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# Nonstandard finite difference schemes for a fractional-order Brusselator system

- Mevlüde Yakıt Ongun
^{1}Email author, - Damla Arslan
^{1}and - Roberto Garrappa
^{2}

**2013**:102

https://doi.org/10.1186/1687-1847-2013-102

© Ongun et al.; licensee Springer 2013

**Received:**24 December 2012**Accepted:**27 March 2013**Published:**12 April 2013

## Abstract

In this paper, we discuss numerical methods for fractional order problems. Some nonstandard finite difference schemes are presented and investigated. The application in the simulation of a fractional-order Brusselator system is hence presented. By means of some numerical experiments, we show the effectiveness of the proposed approach.

## Keywords

- Fractional Order
- Fractional Derivative
- Fractional Calculus
- Fractional Differential Equation
- Discrete Representation

## 1 Introduction

Recently, fractional calculus has gained an increasing popularity due to the wide range of applications in fields including engineering, chemistry, finance, physics, seismology and so on.

Although the discussion on derivatives of noninteger order dates back to almost as far as the development of the classical theory of integer-order differential calculus, only at the end of the nineteenth century it has been realized the great enhancements that could be achieved by exploiting the power of fractional calculus; by means of fractional differential equations (FDEs) it is indeed possible to describe, in a natural way, real-life phenomena with memory effect and systems exhibiting anomalous diffusion [1].

The publication of some cornerstone books completely devoted to fractional calculus (we just cite here the works of Oldham and Spainer [2], Samko, Kilbas and Marichev [3], Miller and Ross [4], Podlubny [5], Diethelm [6] and Mainardi [7]) has successively played a considerable role in disseminating the subject. A variety of recent books [8–13] have also been published to illustrate applications of FDEs and methods for their solution.

In most cases, the solution of a FDE does not exist in terms of a finite number of elementary functions; it is therefore fundamental to device numerical methods in order to practically evaluate approximated solutions by means of difference schemes or other alternative approaches (*e.g.*, see [14–19]).

A major difficulty in the numerical treatment of FDEs is the presence of the long and persistent memory, which is related to the nonlocal nature of fractional derivative operators. From a practical point of view, storing and taking into account all the past history of the solution is usually a very demanding task.

In the presence of nonlinearities, these difficulties are amplified by the need of solving, at each step, some nonlinear algebraic systems whenever implicit schemes are adopted to cope with stability issues.

Nonstandard finite difference (NSFD) schemes have been introduced [20, 21] with the aim of avoiding full implicit schemes, which are computationally expensive, but at the same time preserving some of the main essential physical properties of the solution such as, for instance, positivity, monotonicity or convergence towards a stable steady-state.

The central aim of this work is to apply NSFD schemes within the context of fractional-order problems and study their potentials in replicating some of the main properties of the true solution [22–25]. In particular, the paper is concerned with the numerical simulation, by means of some ad hoc devised NSFD schemes, of a fractional-order Brussellator system; the stability properties of the system are analyzed and we show that the proposed NSFD methods allow to preserve stability in the numerical solution.

The paper is organized as follows. In Section 2, we briefly review the main definitions concerning fractional derivatives and FDEs and we introduce NSFD methods. Section 3 is devoted to discuss the main problems in applying NSFD schemes to fractional-order problems. In Section 4, we analyze stability of the fractional-order Brusselator system, and we present some suitably devised NSFD for this system. By means of some numerical simulations, in Section 5, we show the stability preserving properties of the proposed schemes and we compare the results with those provided by a classical method. Finally, some concluding remarks are given in Section 6.

## 2 Preliminaries and notations

In this section, some basic definitions and properties in the theory of the fractional calculus are presented; moreover, we introduce the main aspects concerning nonstandard discretization methods.

### 2.1 Fractional derivatives and FDEs

where $\mathrm{\Gamma}(\cdot )$ is the Euler gamma function, $m=\lceil \alpha \rceil $ is the smallest integer such that $m>\alpha $ and ${D}^{m}$ and ${d}^{m}/d{t}^{m}$ denote the standard derivatives of integer order.

which does not have a clear physical meaning.

*f*assumed sufficiently smooth at ${t}_{0}$, ${T}_{m-1}[f;{t}_{0}]$ is the $(m-1)$th degree Taylor polynomial for

*f*centered at ${t}_{0}$

*i.e.*,

*i.e.*,

This method has been extensively studied in literature (*e.g.*, see [18, 26]) and the numerical solution obtained by (6) converges to true solution with order 1 as ${h}_{N}\to 0$. Method (6) will form the basis on which NSFD methods will be devised in the subsequent section.

We introduce the following result concerning the weights of the GL discretization scheme (6), which will be used later on.

**Lemma 1**

*Let*$0<\alpha <1$

*and*${w}_{n}^{(\alpha )}$

*the coefficients in the GL operator*.

*Then for any*$n=1,2,\dots $

- 1.
$-1<{w}_{n}^{(\alpha )}<0$,

- 2.
$0<{w}_{n}^{(\alpha -1)}<1$.

*Proof* The proof is an immediate consequence of the recursive relationship stated in (3). □

### 2.2 Nonstandard discretization

NSFD schemes were firstly proposed by Mickens [20, 21] for either ODEs or PDEs and, successively, their use has been investigated in several fields (see, for instance, [27–31]).

*λ*is a, possibly vector, parameter. Given a mesh-grid ${t}_{n}={t}_{0}+hn$, that just for simplicity we assume to be equispaced with step-size $h>0$, NSFD schemes are constructed by following two main steps: (1) the derivative at the left-hand side of (7) is replaced by a discrete representation in the form

*h*and must fulfill the consistency condition

to ensure that the discrete representation in (8) converges to the corresponding continuous derivative as $h\to 0$. Examples of denominator functions fulfilling (10) are *h*, $sin(h)$, $1-{e}^{-h}$, $(1-{e}^{-\lambda h})/\lambda $ and so forth.

Other than by fulfilling the consistency condition (10), there are however some other criteria for choosing a suitable denominator function. The main aim is to achieve what is called the *dynamic consistency*: the solution of the discrete model (9) must satisfy some properties of particular importance for the original continuous model (7) (*e.g.*, positivity, monotonicity, fixed points and so on).

*h*is sufficiently small. It is instead easy to verify that the use of the nonstandard denominator function

ensures this desirable property for any value of *h*.

*f*. For more general problems having the fixed points ${\tilde{y}}_{\ell}$, $\ell =1,2,\dots ,L$, such that $f(t,{\tilde{y}}_{\ell},\lambda )=0$, after denoting

*ϕ*can be chosen as

to ensure that the steady-state of the discrete model preserves the same stability properties of the original continuous model.

## 3 NSFD for fractional differential equations

when the GL discretization (6) is exploited as the underlying method for the construction of NSFD schemes.

Although all the above functions satisfy the consistency condition (12), not all of them preserve the first order convergence of (6). Indeed, it is immediate to observe that to achieve this further goal it is necessary that *p* in (12) satisfies $p\ge 1+\alpha $. Since most of the listed functions do not fulfill this requirement, it is obvious to expect a drop in the convergence order which could reduce to *α*, when $0<\alpha <1$, instead of being 1.

Both of them fulfill the consistency condition (12) but just ${\varphi}_{2}(h,\lambda )$ preserves order of convergence 1 since ${\varphi}_{2}(h,\lambda )={h}^{\alpha}-\frac{\alpha \lambda}{2}{h}^{1+\alpha}+\cdots $ . Moreover, while ${\varphi}_{2}(h,\lambda )$ satisfies for any values of *h* the stability requirement (13) only when *λ* is of small or moderate size, the function ${\varphi}_{1}(h,\lambda )$ fulfills (13) in any circumstance. Thus, ${\varphi}_{1}(h,\lambda )$ is expected to better work for preserving stability whereas ${\varphi}_{2}(h,\lambda )$ seems more suitable for achieving higher accuracy.

## 4 Fractional-order Brusselator model: stability analysis and NSFD schemes

where $x(t)$, $y(t)$ are activator and inhibitor variables and *a*, *μ* are external parameters (the relationship between them determines the system dynamics) [37].

*E*of (14) is the solution of

which can be easily determined as $E=(a,\frac{\mu}{a})$. The following result allows to summarize the dynamics of this equilibrium point [38].

**Theorem 2** *There exists a marginal value* ${\alpha}_{0}$ *such that the equilibrium* *E* *is locally asymptotically stable if* $\alpha <{\alpha}_{0}$ *and it is unstable if* $\alpha >{\alpha}_{0}$.

*Proof*The steady state is locally asymptotically stable if all the eigenvalues

*λ*of the Jacobian matrix

*λ*can be determined by solving the characteristic equation $det(J(E)-\lambda I)=0$. Since the Jacobian matrix of the system (14) at the equilibrium point is

where $trJ=\mu -1-{a}^{2}$ and $detJ={a}^{2}$.

For $0<\alpha <2$ consider the parabola ${tr}^{2}J-4detJ=0$ and introduce the marginal value ${\alpha}_{0}=\frac{2}{\pi}|arg({\lambda}_{i})|$, $i=1,2$. When $\alpha <{\alpha}_{0}$ the system has oscillatory, but stable modes; when $\alpha >{\alpha}_{0}$ unstable and more complicated dynamics arise [38]. □

The value of *α* is thus an additional bifurcation parameter, which switches the stable and unstable states of the system and changes the form of the limit cycle. When $\alpha =1$, the system has a unique limit cycle when $\mu >{a}^{2}+1$ and it has a stable limit cycle for ${(a-1)}^{2}<\mu <{a}^{2}+1$ [38].

Since system (14) does not have a general solution in closed form, numerical methods must be used to approximate its solutions; a major requirement is that the numerical schemes preserve the dynamics of the system.

To discretize the fractional-order nonlinear system (14), we propose and discuss some NSFD schemes applied in combination with the truncation of the GL operator as stated in (11).

*NSFD scheme 1*: As a first nonstandard scheme, we make the replacement of the nonlinear term in the right-hand side of (14) by means of

In our tests as the denominator function, we will use the function ${\varphi}_{1}(h,\mu +1)=\frac{1-{e}^{-{h}^{\alpha}(\mu +1)}}{\mu +1}$ introduced in Section 3.

*NSFD scheme 2*: In our second nonstandard scheme, we use the replacement

and the denominator function ${\varphi}_{2}(h,\mu +1)={(\frac{1-{e}^{-h(\mu +1)}}{\mu +1})}^{\alpha}$ will be used.

*NSFD scheme 3*: In our last nonstandard scheme, we choose

When $\varphi (h)={h}^{\alpha}$, this is a fully explicit scheme and it is the counterpart for FDEs of the classical forward Euler method. Anyway in our experiments, we will use both the denominator functions ${\varphi}_{1}(h,\mu +1)=\frac{1-{e}^{-{h}^{\alpha}(\mu +1)}}{\mu +1}$ and ${\varphi}_{2}(h,\mu +1)={(\frac{1-{e}^{-h(\mu +1)}}{\mu +1})}^{\alpha}$ to compare the behavior and the corresponding schemes will be denoted respectively as NSFD 3a and NSFD 3b.

To study positivity of the numerical approximations let us assume ${x}_{0}\ge 0$, ${y}_{0}\ge 0$ and $a,\mu >0$. A straightforward analysis allows us to identify the conditions under which positive iterations ${x}_{n}$ and ${y}_{n}$ are obtained. We summarize these conditions as follows:

Scheme 1: $\varphi (h)<\frac{\alpha}{\mu +1}$ and $\varphi (h)<\frac{1}{{x}_{n}{x}_{n-1}}$.

Scheme 2: ${x}_{n-1}{y}_{n-1}<\mu +1$ and $\varphi (h)<\frac{\alpha}{{x}_{n}{x}_{n-1}}$.

Scheme 3: ${x}_{n-1}{y}_{n-1}>\mu +1$ and $\varphi (h)<\frac{\alpha}{{x}_{n-1}{y}_{n-1}}$.

## 5 Numerical simulations

We present in this section some numerical simulations. To compare the results obtained with the NSFD schemes investigated in this paper, we use the reference solution provided by the Adams-Bashforth-Moulton (ABM) method implemented in [41], whose stability properties have been investigated in [42].

*N*of steps. The errors $E(N)$ with respect to the reference solution are reported together with the estimation of the order of convergence (EOC) obtained as $\mathit{EOC}={log}_{2}(E(N)/E(2N))$.

**Errors and EOC with some values of**
N
**at**
$\mathit{T}\mathbf{=}\mathbf{40}$
**for**
$\mathbf{(}\mathit{a}\mathbf{,}\mathit{\mu}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{)}$
**,**
$\mathit{\alpha}\mathbf{=}\mathbf{0.8}$
**and**
$\mathbf{(}{\mathit{x}}_{\mathbf{0}}\mathbf{,}{\mathit{y}}_{\mathbf{0}}\mathbf{)}\mathbf{=}\mathbf{(}\mathbf{0.9}\mathbf{,}\mathbf{2.1}\mathbf{)}$

N | NSFD 1 | NSFD 2 | NSFD 3a | NSFD 3b | ||||
---|---|---|---|---|---|---|---|---|

Error | EOC | Error | EOC | Error | EOC | Error | EOC | |

320 | 3.05(−4) | 1.45(−4) | 2.95(−4) | 1.54(−4) | ||||

640 | 1.70(−4) | 0.842 | 7.33(−5) | 0.985 | 1.67(−4) | 0.823 | 7.32(−5) | 1.072 |

1,280 | 9.55(−5) | 0.831 | 3.61(−5) | 1.022 | 9.39(−5) | 0.827 | 3.64(−5) | 1.007 |

2,560 | 5.42(−5) | 0.818 | 1.80(−5) | 1.005 | 5.32(−5) | 0.818 | 1.82(−5) | 1.003 |

5,120 | 3.09(−5) | 0.811 | 8.98(−6) | 1.002 | 3.04(−5) | 0.807 | 9.07(−6) | 1.002 |

10,240 | 1.77(−5) | 0.807 | 4.49(−6) | 1.001 | 1.74(−5) | 0.803 | 4.53(−6) | 1.001 |

As expected from the discussion in Section 3, a drop in the order of convergence is achieved by using the denominator function ${\varphi}_{1}$ whilst the function ${\varphi}_{2}$ allows to preserve order 1. Moreover, all the schemes allow to obtain quite accurate results.

## 6 Concluding remarks

In this paper, some NSFD schemes have been investigated for the numerical solution of the fractional-order Brusselator differential system. Some different denominator functions and nonlocal terms have been proposed and the results have been compared with a classical Adams-Bashforth-Moulton method for FDEs. From the numerical experiments, we observed that NSFD schemes allow to replicate quite well the behavior of the true solution, and hence they are a useful tool for detecting the main stability properties for the problem under investigation and for similar problems.

## Declarations

### Acknowledgements

M.Y. Ongun and D. Arslan would like to acknowledge the partly financial support received from the Scientific Research Project Commission, SDU, Turkey, Project No: 2695-YL-11. The work of R. Garrappa has been carried out under the PRIN-MIUR project 2009F4NZJP.

## Authors’ Affiliations

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