Numerical solutions of the variable-order space fractional activator-inhibitor reaction-difusion systems

Abstract

Reaction-difusion equations containing fractional derivatives can provide adequate mathematical models for explaining anomalous difusion and transport dynamics in complex systems that can not be adequately modeled by standard numerical order equations. Researchers have recently found that many physical processes display fractional order dynamics that difers with time or space. The continuity of order in the fractional calculation allows the order of the fractional operator to be regarded as a variable. The Samko-Ross variable-order fractional operator can be viewed as a generalization of the RiemannLiouvile type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for reaction-difusion activator-inhibitor systems using this definition have not yet appeared in the literature. In this paper, we provide a numerical method to get the approximate solutions of the variable-order space-fractional Activator-inhibitor systems, namely the reaction-difusion system with cubic nonlinearity and Brusselator model on a 1-D space-domain, we adopt the Riesz variable-order space fractional derivative. Numerical simulations demonstrate that the finite diference approach is computationally eficient.

Keywords

• Variable-order calculus
• activator-inhibitor systems
• numerical simulation

1. Introduction

In recent decades, fractional derivatives have been commonly used in the modeling of complex physical and mechanical phenomena in wide classes of complex media with hereditary, fractal and non-markovian properties, is a field of rapidly growing interest with applications in many diferent fields. These include the study of biology [1], hydrology [2], biochemistry [3], finance [4], and physics [5]. But researchers have found that many important dynamic processes show fractional order behavior that may change with time and / or space. This fact suggests that diferential calculus is a natural candidate to provide an efective mathematical framework to describe the complex dynamic problems that appear in diferent biological and chemical models, such as viscoelastic materials, anomalous difusion. In reaction difusion systems which can be described by two variables, one has the notion of an activator and an inhibitor. In that case, be the difusion coeficient of the inhibitor greater than the difusion coeficient of the activator if spontaneous steady-state patterns are to occur. Thus, pattern formation is said to occur as a result of "short range activation" and "long range inhibition", the activator-inhibitor reaction-difusion systems model was used to simulate pattern forming processes. In this work we show the behavior of solutions for activator-inhibitor models with variable superdifusion by remplacing the second derivative in space by the variable order Riesz fractional derivative. The studies of dynamics and numerical simulations of fractional-space activator–inhibitor reaction–difusion systems have recently met with the interest of researchers (cf. [6, 7]).

2. Variable–order fractional derivative

In [8, 9] Samko and Ross directly generalized the Riemann-Liouvile and Marchaud fractional integration and diferentiation of the case of variable order and discussed some properties and the inversion formula . It has become a controversial research and has raised widespread concern in the last years.

This section is devoted to the most important definitions of variable–order direvatives. 2.1. Variable–order Riemann–Liouville fractional derivatives (, )

(, ) = (, ) (, ) =

1 (−1) [ Γ( − (, )) ∫

[ Γ( − (, )) ∫

( − ) − (, )−1 (, ) ( − ) − (, )−1 (, ) ] = ] = where Γ(.) is gamma function and − 1 < (, ) ⩽ , ∈ ℕ . 2.2. Variable-order Grünwald-Letnikov fractional derivatives

Left derivative:

Right derivative:

Left derivative:

Right derivative: where

(, ) ( ) =

Γ( (, ) + 1) Γ( + 1)Γ( (, ) − + 1) and − 1 < (, ) ⩽ , ∈ ℕ

lim ℎ→0,ℎ = − ℎ

− (, ) ∑(−1) (, ) (, ) (, ) =

lim ℎ→0,ℎ = − ℎ − (, ) ∑(−1) (, ) ( ( ) (, − ℎ ) ) (, + ℎ ) 2.3. Variable–order Riesz fractional derivative (, ) (, ) | | (, ) = −

1 2 cos ( (, ) 2 ) [ (, ) (, ) + (, ) (, )] .

3. General Activator-inhibitor model

Consider a 2-species reaction-difusion system

= Δ + (, ), Assume there exists a constant steady state = ( , ), = Δ + (, ) ( , ) = ( , ) = 0

The Jacobian of the previous system is defined by

| = ( 2111 2122) = ⎝⎜⎜⎛ || || ⎠⎟⎟⎞

We say that the reaction difusion system is the inhibitory activator system if the coeficients of its Jacobian matrix have the following relationships:

The species that achieve > 0, ( < 0) is called activator (inhibitor), respectively ( = 1, 2). 4. The variable order space fractional Activator-inhibitor models 4.1. Variable order space fractional reaction–difusion model with cubic nonlinearity We consider the reaction-difusion model with cubic nonlinearity in one-dimensional space, by replacing the classical spatial diferential operators by variable order Riesz fractional analogues , we obtain the following system (, ) = | | (, ) + − + , and the initial and boundary conditions in in [0, ] × [, ] , (0, ) = 0( ), (0, ) = 0( ), ∈ ℝ, ⩾ 0 and ⩾ 0 are external parameters.

Here , represent the concentrations of two species having the difusion rates , > 0,

4.2. Variable order space fractional Brusselator model

We consider the well-known reaction-difusion model Brusselator in one-dimensional space, by replacing the classical spatial diferential operators by variable order Riesz fractional analogues , we get the following system ( 4 ) ( 5 ) ( 6 ) ( 7 ) ( 8 ) (, ) + − ( + 1) + 2

, − 2 , in in

5. Explicit Euler Approximation

The species , represent the concentrations of two intermediary reactants having the dif0 are fixed concentrations, and , are external We show the approximate solutions for systems ( 1 )-( 4 ) and ( 5 )-( 8 ) by applying the explicit finite diference method which described in [10].

We consider the numerical approximation in the time domain [0, ] and the space domain [, ]. Let = Δ (0 ⩽ ⩽ ), = 0, … , ,

= + Δ ( ⩽ ⩽ ), = 0, … , , where the time step is Δ = and the space step is − . We denote that wald= f ormulas for the variable-order Riemann-Liouville fractional derivatives approximation Δ = ( , ) = ( , ) = − 2 sec( ) . The approximate Grün, show as follow: , ( , ) =

( , ) ≈ (Δ )− +1 ∑ ( , ) =

( , ) ≈ (Δ )− −1 +1 =0 where ( ) is the Grünwald weights defined by ( 0) = 1, ( ) = ( 1 − + 1

) ( −1), ( = 1, 2, … ) Therefore, the equations ( 1 )-( 2 ) and ( 5 )-( 6 ) can be discretized as follow: ( , ( ,

+1 ( 1 ) ∑ =0 +1 =0 +1 = + ( ) +1, +1− + ,

( 2 ) +1 = + ( 1 ) ∑ ( ) +1, +1− + , ( 2 ) ∑

The stability and convergence of the explicit Euler approximation are descused in [10].

6. Numerical experiments

tively Δ = 240 , Δ = (Δ2 )2 − 0.001.

In this section, we show approximate solutions for the systems ( 1 )-( 4 ) and ( 5 )-( 8 ) to demonstrate the changes in solutions behaviour that arise when the exponent is varied from integer order to fractional order to variable order, and to identify the diferences between solutions. The computer algorithm for numerical method ( 9 )-( 10 ), was written in Matlab, throughout the simulations we took the following values: = 0, = 4, = 20, spatial and time steps respec

6.1. Reaction-difusion model with cubic nonlinearity

We take, = −0.1, = 0.05, = 1, = 0.0503, = 0.07504 and the initial conditions: { 0( ) =

0.0503 − 10−3 cos(2 ) + 10−3 cos(5 ) 0( ) = 0.02504 + 10−3 cos(2 ) + 10−3 cos(5 ) The figure 1 shows the approximate solution for system ( 1 )-( 4 ) for (, ) = 2, when the derivative is fractional order (, ) = 1.35 we can see the sulotions in figure 2. The behaviour of the solution is particularly interesting for the case (, ) = 1.5 + 0.4 cos( ) ∗ sin(2 ), see figure 3.

6.2. Brusselator model

Figure 4 shows the behavior of the numerical solution for system ( 5 )-( 8 ) with (, ) = 2, when the derivative is fractional order (, ) = 1.35 we can see the sulotions in figure 5. The behaviour of the solution is particularly interesting for the case (, ) = 1.5 + 0.4 cos( ) ∗ sin(2 ), see figure 6.

7. Conclusions

In this work, we have got interesting behavior of solutions for activator-inhibitor models with variable superdifusion by replacing the classical second derivative in space by the variable order Riesz fractional derivative of order 1 < (, ) ⩽ 2, it seems that our numerical results will open horizons for analytical studies about such types of models and guide them by guessing.

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