There exist \(m>0\) and a continuous function \(F_{c}:\mathbb {R}^{2}_{+}\rightarrow\mathbb{R}_{+}\) such that \(\mathcal {F}(x,y,t), \mathcal{G}(x,y,t)\le\exp(mt)F_{c}(x,y)\) for all \(x,y,t\ge0\);
Proof
It suffices to establish the corresponding result for (2.3). See a similar proof in [15], Theorem 3.3.1, which utilized Banach’s fixed point theorem for establishing the result. □
2.1 Classical twocomponents reactiondiffusion systems
2.2 General twocomponents fractional reactiondiffusion systems
3 Fractional derivatives and adaptive numerical approaches

In the case of an infinite system, \(x\in(\infty, \infty)\), here R is a subset of \((\infty, \infty)\).

\(x\in[0, L], \frac{\partial u_{i}}{\partial x}(0,t)=\frac {\partial u_{i}}{\partial x}(L,t)=0, i=1,2,\ldots,n\), noflux or Neumann boundary condition for a finite system.

\(x\in[0, L], \mathbf{u}(0, t)=\mathbf{u}(L, t)=\mathbf{u}_{a}, i=1,2,\ldots ,n\), called the Dirichlet or fixed concentration boundary condition, also for a fixed system.
3.1 Integral representation of fractional derivative
In what follows, we shall describe the most two popular integral representation of fractional derivative via the Caputo and Riesz integral representations of the diffusion equation.
3.1.1 Caputo spacefractional derivative
3.1.2 Riesz spacefractional derivative
3.2 Numerical techniques for fractional diffusion equation
In this section, we do not intend to go into details, but we report a brief survey of some of the numerical approaches that have been used. Several numerical techniques have been used in the literature to circumvent the nonlocal restrictions associated with the spacefractional operators. Some of these methods include finite element, finite difference and spectral methods, to mention a few.
3.2.1 Finite difference method
Over the years, many time dependent partial differential equations have combined loworder nonlinear with higherorder linear terms. Until now, numerical simulation of most physical models still relied on a loworder finite difference scheme to discretize both the space and timefractionalorder derivatives, which are largely encountered in the fields of computational physics and mathematics. While the finite difference scheme remains simple, straightforward and easytocode for the integration of integerorder (classical) differential equations, its applicability is reduced for fractionalorder differential equations as it results to systems of linear equation defined by dense or large full matrices.
3.2.2 Finite element method
The major advantage of adopting a finite element technique is due to the flexibility it offers. For instance, the ability to model complex geometries, and to use local refinement to improve approximations. The main obstacle to circumvent is the nonlocal nature of the fractional operator, and direct application of finite element would result in a large and dense matrix [31]. The construction of such a matrix often results in serious difficulties, particularly in efficiency. When using a finite element technique to approximate a fractional Laplacian operator, it is not a trivial exercise to obtain reliable results, because of the dense matrix structure; to get optimal convergence would tremendously amount to an increase in the radius of truncation [27, 32].
3.2.3 Fourier spectral method
A little attention has been given to spectral method despite its desired ability to achieve higherorder convergence (accuracy and efficiency) over the existing loworder schemes such as finite difference and finite element schemes when applied to solve fractionalinspace reactiondiffusion equations. Application of spectral methods for the solution of classical reactiondiffusion problems is considered to be relatively simple and, being a subject of mathematical and theoretical studies for some years, has been by now examined almost completely. However, the case of fractionalorder reactiondiffusion is still poorly understood with just a handful of papers addressing the problems with spectral methods. Some of the little work done on spectral methods is [19, 32–34].
3.2.4 Time stepping method
3.2.5 Convergence analysis
The relative error for fractional diffusion equation for various values of discretization, at \(\pmb{D=0.25}\) and \(\pmb{t=5}\)
N  40  50  60  70  80 

Fourier spectral  1.2894e–08  1.5238e–08  2.2660e–08  3.1952e–08  4.1239e–08 
Finite difference  1.0017e–04  3.9125e–04  2.6447e–03  2.9589e–03  1.4286e–02 
Ratio  7.7687e+03  2.5676e+04  1.1671e+05  9.2605e+04  3.4642e+05 
4 Numerical experiments
4.1 Twodimensional results
4.2 Threedimensional results
5 Conclusion
For many years, the finite difference method has been considered as the mainstay in the numerical treatment of nonlinear partial differential equations. In this paper, we introduce the Fourier spectral method as a better alternative approach to the finite difference scheme, which is capable of removing the stiffness issues associated with the nonintegerorder spatial derivative of fractional reactiondiffusion equations. For the temporal discretization, we employ a fractional exponential integrator whose formulation is based on fourth order exponential time differencing to advance the resulting coupled system of ordinary differential equations in time. Two notable examples of reactiondiffusion systems taken from the literature are considered and formulated in spacefractional form. Our simulation results for the chosen examples show that pattern formations in the subdiffusive \((0<\alpha<1)\) and superdiffisive \((1<\alpha<2)\) scenarios are practically the same case as with the standard reactiondiffusion problems. The dynamic richness of our numerical techniques is explored in two and three space dimensions. The methodology in this research can be extended to solve multicomponents integer and nonintegerorder systems.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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