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Mathematical analysis and numerical simulation of twocomponent system with nonintegerorder derivative in high dimensions
Advances in Difference Equations volume 2017, Article number: 223 (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.
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.
Model equation
In this section, we consider a two species scaled and nondimensionalized system in the form [12–14]
where \(u(x,t), v(x,t)\) are used to describe the concentration of species at spatial position x and time t, due to the presence od diffusion, with respective diffusion coefficients \(D_{u}>0, D_{v}>0\). The nonlinear functions describing the reaction kinetics are given by \(\mathcal{F}\) and \(\mathcal{G}\). System (2.1) can be solved using any of the boundary conditions, namely Neumann (zeroflux), Dirichlet, periodic or Robin type on bounded domain \(\Omega\subset \mathbf{R}^{n}\). The chemical species concentrations are specified at \(t=0, \forall x\in\Omega\).
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]
Let operator A be linear in a Banach space, denoted \(\mathcal{B}\), and assume A is dense and closely defined. If there exist \(a,\omega \in(0,\pi), \mathcal{M}\ge1\), such that
and
where A is defined as a sectorial operator.
Theorem 2.2
If A is given as a sectorial operator, then −A is the corresponding infinitesimal generator of an analytic semigroup, \(G(t)\). If \(\operatorname{Re}\lambda >a, a\in\mathbb{R}\), whenever \(\lambda\in\sigma(A)\) then, for any \(t>0\),
Hence,
Proof
The reader is referred to Theorem 1.3.4 in [15]. □
Lemma 2.3
If u and v are continuous from \([0,T]\) to \(L^{p}(\Omega)\), then the integrals
exist, \(G_{1}\) and \(G_{2}\) are the respective analytical semigroup of the Laplacian operators \(\mathcal{L}=\Delta\) and \(D(\mathcal{L})\), \(I_{1}\) and \(I_{2}\) are continuous on \([0,T)\) with \(I_{1}(t)\), \(I_{2}(t)\in \mathcal{D}(\mathcal{L})\) and \(I_{1}(t),I_{2}(t)\rightarrow0^{+}\) in \(L^{p}\) for \(t\rightarrow0^{+}\).
Proof
If the reactiondiffusion system (2.1) has a classical solution, then u and v satisfy
The continuity of \(u, v\) indicates continuity of \(t\mapsto\mathcal {F}(u(t),v(t),t)\) and \(t\mapsto\mathcal{G}(u(t),v(t),t)\), so that the linear problems \(\partial_{t} U\Delta U=\mathcal{F}(u(t),v(t),t)\) and \(\partial_{t} VD\Delta V=\mathcal{G}(u(t),v(t),t)\) and \(U(0)=u_{0}, V(0)=v_{0}\) have a unique solution. For details of the proof, see Lemma 3.2.1 in [15]. □
Theorem 2.4
Assume the following conditions.
 (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. □
Classical twocomponents reactiondiffusion systems
A Turing instability (diffusiondriven instability) arises when a homogeneous steady state solution of the reaction system of the form (2.1) is linearly stable to some perturbations in the absence of the diffusion terms (\(D_{u}, D_{v}\)) but linearly unstable in the presence of diffusion to small spatial perturbations. A spatially uniform steady state of the system (2.1) is the state \((u^{*},v^{*}):\mathcal{F}(u^{*},v^{*})=\mathcal{G}(u^{*},v^{*})=0 \) in such a way that \(u=u^{*}, v=v^{*}\) satisfies the boundary conditions. For instance, using a zeroflux boundary condition on a rectangular domain, for diffusiondriven instability to occur, the conditions
must be satisfied. We evaluate the partial derivatives of \(\mathcal {F}\) and \(\mathcal{G}\) at the stationary uniform state \((u^{*},v^{*})\), that is, the zeros of \(\mathcal{F}\) and \(\mathcal{G}\). The linear stability for the time evolution of the perturbations
about the equilibrium steady state \((u^{*},v^{*})\) in the standard twocomponents system are given by
where \(D=D_{u}/D_{v}\) denotes the diffusivity ratio, \(\lambda>0\) is a scaling variable which can be defined as the relative strength of the reaction kinetics or as the linear size of the spatial domain, and
Next, for simplicity, we adopt the Laplace transform technique in the case of anomalous diffusion to find the Turing conditions. On applying a temporal Laplace and spatial Fourier transform, we obtain
where \(k,r\) are the Fourier and Laplace transform variables, the tilde and hat symbols denote, respectively, the Fourier and Laplace transformed variables. By finding the inverse of the Laplace transforms, we obtain the perturbations temporal growth
after factorizing the denominator and using partial fractions. If the roots in (2.6) are found to be distinct, the inverted canonical expression can be written in the form
with the corresponding inverse Laplace transform as
For any of the roots to be positive (that is, having a real component greater than zero), the homogeneous steady state becomes linearly unstable, but linearly stable if otherwise. In what follows, we quickly summarized the conditions for a Turing (diffusiondriven) instability as (i) \(\Re(s_{1}(k=0))<0, \Re(s_{2}(k=0))<0\), and (ii) \(\Re (s_{1}(k>0))>0, \Re(s_{2}(k>0))>0\), with \(s_{q}\) defined as the zeros of the quadratic equation
It follows from equation (2.6) that, with the conditions \(a_{11}+a_{22}<0\) and \(a_{11}a_{22}a_{12}a_{21}>0\), condition (i) is consistent, but not so for (ii) in conjunction with (i) if \(D=1\). Conditions (i) and (ii) cannot be attained simultaneously if \(D\le1\). In the physical sense, it implies that the necessary condition for a diffusiondriven (Turing) instability to give rise to pattern formation in a twocomponent system is that one of the species must diffuse faster than the other [12].
General twocomponents fractional reactiondiffusion systems
So far, we have examined the classical reactiondiffusion system. Here, we now consider the fraction reactiondiffusion system, a special case of (2.1) written in the general form
where \(0<\alpha\le2\) is termed the fractional power or mostly regarded as the anomalous diffusion exponent of the species’ \(u(x,t),v(x,t)\) concentrations. The parameter λ and functions \(\mathcal{F}\) and \(\mathcal{G}\) remain as earlier defined, and as illustrated in [17–19], we have
with a similar expression for v, is the generalization of the diffusion operator from standard to fractional. When solving the system with the Laplace transform, the term \(\mathcal{L}^{1} \{ \frac {\partial^{\alpha1}}{\partial t^{\alpha1}}\nabla^{2} u(x,t)_{t=0} \}\) actually preludes the introduction of nonphysical terms. Proceeding as earlier mentioned, a perturbation about point \((u^{*},v^{*})\) results in the linearized equations
By adopting the techniques of spatial Fourier and temporal Laplace transforms, we get
which we decouple as
Thus, we obtain the conditions for the diffusiondriven case in the fractional reactiondiffusion system by finding the inverse of the Laplace transforms. So in this work, we classified the scaling of the fractional power α into subdiffusion when \(0<\alpha<1\) [20, 21], advection flow equation when \(\alpha=1\) [19, 22], and superdiffusion when \(1<\alpha<2\) [10, 19]. In numerical experiments, our results will be based on these classifications.
Fractional derivatives and adaptive numerical approaches
By replacing Fick’s law for the flux U (which governs the rate at which mass is transported through an unit area against the concentration gradient) by its fractional derivative, a spacefractional diffusion equation can be derived in the form [23]
with the initial function
subject to any of the boundary conditions:

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.
Here \(u_{i}(t,\mathbf{x})\in\mathbf{R}^{n}\), \(\mathcal{ F}_{i}:\mathbf {R}^{n}\rightarrow\mathbf{R}\) and D is the diffusion tensor, and \(\Delta^{\alpha}= (\frac{\partial^{\alpha}}{\partial x^{\alpha}}, \frac{\partial^{\alpha}}{\partial y^{\alpha}},\frac{\partial^{\alpha}}{\partial z^{\alpha}} )^{\mathbf{T}}\) is the RiemannLiouville fractional gradient, for
with \(\frac{\partial^{\alpha}}{\partial y^{\alpha}}\) and \(\frac{\partial^{\alpha}}{\partial z^{\alpha}}\) having similar expressions.
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.
Caputo spacefractional derivative
A spacefractional diffusion equation takes the form
where \({}_{a}^{c} D_{x}^{\alpha}\) defines the fractional derivative in the Caputo sense [10, 11]
If we set \(h=(ba)/N, x=x_{l}=a+lh, u_{0}=u(xlh)=u(a), u_{1}=u(x(l1)h)=u(a+h),\ldots,u_{l}=u(x)=u(a+lh)\). Then we can approximate the fractional derivative term with \(n=2\) with
By mimicking the approach suggested in [24, 25], we can recast the fractionalinspace diffusion equation (3.3) to become an ODE,
for \(l=1,2,\ldots,N\), and \(u_{l}=u(x,t)\).
Riesz spacefractional derivative
In a similar fashion, a spacefractional diffusion equation can be taken as
where \({}_{0}^{R} D_{x}^{\alpha}\) is the Riesz fractional derivative, with expression
and
In this sense, \(_{x}I_{\pm}^{\alpha}\) is used to represent the RiemannLiouville (or Weyl) fractional integrals, defined as
and, recovering the Riesz potential, we have
In the interval \(1<\alpha<2\), which corresponds to the superdiffusive scenario, we define the pseudodifferential operator as
The integral operator \(_{x}I_{\pm}^{\alpha}\) is represented by \(_{x}I_{\pm }^{\alpha}=\frac{d^{2}}{dx^{2}}(_{x}I_{\pm}^{2\alpha})\), so that
In the spirit of Liu et al. [24, 25], we have the following equivalent ODE for (3.4):
Many others integral representations of the fractional derivatives are well classified in [1, 2, 4, 9–11, 21, 26].
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.
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.
A finite difference (FD) discretization of the fractionalorder differential equation can be obtained by using the wellknown shifted GrünwaldLetnikov approximation of the RiemannLiouville derivatives [1, 27, 28]. Here, we provide a further approximation by applying the FD method to solving the spacefractional diffusion equation
subject to the boundary conditions \(u(0,t)=u(1,t)=0\) and the initial function \(u(x,0)=f(x)\) in onedimension. We consider the cases \(0<\alpha<1\) and \(1<\alpha<2\) useful for many applications. The scenario of noninteger \(1<\alpha<2\) is used to model the superdiffusive case in which a cloud of particles spreads at a faster rate than in the classical diffusion model. It should be noted that we have the classical advective flow equation when \(\alpha=1\) and standard diffusion equation whenever \(\alpha=2\).
By using the FD method to approximate space derivatives first in the classical medium \(\alpha=2\), equation (3.5) can be written in the matrix form
where \(\delta=D/h^{2}\) and
We take \(\mathbf{A}/h^{2}\) to be approximation of differential operator \(T=\partial^{2}/\partial x^{2}\), then we set \(m(T^{\alpha/2})\approx 1/h^{\alpha}\mathbf{A}^{\alpha}\) (an approximate matrix representation of T) and we approximate (3.5) by the equation
with \(\bar{\delta}=D/h^{\alpha}\). The solution of fractionalinspace diffusion (3.8) is given by \(\mathbf{u}(t)=\exp(\bar{\delta }\mathbf{A}^{\alpha}t)\mathbf{u}(0)\). But if A is a symmetric positive definite matrix, the solution of (3.8) takes the form
where
are the eigenvalues of A, \(\mathbf{B}\in\mathbb{R}^{(N1)\times(N1)}\).
Finite element method
With the finite element method, the exact solution is usually approximated with expansion in terms of piecewise polynomials \(\phi_{j}(x)\):
The basis function \(\phi_{j}(x)\) has a compact support which corresponds to the node j of a mesh (or grid) that divides the computational domain in \(N1\) segments with steplength \(h=L/(N1)\). The position of node j is denoted \(x_{j}\), and computing the fractional derivative of the solution then amounts to computing the fractional derivative of all basis functions \(\phi_{j}(x)\); see Hanert [29, 30] for details.
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].
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].
For spectral representation, we let H be the real Hilbert space in \(\mathcal{L}_{2}(0,L)\), and consider the operator \(T:\mathcal {H}\rightarrow H\) defined by \(T\varphi=\Delta\varphi=\frac {d^{2}\varphi}{dx^{2}}\) on \(\mathcal{H}=\{\varphi\in H; \varphi',\varphi''\in\mathcal {L}_{2}(0,L) \text{ are continuous with boundary condition } \mathcal {B}(\varphi)=0 \}\). Then \(T\varphi_{n}=\lambda_{n}\varphi_{n}, n=1,2,\ldots\) . For any \(\varphi\in H\),
So, if ϖ is a continuous function on \(\mathbb{R}\), then
provided \(\sum_{n=1}^{\infty} \vert \varpi(\lambda_{n})c_{n} \vert ^{2},\infty\).
For the space discretization, we consider the onedimensional fractionalinspace diffusion equation (3.5) above, subject to initial condition \(u(x,0)=u_{0}(x)\) and homogeneous boundary conditions clamped at the extreme ends of spatial domain \(x\in[a,b]\). By relating (3.9) with the technique in [35], the fractional Laplacian \((\Delta)^{\alpha}\) can be defined in the space of functions
where
Therefore for any \(u\in H_{0}^{\alpha}\), the Laplacian \((\Delta )^{\alpha}\) is defined by
where \(\varphi_{j}\) and \(\lambda_{j}\) will depend on the specified boundary conditions.
For homogeneous Dirichlet boundary condition we obtain
and for homogeneous Neumann boundary condition we get
Next, we want to extend this approach to higherdimensional twocomponent fractionalinspace reactiondiffusion system (2.7) in the presence of reaction (source) terms; by applying the fast Fourier transform we obtain
where U and V are the double Fourier transforms for species \(u(x,y,t)\) and \(v(x,y,t)\). In other words,
With \(\Omega^{\alpha}=\omega_{x}^{\alpha}+\omega_{y}^{\alpha}\), we explicitly remove the issue of stiffness in the linear piece using integrating factors, by setting
such that
At this point, we can now discretize the square domain on equispaced number of points \(N_{x}\) and \(N_{y}\) in the spatial directions x and y. We adapt the discrete fast Fourier transform (DFFT) [35] to transform (3.15) to a system of ODEs
where \(u_{i,j}=u(x_{i},y_{j})\), \(v_{i,j}=v(x_{i},y_{j})\) and \(\Omega^{\alpha }_{i,j}=\chi_{x}^{\alpha}(i)+ \chi_{y}^{\alpha}(j)\). Boundary conditions are now clamped at the ends of the domain. At this junction, the whole system has been converted to ODEs, the stiffness property associated with the spatial derivatives is gone. It should be noted that any explicit higherorder scheme can be employed to advance in time. High order time integration methods can be naturally used in conjunction with high resolution spectral methods to obtain efficient and accurate approximations of the phase field models.
Time stepping method
For the temporal discretization, we employ the modified Krogstad [36, 37] version of the exponential time differencing RungeKutta method that was earlier proposed by Cox and Matthews [38]. It reads
where \(W=W(u,v)\), L, N defined as linear and nonlinear operators with variable coefficients
and the constants
To overcome the pronounced cancelation errors associated with the higherorder exponential time differencing RungeKutta method, the three coefficients in (3.19) suffer serious cancelation errors when the eigenvalues of L are close to zero. The vulnerability cancelation errors in the higherorder variants can render the whole schemes useless. As a result, modified exponential time differencing (ETD) schemes are proposed by using the complex contour integrals
here Γ is the contour that encloses the eigenvalues of l [39]. Details of derivation and stability analysis of scheme (3.17), which we denote for brevity in this paper ETD4RK, can be found in [13, 19, 37–43]. In the past, ETD schemes have also been used by some authors under different names [44–46].
Convergence analysis
Here, by convergence analysis we examined the tradeoff between the finite difference and Fourier spatial discretizations in terms of accuracy and efficiency. We test the performance of the schemes by reporting the error norm \(\ell_{\infty}\) in the numerical experiment at \(t=2\) as
where ũ and u are the respective approximate and exact solutions, and N is the number of computational grids.
By using (3.10 and (3.12), a straightforward analytical solution of (3.5) can be presented as
with \(\bar{u}_{j}(0)=\langle u_{0}(x),\varphi_{j}(x)\rangle\). Eigenvalues and eigenfunctions will depend on any of the given boundary conditions (3.13) and (3.14). Fourier spectral methods denote the truncated series expansion of (3.21) when a finite number of orthonormal trigonometric eigenfunctions \(\{\varphi_{j}\}\) are considered, that is,
For the space convergence, the solution of diffusion equation (3.5) using a finite difference or finite element matrixbased technique can be approximated by
where L denotes the corresponding eigenvectors matrix, and u represents the vector node of u [32, 47]. Obviously, both (3.22) and (3.23) are exact in time, to study the convergence of the two schemes used for numerical approximation of (3.5) we proceed by using a simple example. Hence, we consider for simplicity a onedimensional fractionalinspace diffusion benchmark problem with simple source term [10, 14, 32], which consists of finding \(u(x,t)\) such that
where
with initial condition computed as \(u(x,0)=zeros(N, 1)\), \(u(0,t)=0\), \(u(1,t)=0\) and the diffusion constant \(D>0\). The exact solution is given by \(u(x,t)=t^{\alpha}\sin^{3}(2\pi x)\).
Convergence results for finite differences and Fourier spectral methods are at varying α are displayed in Figure 1 \((ab)\), respectively. Also in Figure 1, we present the numerical results justifying the performance of both finite difference and Fourier spectral methods at some instances of α. Panel (c) shows the comparison between the exact (solidlines) and numerical (superimposed stars, circle and crosses) solutions for \(\alpha=(0.5, 1.1, 1.5)\). Plot (d) is obtained at different instances of time \(t=(1, 2, 4, 8)\) at fractional value \(\alpha=1.25\). Comparison in panels (a) and (b) for \(t=1\) and the relative error results in Table 1 justify well enough abandoning the finite difference method in the remaining part of this work.
Numerical experiments
In this section, we illustrate the mathematical techniques presented in the previous sections with some numerical examples in one, two, and probably three dimensions. We first consider the fractionalinspace reactiondiffusion system (2.7) and focus on the rescaled BeddingtonDeAngelis functional response with logistic growth term [48–51],
In order to give a good working guideline on the appropriate choice of parameters when numerically simulating the full fractional reactiondiffusion system, it is necessary to consider the local dynamics of the system [52]. By considering the nullclines \(\mathcal{F}(u,v)=0\) and \(\mathcal{G}(u,v)=0\), it is not difficult to see that model (2.7) with kinetics (4.1) possesses three equilibrium points in the absence of diffusion. The first point when \((u^{*},v^{*})=(0,0)\) corresponds to total washout of both populations, while \((u^{*},v^{*})=(1,0)\) or vise versa corresponds to existence of one population. The third equilibrium point
is a trivial state which corresponds to the coexistence of both prey and predator. It is the dynamics in the biologically meaningful region \(u\ge0, v\ge0\) which is of great interest.
The Jacobian or community matrix that corresponds to the interior equilibrium point \((u^{*},v^{*})\) is given by
where \(\mathcal{A}=(1u^{*})\) and \(\mathcal{B}=(\beta+u^{*}+v^{*})\). To ensure that the point \((u^{*},v^{*})\) satisfies \(\vert \mathbf{J}(u^{*},v^{*}) \vert >0\), it is necessary to keep β in the interval \((0,\frac{\gamma \delta}{\delta} )\) when numerically simulating the full system.
For the second example, we shall focus on the specific type II functional response [53],
The equilibrium point \((u^{*},v^{*})\) corresponding to the existence of both species is given by
Details of the linear stability analysis of kinetics (4.3) can be found in [52, 53].
It is clear from the results obtained in Figure 2 that the two species oscillate in phase. The populations profile in (a) which correspond to the coexistence of the species are unstable with initial time but become stable as time is increasing. It is otherwise with kinetics (4.3) in plot (c), where the species are stable with initial time and become unstable with spatiotemporal oscillations at bigger time, say \(t=1{,}000\).
Twodimensional results
In order to give a better illustration of the theoretical results and some biological wave phenomena of the fractionalinspace system describing spatial predatorprey interactions, we present in this subsection some numerical results in two space dimensions (2D). We let \(x,y\in\mathbf{R}^{2}, \Delta^{\alpha}=(\partial^{\alpha}/\partial x^{\alpha}+ \partial^{\alpha}/\partial y^{\alpha})\) and experiment on a growing domain in the interval \([L, L ]\) subject to Neumann boundary conditions clamped at the domain extreme ends, and initial conditions specifically chosen as
so as to induce a nontrivial spatiotemporal dynamics.
In the experiment with kinetics (4.1), it was found that the two species have similar pattern distributions. As a result, we report only that of species u as shown in Figure 3. The effects of fractional power α are shown to be more pronounced in the subdiffusive case when \(\alpha=0.5\) and superdiffusive scenario at \(\alpha=1.5\). The unsteady oscillatory patterns indicate the presence of a Turing instability. The evolution of more complex spiral patterns is shown in Figure 4 for kinetics (4.3) at different instances of fractional index α at time \(t=1{,}000\). The upper and lower rows represents the distribution patterns of species u and v, respectively, with varying values of α. We further report some complex spiral patterns distribution for v in Figure 5 in the fractional regime \(1<\alpha<2\), which we classify as a superdiffusive case. It should be mentioned that other Turing patterns can be observed, depending on the choice of initial and parameter values.
Threedimensional results
In the spirit of [54–56], we further explore some of the dynamic richness of fractionalinspace reactiondiffusion, we extend our numerical experiments to three space dimensions (3D). As usual, we let \(x,y,z\in\mathbf{R}^{3}, \Delta^{\alpha}=(\partial ^{\alpha}/\partial x^{\alpha}+ \partial^{\alpha}/\partial y^{\alpha}+\partial^{\alpha}/\partial z^{\alpha})\) denote the Laplacian operator with fractional order α. As discussed in Section 3, our numerical simulation results apply to general boundary conditions, but for the sake of ease of exposition we first experiment on a growing domain of size \([L, L]\) subject to Neumann boundary conditions and also consider a solution in the domain size \([0, L]\) subject to the Dirichlet or fixed boundary condition. For both kinetics, we focus on the initial functions
In Figure 6, we observed different amazing patterns for the distributions of species v. We considered two important boundary conditions to explore the variability in 3D patterns for the fractionalinspace reactiondiffusion system. We should also inform the reader that other pattern scenarios are possible and subject to the choice of initial functions, boundary conditions, fractional power index value and the domain size.
Similarly in Figure 7, we experiment with system (2.7) with kinetics (4.3). It was also found that the distributions of species u and v are almost of the same type. As a result, we present the evolutions of species u in 3D with both Neumann and Dirichlet boundary conditions as shown in the upper and lower rows, respectively. Other patterns are possible depending on the choice of parameters and initial conditions.
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.
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DOI
MSC
 34A34
 35A05
 35K57
 65L05
 65M06
 93C10
Keywords
 Fourier spectral method
 exponential integrator
 fractional reactiondiffusion
 nonlinear PDEs
 numerical simulations
 Turing instability