RETRACTED ARTICLE: Fractional order Lengyel–Epstein chemical reaction model

Abstract

The proposed model in this study is the fractional order differential equations system with multi-orders of the dimensionless Lengyel–Epstein model being the oscillating chemical reactions. It is founded the positive equilibrium point. Additionally, the stability of the positive equilibrium point obtained from this system is analyzed. The results founded from this qualitative analysis are corroborated by numerical simulations drawn by various programs using two different techniques.

1 Introduction

The oscillating chemical reactions as the Belousov–Zhabotinsky reaction and the Briggs–Rauscher reaction are outstanding. The mathematical models of these reactions are analyzed mathematically; then again; these models are muddled. In this paper, we are interested in a fractional version of the Lengyel–Epstein reaction diffusion system as a model of the chlorite-iodide malonic acid (CIMA) chemical reaction. The considered model has attracted the interest of many researchers since its inception in 1991. The reason for this interest is the fact that CIMA reaction is one of the earliest experiments that confined the theoretical propositions of Alan Turing in 1952, Concerning the chemical basis for morphogenesis and more generally pattern formation (Mansouri et al. 2019). The Lengyel–Epstein reaction involved chlorine dioxide (${\text{ClO}}_{2}$), iodine (${\text{I}}_{2}$) and malonic acid (${\text{MA}}$) is simpler reaction according to these. The reactions (Epstein and Pojman 1998) are:

$$ \left\{ {\begin{array}{*{20}l} {{\text{MA}} + {\text{I}}_{2} \to {\text{IMA}} + {\text{I}}^{ - } + {\text{H}}^{ + } , } \hfill & {({\text{i}})} \hfill \\ {{\text{ClO}}_{2} + {\text{I}}^{ - } \to {\text{ClO}}_{2}^{ - } + \frac{1}{2}{\text{I}}_{2} , } \hfill & {({\text{ii}})} \hfill \\ {{\text{ClO}}_{2}^{ - } + 4{\text{I}}^{ - } + 4{\text{H}}^{ + } \to {\text{Cl}}^{ - } + 2{\text{I}}_{2} + 2{\text{H}}_{2} {\text{O}}.} \hfill & {({\text{iii}})} \hfill \\ \end{array} } \right. $$

(1)

• 1. The iodization of malonic acid (MA) is given in (1(i)).

• 2. The oxidation of iodide ions by free chlorine dioxide radical is given by (1(ii)).

• 3. A reaction between chlorite and iodide ions produced in the 1(i) and 1(ii) process to generate iodine is given by (1(iii)).

Considering the empirical rate laws corresponding to these processes and ignoring constant factors, the model for this reaction was reduced to the conventional Lengyel–Epstein model with two independent variables $u$ and $v$ which related to the iodide concentration (${\text{I}}^{ - }$) and the chlorite concentration (${\text{ClO}}_{2}^{ - }$). This model is the system which occurred from two differential equations. The change rates of the concentrations of variables must be balanced by means of analyses and it is difficult that the reactions transform to the differential equations. Let $U$ and $V$ is the concentrations of iodide and chlorite, respectively. Therefore, the proposed rate equations are given by

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}U}}{{{\text{d}}t}} = \frac{{k_{1} \left[ {\text{MA}} \right]\left[ {{\text{I}}_{2} } \right]}}{{k_{2} + \left[ {{\text{I}}_{2} } \right]}} - k_{4} \left[ {{\text{ClO}}_{2} } \right]U - 4\left( {k_{5} \left[ {{\text{H}}^{ + } } \right]UV + k_{6} \left[ {{\text{I}}_{2} } \right]\frac{UV}{{\alpha + U^{2} }}} \right)} \hfill \\ {\frac{{{\text{d}}V}}{{{\text{d}}t}} = k_{4} \left[ {{\text{ClO}}_{2} } \right]U - \left( {k_{5} \left[ {{\text{H}}^{ + } } \right]UV + k_{6} \left[ {{\text{I}}_{2} } \right]\frac{UV}{{\alpha + U^{2} }}} \right),} \hfill \\ \end{array} } \right. $$

(2)

where the $k_{i}$ for $k = 1,2,4,5,6$ are the rate constants and $\alpha$ (mass/volume)² is a constant too. The rate conditions have connected to all particles and particles associated with the procedure (Chicone 2010). Nonetheless, it very well may be indicated all through investigations that the iodide and chlorite fixations must change all the more quickly than other molecules. In this sense, it is attainable to assume for effectiveness which it stays consistent the grouping of different molecules as long as response. Subsequently, this presumption does not need to be applied. Nevertheless, we have to choose to approach real values in all mathematical modeling. In the wake of picking the correct course for maintainability, the $k_{5}$ term will come to disregard too as in the Lengyel and Epstein (1991) for no conspicuous reason aside from maybe, since it is sure that the hydrogen focus (${\text{H}}^{ + }$), which is the lightest core, is little. The previously mentioned theories cause to the differential equation’s system as the accompanying;

$$ \begin{array}{*{20}l} {\frac{{{\text{d}}U}}{{{\text{d}}t}} = M - NU - 4P\left( {\frac{UV}{{\alpha + U^{2} }}} \right)} \hfill \\ {\frac{{{\text{d}}V}}{{{\text{d}}t}} = PU - P\left( {\frac{UV}{{\alpha + U^{2} }}} \right),} \hfill \\ \end{array} $$

(3)

$M,N,P > 0$.

For a better application in mathematical modeling, it is necessary that mathematical models are to be made dimensionless. This situation is succeeded by changing the variables

$$ U = \sqrt \alpha u,\quad V = \frac{\alpha B}{C},\quad t = \frac{\tau }{B} $$

(4)

and two lumped parameters which are dimensionless are

$$ l = \frac{M}{N},\quad m = \frac{C}{B\sqrt \alpha },\quad l,m > 0. $$

(5)

After all these operations, the Lengyel–Epstein model is as the following (Chicone 2010):

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}u}}{{{\text{d}}\tau }} = l - u - \frac{4uv}{{1 + u^{2} }}} \hfill \\ {\frac{{{\text{d}}v}}{{{\text{d}}\tau }} = mu\left( {1 - \frac{v}{{1 + u^{2} }}} \right)} \hfill \\ \end{array} } \right.. $$

(6)

Kinetic studies, in solution, are a controlling tool in the investigation of the reaction mechanisms, permitting to deduce vital information of the procedures that occur before the determining step of the speed. Through a kinetic study one can determine the speed law of a reaction as well as its rate constant. One of the approaches used for this is the use of integrated equations. In this approach, it is sought to check whether the change in the concentration of one of the reactants or products follows a first or second order kinetics or, more rarely, a kinetics with higher orders or even zero order. In a chemical reaction, reactants pass through a transition state region along the reaction coordinate between reactants and products, where chemical bonds are broken and reformed. This transition state, first proposed by Polanyi and Eyring in the early 1930s. Many researcher dictates almost every aspect of a chemical reaction, and has played a central conceptual role in the development of chemistry as a branch of science. Therefore, observing and understanding the transition state have been regarded as the ‘‘Holy Grails’’ of chemistry (Wang et al. 2018).

Model (6) is an integer order system, that is, the first-order derivative with respect to time variable $\tau$. The first-order derivative to the variable $\tau$ implies the transient change rate of these reactions. However, due to the complexity of biochemical reactions, chemical reaction processes are often affected by or depend on the history of chemical reactions. Thus, this phenomenon can be described by fractional order differential equations. In fact, fractional calculus is an old mathematical topic developed as a pure theoretical field of mathematics for more than three centuries. When one observes an order not whole, the first idea that occurs is to look for what is wrong. This is because the appearance of non-integer orders is not usually addressed clearly in the books and courses of chemical kinetics. However, the appearance of a non-integer order is a consequence of a mechanism somewhat more complex than those considered to arrive at the integrated velocity laws, before quite common occurrence. Introducing fractional time derivatives has recently been shown to model natural phenomena more accurately especially in chemical reactions.

Fractional calculus has a longer history than integer calculus. It is a dominant tool for finding elucidation of nonlinear problems. However, the application of fractional derivatives and integrals has been infrequent until the last century. Fractional order derivatives allow us to deal comfortably with memory effects in a dynamical system and thus it can be successfully applied in some fields such as viscoelastic damping, anomalous diffusion process, signal processing, electrochemistry, fluid flow, chemistry and so on. In many previous studies, the existence or no-existence of periodic solution and limit cycle of fractional order equation (system) are important research directions (Yuan and Kuang 2017). In recent years, even fractional order models of happiness and love have been developed and they are claimed to give better representation than the integer order dynamical approach. Highly remarkable scientific books provide the main theoretical tools for the qualitative analysis of fractional order dynamical systems and at the same time show the interconnection as well as contrast between classical differential equations and fractional differential equations. As of late, a substantial amount of writing has been produced concerning the use of fractional differential equations in nonlinear dynamics (El-Saka and El-Sayed 2013; El-Saaka et al. 2009; El-Sayed et al. 2007; Podlubny 1999; Podlubny and El-Sayed 1996).

Recently, most of the dynamical structures dependent on the non-fractional order calculus have been changed into the fractional order domain on account of the extra degrees of opportunity and the adaptability which can be connected to conclusively fit the test data much better than anything the integer order modeling. Similarly, fractional order models have memory so fractional order differential equation gives us an increasingly reasonable approach to show FOLECR. Recently Zafar et al. (2017a, b, c, d, 2018) composed numerous papers on fractional order models. Zafar et al. (2017a) utilizes the Adams Bashforth-Moulton (PECE) technique to explain the fractional order Bovine Babesiosis Disease and Tick Population numerically. Afterward, he has utilized three distinctive fractional order methods for explaining the HIV/AIDS epidemic fractional order model (Zafar et al. 2017c). He has likewise chipped away at fractional order dengue internal transmission model and unravel it numerically utilizing three fractional order strategies (Zafar et al. 2018). The primary reason should be that the type of the equilibrium point of the subsystem keeps unchanged with the variation of the fractional order. However, the oscillation period may become much longer with the decrease of fractional order $\alpha$. The differences between the periods of integer order and fractional order systems lie in the fact, power law stability (Li et al. 2008) is used define the asymptotically stability of fractional order system instead of traditional exponential stability. In our work, we aim to propose and study the dynamics of the time fractional system corresponding to Lengyel–Epstein model.

This article is organized into five sections. The introduction is the first section in which we elaborate some history of fractional calculus. In Sect. 2, we will elaborate notations related to the concept of FDEs. In Sect. 3, we ponder on the fractional order model linked with the dynamics of FOLECR. Qualitative dynamics of the considerable system is resolute using elementary reproduction number. In Sect. 4, numerical imitations are offered to validate the main outcomes and conclusion is drawn in Sect. 5.

2 Asymptotic stability of their equilibrium points

Definition 2.1

The fractional integral of order $\beta \in R^{ + }$ of the function $g\left( t \right)$, $t > 0$ is defined by

$$ I^{\beta } g\left( t \right) = \int\limits_{0}^{t} {\frac{{\left( {t - s} \right)^{\beta - 1} }}{\varGamma \left( \beta \right)}} g\left( s \right){\text{d}}s. $$

(7)

And the fractional derivative of order $\alpha \in \left( {n - 1,n} \right]$ of $g\left( t \right)$, $t > 0$ is defined by

$$ D^{\alpha } g\left( t \right) = I^{n - \alpha } D^{n} g\left( t \right) = \mathop \int \limits_{0}^{t} \frac{{\left( {t - s} \right)^{n - \alpha - 1} }}{{{{\varGamma }}\left( {n - \alpha } \right)}} g^{\left( n \right)} \left( s \right){\text{d}}s,\quad D = \frac{\text{d}}{{{\text{d}}t}} $$

(8)

Theorem 2.1

The fractional differential equation’s system with multi-order is as the following

$$ D_{*}^{\check{\alpha}} = {\check{g}} \left( {\check{t}} ,{\check{x} } \right),\quad {\check{x}} \left( 0 \right) = {\check{x}}_{0} , $$

(9)

where ${\check{x}} = \left[ {x_{1} \left( t \right),x_{2} \left( t \right), \ldots ,x_{n} \left( t \right)} \right]^{\text{T}} \in R^{n}$, ${\check{g}} = \left[ {g_{1} ,g_{2} ,g_{3} , \ldots ,g_{n} } \right]^{\text{T}} \in R^{n}$, $g_{i} :\left[ {0, + \infty } \right) \times R^{n} \to R$, $i = 1,2, \ldots ,n$, ${\check{\alpha }} = \left[ {\alpha_{1} , \alpha_{2} , \alpha_{3} , \ldots ,\alpha_{n} } \right]^{\text{T}}$is the multi-order of system (9) and $D_{*}^{\check{\alpha }} = \left[ {D_{*}^{{\alpha_{1} }} ,D_{*}^{{\alpha_{2} }} , \ldots ,D_{*}^{{\alpha_{n} }} } \right]$, $D_{*}^{{\alpha_{i} }}$denotes $\alpha_{i}$th-order fractional derivative in the Caputo sense. In this sense, $D_{*}^{\check{\alpha}} {\check{x}} \left( t \right) = \left[ {D_{*}^{{\alpha_{1} }} x_{1} \left( t \right),D_{*}^{{\alpha_{2} }} x_{2} \left( t \right), \ldots ,D_{*}^{{\alpha_{n} }} x_{n} \left( t \right)} \right]$. From a mathematical point of view, the multiple orders can be any real vector even complex one. In engineering applications, $\alpha_{i}$often lies in (0, 1), and is a rational number related to physical measure. In this paper we presume $\alpha_{i}$to a rational number in (0, 1) (Odibat 2010; Deng 2007).

Theorem 2.2

It is assumed that $J\left( E \right)$is Jacobian matrix evaluated at equilibrium point $E$. This point of the system (9) is asymptotically stable if all the eigenvalues obtained from the polynomial $\det \left( {{\text{diag}}\left( {\lambda^{{M\alpha_{1} }} , \lambda^{{M\alpha_{2} }} , \ldots ,\lambda^{{M\alpha_{n} }} } \right) - J\left( E \right)} \right) = 0$satisfy $\left| {{ \arg }\left( \lambda \right)} \right| > \frac{\gamma \pi }{2}$ (Odibat 2010).

Lemma 2.1

Consider Definition 2.1, also, let $\alpha_{1} = \alpha_{2} = \alpha \in \left( {0,1} \right]$and $\beta \in \left( {0,1} \right)$if $f \in C\left[ {0,T} \right]$, then $\left. {I^{\beta } f\left( t \right)} \right|_{t = 0} = 0$. In this respect, taking into consider the following system (El-Saka and El-Sayed 2013; El-Saaka et al. 2009; Podlubny and El-Sayed 1996; Deng 2007; Matignon 1996; Ahmed et al. 2007):

$$ \left\{ {\begin{array}{*{20}c} {D^{\alpha } u_{1} \left( t \right) = g_{1} \left( {u_{1} , u_{2} } \right)} \\ {D^{\alpha } u_{2} \left( t \right) = g_{2} \left( {u_{1} , u_{2} } \right)} \\ \end{array} } \right. $$

(10)

with the initial condition

$$ u_{1} \left( 0 \right) = u_{10} ,\quad u_{2} \left( 0 \right) = u_{20} . $$

(11)

For evaluate the equilibrium points, we have presumed that $D^{\alpha } u_{i} \left( t \right) = 0 \Rightarrow g_{i} \left( {u_{1}^{\text{eq}} , u_{2}^{\text{eq}} } \right) = 0$ for $i = 1,2$. In this sense, we have the equilibrium point $\left( {u_{1}^{\text{eq}} , u_{2}^{\text{eq}} } \right)$ of system (10).

The Jacobian matrix of the system (10) is

$$ J = \left[ {\begin{array}{*{20}c} {\frac{{\partial g_{1} }}{{\partial u_{1} }}} &\quad {\frac{{\partial g_{1} }}{{\partial u_{2} }}} \\ {\frac{{\partial g_{2} }}{{\partial u_{1} }}} & \quad{\frac{{\partial g_{2} }}{{\partial u_{2} }}} \\ \end{array} } \right]. $$

If all the eigenvalues $\lambda_{1}$ and $\lambda_{2}$ obtained from the equation $J\left( {u_{1}^{\text{eq}} , u_{2}^{\text{eq}} } \right) = 0$ satisfies the conditions

$$ \left| {{ \arg }\left( {\lambda_{1} } \right)} \right| > \frac{\alpha \pi }{2},\;\left| {{ \arg }\left( {\lambda_{2} } \right)} \right| > \frac{\alpha \pi }{2}, $$

(12)

then we say that the equilibrium point $\left( {u_{1}^{\text{eq}} , u_{2}^{\text{eq}} } \right)$ is locally asymptotically stable point for system (10). The stability region of the fractional order system by $\alpha$-order is shown in Fig. 1 (in which $\sigma$, $\omega$ refer to the real and imaginary parts of the eigenvalues, respectively, and $j = \sqrt { - 1}$). From here, it is seen that the stability region of the fractional order case is broader than the stability region of the integer order case (Ahmed et al. 2006; Murray 2002).

Fig. 1

Stability region of fractional order system in (16)

The characteristic equation obtained from $J\left( {u_{1}^{\text{eq}} , u_{2}^{\text{eq}} } \right) = 0$ is such as the following generalized polynomial:

$$ P\left( \lambda \right) = \lambda^{2} + a_{1} \lambda + a_{2} = 0. $$

(13)

Let us consider the conditions (12) and the polynomial (13). Therefore, the conditions for locally asymptotically stability of the equilibrium point $\left( {u_{1}^{\text{eq}} , u_{2}^{\text{eq}} } \right)$ are either Routh–Hurwitz conditions $(a_{1} , a_{2} > 0)$ (Hale and Koçak 1991).

Or

$$ a_{1} < 0,\;4a_{2} > \left( {a_{1} } \right)^{2} ,\;\left| {\tan^{ - 1} \left( {\frac{{\sqrt {4a_{2} - \left( {a_{1} } \right)^{2} } }}{{a_{1} }}} \right)} \right| > \frac{\alpha \pi }{2}. $$

(14)

Theorem 2.3

(Routh–Hurwitz Criteria) The characteristic polynomial is

$$ P\left( \lambda \right) = \lambda^{n} + a_{1} \lambda^{n - 1} + a_{2} \lambda^{n - 2} + \cdots + a_{n - 1} \lambda + a_{n} , $$

where the $a_{i}$ coefficients for $i = 1, 2, \ldots , n$ are real constants. The n-Hurwitz matrices by the coefficients $a_{i}$ of the upper polynomial are

$$ H_{1} = \left( {a_{1} } \right),\;H_{2} = \left( {\begin{array}{*{20}c} {a_{1} } & \quad 1 \\ {a_{3} } &\quad {a_{2} } \\ \end{array} } \right),\;H_{3} = \left( {\begin{array}{*{20}c} {a_{1} } &\quad 1 &\quad 0 \\ {a_{3} } &\quad {a_{2} } &\quad {a_{1} } \\ {a_{5} } &\quad {a_{4} } &\quad {a_{3} } \\ \end{array} } \right), $$

$$ H_{n} = \left( {\begin{array}{*{20}l} {a_{1} } \hfill &\quad 1 \hfill &\quad 0 \hfill &\quad 0 \hfill &\quad \cdot \hfill &\quad 0 \hfill \\ {a_{3} } \hfill &\quad {a_{2} } \hfill &\quad {a_{1} } \hfill &\quad 1 \hfill &\quad \cdot \hfill &\quad 0 \hfill \\ {a_{5} } \hfill &\quad {a_{4} } \hfill &\quad {a_{3} } \hfill &\quad {a_{1} } \hfill &\quad \cdot \hfill &\quad 0 \hfill \\ \cdot \hfill &\quad \cdot \hfill &\quad \cdot \hfill &\quad \cdot \hfill &\quad {} \hfill &\quad \cdot \hfill \\ \cdot \hfill &\quad \cdot \hfill & \quad \cdot \hfill & \quad {} \hfill &\quad \cdot \hfill &\quad \cdot \hfill \\ 0 \hfill &\quad 0 \hfill &\quad 0 \hfill &\quad 0 \hfill &\quad \cdot \hfill &\quad {a_{n} } \hfill \\ \end{array} } \right), $$

where $a_{j} = 0$ if $j > n$. The roots of polynomial $P\left( \lambda \right)$ are negative or have negative real parts, iff the determinants of all Hurwitz matrices are positive: $H_{j} > 0$, $j = 1,2, \ldots ,n$. For simplicity the Routh–Hurwitz criteria for polynomial of degree $n = 2, 3, 4$ and $5$ are summarized as the following:$n = 2$: $a_{1} ,a_{2} > 0$,$n = 3$: $a_{1} ,a_{3} > 0$ and $a_{1} a_{2} > a_{3}$,$n = 4$: $a_{1} ,a_{3} ,a_{4} > 0$ and $a_{1} a_{2} a_{3} > a_{3}^{2} + a_{1}^{2} a_{4}$,$n = 5$: $a_{1} ,a_{2} ,a_{3} ,a_{4} ,a_{5} > 0$, $a_{1} a_{2} a_{3} > a_{3}^{2} + a_{1}^{2} a_{4}$ and $\left( {a_{1} a_{4} - a_{5} } \right)\left( {a_{1} a_{2} a_{3} - a_{3}^{2} - a_{1}^{2} a_{4} } \right) > a_{5} \left( {a_{1} a_{2} - a_{3} } \right)^{2} + a_{1} a_{5}^{2}$.

This criterion has given necessary and sufficient conditions for the roots of the characteristic polynomial (with real coefficients) to lie in the left half of the complex plane (Allen 2007).

3 Fractional order Lengyel–Epstein chemical reaction model

The proposed model in this study is fractional multi-order differential equation system form of Lengyel–Epstein model suggested in (6). Therefore, the fractional order model given in (6) is

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}^{{\alpha_{1} }} u}}{{{\text{d}}\tau^{{\alpha_{1} }} }} = l - u - \frac{4uv}{{1 + u^{2} }}} \hfill \\ {\frac{{{\text{d}}^{{\alpha_{2} }} v}}{{{\text{d}}\tau^{{\alpha_{2} }} }} = mu\left( {1 - \frac{v}{{1 + u^{2} }}} \right)} \hfill \\ \end{array} } \right. $$

(15)

with initial condition $u\left( 0 \right) = u_{10}$, and $v\left( 0 \right) = v_{10}$. Besides that, let us assumed that $\alpha_{1} , \alpha_{2} \in \left( {0,1} \right]$, $\alpha_{1} = \frac{{k_{1} }}{{m_{1} }}$, $\alpha_{2} = \frac{{k_{2} }}{{m_{2} }}$, $\gamma = \frac{1}{p}$, $p\alpha_{1} , p\alpha_{2} > Z^{ + } - \left\{ 1 \right\}$ and the smallest common multiple of $m_{1}$ and $m_{2}$ is $p$. Stability conditions of equilibrium point obtained from $\frac{{{\text{d}}^{{\alpha_{1} }} u}}{{{\text{d}}\tau^{{\alpha_{1} }} }} = \frac{{{\text{d}}^{{\alpha_{2} }} v}}{{{\text{d}}\tau^{{\alpha_{2} }} }} = 0$ for system (15) are that $\det \left( {{\text{diag}}\left( {\lambda^{{p\alpha_{1} }} , \lambda^{{p\alpha_{2} }} } \right)} \right) = 0$ and satisfy $\left| {{ \arg }\left( \lambda \right)} \right| > \frac{\gamma \pi }{2}$.

3.1 Qualitative analysis

System (15) can be written as

$$ \left\{ {\begin{array}{*{20}l} {\frac{{{\text{d}}^{{\alpha_{1} }} u}}{{{\text{d}}\tau^{{\alpha_{1} }} }} = l - u - \frac{4uv}{{1 + u^{2} }} = G\left( {u,v} \right)} \hfill \\ {\frac{{{\text{d}}^{{\alpha_{2} }} v}}{{{\text{d}}\tau^{{\alpha_{2} }} }} = mu\left( {1 - \frac{v}{{1 + u^{2} }}} \right) = H\left( {u,v} \right)} \hfill \\ \end{array} } \right. $$

(15a)

in $R^{ + } \times \varPsi$.

Where $\varPsi$ is a bounded domain in ${\mathbb{R}}^{2}$ with smooth boundary $\partial \varPsi$, $0 < \alpha_{1} ,\alpha_{2} \le 1$ is the fractional order with Caputo fractional derivative over $\left( {0,\infty } \right)$ and $l$ and $m$ are strictly positive constants.

Theorem 3.1

“The feasible region $\varGamma$defined by

$$ \varGamma = \{ \left( {u,v} \right)^{\text{T}} \in {\mathbb{R}}_{ + }^{2} |0 \le u, 0 \le v\} $$

with initial conditions $u\left( 0 \right) > 0$, and $v\left( 0 \right) > 0$is positively invariant”.

Proof

Here we will prove the Theorem 3.1

$$ \left. {\frac{{{\text{d}}^{{\alpha_{1} }} u}}{{{\text{d}}\tau^{{\alpha_{1} }} }}} \right|_{u = 0} = l > 0, $$

$$ \left. {\frac{{{\text{d}}^{{\alpha_{2} }} v}}{{{\text{d}}\tau^{{\alpha_{2} }} }}} \right|_{v = 0} = mu > 0. $$

So the described model is positively invariant. This completes the proof.

Lemma 3.1

“An equilibrium point $\left( {u^{*} , v^{*} } \right)$of (15a) is locally asymptotically stable iff$\left| {{ \arg }\left( {\lambda_{i} } \right)} \right| > \alpha_{i} \frac{\pi }{2},\;i = 1,2$”, where $\lambda_{i}$are the eigenvalues of the Jacobian matrix $J\left( {u^{*} , v^{*} } \right)$and ${ \arg }\left( . \right)$denotes the argument of a complex number.

Proposition 3.1

System (15) has a unique equilibrium $\left( {u^{*} , v^{*} } \right) = \left( {\mu ,1 + \mu^{2} } \right)$with $\mu = l/5$, subject to ${\varPhi} = \left( {\frac{{3\mu^{2} - 5 - m\mu }}{{1 + \mu^{2} }}} \right) - \frac{20m\mu }{{1 + \mu^{2} }} \ge 0$.

$\left( {u^{*} , v^{*} } \right)$is asymptotically stable if ${\text{tr}}J < 0$and unstable if ${\text{tr}}J > 0$, where

$$ J = \left( {\begin{array}{*{20}c} {\frac{{3\mu^{2} - 5}}{{1 + \mu^{2} }}} & {\frac{ - 4\mu }{{1 + \mu^{2} }}} \\ {\frac{{2m\mu^{2} }}{{1 + \mu^{2} }}} & {\frac{ - m\mu }{{1 + \mu^{2} }}} \\ \end{array} } \right) $$

Alternatively, if $\varPhi < 0$, then $\left( {u^{*} , v^{*} } \right)$is asymptotically stable if ${\text{tr}}J \le 0$or $\left| {{ \arg }\left( {\lambda_{1} } \right)} \right| > \frac{{\alpha_{1} \pi }}{2}$and $\left| {{ \arg }\left( {\lambda_{2} } \right)} \right| > \frac{{\alpha_{2} \pi }}{2}$, where $\lambda_{1} = \frac{1}{2}\left[ {\frac{{3\mu^{2} - 5 - m\mu }}{{1 + \mu^{2} }} + i\sqrt { - \varPhi } } \right]$and $\lambda_{2} = \frac{1}{2}\left[ {\frac{{3\mu^{2} - 5 - m\mu }}{{1 + \mu^{2} }} - i\sqrt { - {{\varPhi }}} } \right]$.

Proof

To find the equilibrium point of (15), put $\frac{{d^{{\alpha_{1} }} u}}{{d\tau^{{\alpha_{1} }} }} = \frac{{d^{{\alpha_{2} }} v}}{{d\tau^{{\alpha_{2} }} }} = 0$. So we have

$$ \left\{ {\begin{array}{*{20}l} {l - u - \frac{4uv}{{1 + u^{2} }} = 0} \hfill \\ {mu\left( {1 - \frac{v}{{1 + u^{2} }}} \right) = 0} \hfill \\ \end{array} } \right.. $$

(16)

From the second equation of (16), we have either $u = 0$ or $1 - \frac{v}{{1 + u^{2} }} = 0$.

If $u = 0$, then the first equation of (16) gives $l = 0$, which is not true because $l > 0$. So $u \ne 0$. So we have $v = 1 + u^{2}$. So the equilibrium point is $E_{0} \left( {u^{*} , v^{*} } \right) = \left( {\mu ,1 + \mu^{2} } \right)$ with $\mu = l/5$.

For the stability analysis, the functions on the right hand side of the system (15) can be determined as below:

$$ \left\{ {\begin{array}{*{20}l} {G\left( {u,v} \right) = l - u - \frac{4uv}{{1 + u^{2} }}} \hfill \\ {H\left( {u,v} \right) = mu\left( {1 - \frac{v}{{1 + u^{2} }}} \right)} \hfill \\ \end{array} } \right.. $$

(17)

The Jacobian of (17) is

$$ J\left( {u,v} \right) = \left( {\begin{array}{*{20}c} {G_{u} } &\quad {G_{v} } \\ {H_{u} } &\quad {H_{v} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} { - 1 - \frac{{4v\left( {1 - u^{2} } \right)}}{{\left( {1 + u^{2} } \right)^{2} }}} &\quad { - \frac{4u}{{1 + u^{2} }}} \\ {m - \frac{{mv\left( {1 - u^{2} } \right)}}{{\left( {1 + u^{2} } \right)^{2} }}} &\quad { - \frac{mu}{{1 + u^{2} }}} \\ \end{array} } \right). $$

(18)

Then the Jacobian matrix (18) at $E_{0} \left( {u^{*} , v^{*} } \right)$, we have

$$ J\left( {E_{0} } \right) = \left( {\begin{array}{*{20}c} {\frac{{3\mu^{2} - 5}}{{1 + \mu^{2} }}} &\quad {\frac{ - 4\mu }{{1 + \mu^{2} }}} \\ {\frac{{2m\mu^{2} }}{{1 + \mu^{2} }}} &\quad {\frac{ - m\mu }{{1 + \mu^{2} }}} \\ \end{array} } \right). $$

(19)

Its determinant and trace are given by

$$ \det J\left( {E_{0} } \right) = \frac{5m\mu }{{1 + \mu^{2} }} $$

and

$$ {\text{tr}}J\left( {E_{0} } \right) = \frac{{3\mu^{2} - 5 - m\mu }}{{1 + \mu^{2} }}, $$

respectively.

The characteristic equation of the Jacobian matrix is

$$ \lambda^{2} - \left( {{\text{tr }}J} \right)\lambda + \det J = 0, $$

and its discriminant is

$$ \varPhi = \left( {{\text{tr}} J} \right)^{2} - 4\det J. $$

We study the different cases separately. First if ${{\varPhi }} > 0$, then the eigenvalues $\lambda_{1}$ and $\lambda_{2}$ are real and can be written as$\lambda_{1} = \frac{1}{2}\left[ {{\text{tr}} J + \sqrt \varPhi } \right]$ and $\lambda_{2} = \frac{1}{2}\left[ {{\text{tr}} J - \sqrt \varPhi } \right]$.

Note that $\det J > 0$. Hence the negativity of the eigenvalues rests on the sign of the trace ${\text{tr }}J$.

• (a) If ${\text{tr }}J < 0$, then ${\text{tr}} J - \sqrt \varPhi < 0$, leading to

  $$ \lambda_{2} = \frac{1}{2}\left[ {{\text{tr}} J - \sqrt \varPhi } \right] < 0 $$

  and therefore, $\arg \left( {\lambda_{2} } \right) = \pi$. Since both eigenvalues are real, the trace is negative, and the determinant is positive, it is clear that $\arg \left( {\lambda_{1} } \right) = \pi > \frac{{\alpha_{1} \pi }}{2}$,$\arg \left( {\lambda_{2} } \right) = \pi > \frac{{\alpha_{2} \pi }}{2}$ with $0 < \alpha_{1} ,\alpha_{2} \le 1$. It follows that $\left( {u^{*} , v^{*} } \right)$ is asymptotically stable.

• (b) If ${\text{tr }}J > 0$, then ${\text{tr}} J - \sqrt \varPhi > 0$, leading to

  $$ \lambda_{2} = \frac{1}{2}\left[ {{\text{tr}} J - \sqrt \varPhi } \right] > 0 $$

  and thus $\arg \left( {\lambda_{2} } \right) = 0$. So $\left( {u^{*} , v^{*} } \right)$ is asymptotically unstable.

• (c) If ${\text{tr }}J = 0$, then $\varPhi > 0$, leading to $- 4\det J > 0$ which is a contradiction. Hence this case does not show up.

Next we consider the case of the discriminant $\varPhi$ being equal to zero. Since $\det J > 0$, then it is impossible that ${\text{tr }}J = 0$. The eigenvalues reduce to

$$ \lambda_{1,2} = \frac{1}{2}\left[ {{\text{tr }}J} \right] $$

The sign of the eigenvalues is identical to that of the trace. Consequently $\left( {u^{*} , v^{*} } \right)$ is asymptotically stable for all $0 < \alpha_{1} ,\alpha_{2} \le 1$, if ${\text{tr }}J < 0$ and unstable if ${\text{tr }}J > 0$.

Finally if the discriminant $\varPhi < 0$, then

$$ \lambda_{1,2} = \frac{1}{2}\left[ {{\text{tr}} J \pm \sqrt \varPhi } \right] $$

or

$$ \lambda_{1,2} = \frac{1}{2}\left[ {{\text{tr}} J \pm i\sqrt { - \varPhi } } \right]. $$

Now we have three cases:

• If ${\text{tr }}J < 0$, then the system is asymptotically stable at the equilibrium point.

• If ${\text{tr}} J = 0$, then $\left| {{ \arg }\left( {\lambda_{1,2} = \frac{1}{2}\left[ { \pm i\sqrt { - {{\varPhi }}} } \right]} \right)} \right| = \frac{\pi }{2}$. Hence for $0 < \alpha_{1} ,\alpha_{2} < 1$, the system is asymptotically stable at the equilibrium point.

• If $tr J > 0$, then the system is asymptotically stable at the equilibrium point.

The proof is complete.

Now let us move to the complete solution of (15). For this we are going to use the eigenfunction expansion method.

From the Eq. (19), we have

$$ \det \left( {{\text{diag}}\left( {\lambda^{{p\alpha_{1} }} , \lambda^{{p\alpha_{2} }} } \right) - J\left( {E_{0} } \right)} \right) = 0. $$

Then we have

$$ \lambda^{{p(\alpha_{1} + \alpha_{2} )}} + \lambda^{{p\alpha_{1} }} \left( {\frac{m\mu }{{1 + \mu^{2} }}} \right) + \lambda^{{p\alpha_{2} }} \left( {\frac{{5 - 3\mu^{2} }}{{1 + \mu^{2} }}} \right) + \frac{5m\mu }{{1 + \mu^{2} }} = 0, $$

with $\mu = l/5$. Thus, we have

$$ \lambda^{{p(\alpha_{1} + \alpha_{2} )}} + \lambda^{{p\alpha_{1} }} \left( {\frac{lm}{{5\left( {1 + \left( {\frac{l}{5}} \right)^{2} } \right)}}} \right) + \lambda^{{p\alpha_{2} }} \left( {\frac{{5 - 3\left( {\frac{l}{5}} \right)^{2} }}{{1 + \left( {\frac{l}{5}} \right)^{2} }}} \right) + \frac{lm}{{1 + \left( {\frac{l}{5}} \right)^{2} }} = 0. $$

(20)

Stability conditions of equilibrium point for system (15) are that the satisfy $\left| {{ \arg }\left( \lambda \right)} \right| > \gamma \frac{\pi }{2}$.

For a special case $\alpha_{1} = \frac{1}{p}$, $\alpha_{2} = \frac{1}{p}$, so (20) gives

$$ \lambda^{2} + \lambda \left( {\frac{lm}{{5\left( {1 + \left( {\frac{l}{5}} \right)^{2} } \right)}}} \right) + \lambda \left( {\frac{{5 - 3\left( {\frac{l}{5}} \right)^{2} }}{{1 + \left( {\frac{l}{5}} \right)^{2} }}} \right) + \frac{lm}{{1 + \left( {\frac{l}{5}} \right)^{2} }} = 0. $$

(21)

So we have from (21),

$$ { \det }\left( J \right) = \frac{lm}{{1 + \left( {\frac{l}{5}} \right)^{2} }},\;{\text{and}}\;{\text{tr}}\left( J \right) = \frac{lm}{{5\left( {1 + \left( {\frac{l}{5}} \right)^{2} } \right)}} + \frac{{5 - 3\left( {\frac{l}{5}} \right)^{2} }}{{1 + \left( {\frac{l}{5}} \right)^{2} }} = - \frac{{5lm - 3l^{2} + 125}}{{l^{2} + 25}}. $$

4 Numerical simulation

Over here we have used the Adams Bashfort-Moulton (PECE) method to see the behavior of the system (15) by varying the parameters and order of the system. We apply the predictor–corrector PECE method, so the system (15) takes the following format:

$$ \begin{aligned} u^{\aleph + 1} & = u\left( 0 \right) + \frac{{h^{{\alpha_{1} }} }}{{{{\varGamma }}\left( {\alpha_{1} + 2} \right)}}\left\{ {l - u_{\aleph + 1}^{p} - 4\frac{{u_{\aleph + 1}^{p} v_{\aleph + 1}^{p} }}{{1 + \left( {u_{\aleph + 1}^{p} } \right)^{2} }}} \right\} \\ & \quad + \frac{{h^{{\alpha_{1} }} }}{{{{\varGamma }}\left( {\alpha_{1} + 2} \right)}}\mathop \sum \limits_{d = 0}^{\aleph } a_{j,\aleph + 1} \left\{ {l - \left( u \right)_{d} - 4\frac{{\left( u \right)_{d} \left( v \right)_{d} }}{{1 + \left( {\left( u \right)_{d} } \right)^{2} }}} \right\}, \\ \end{aligned} $$

$$ \begin{aligned} v^{\aleph + 1} & = v\left( 0 \right) + \frac{{h^{{\alpha_{2} }} }}{{{{\varGamma }}\left( {\alpha_{2} + 2} \right)}}\left\{ {m u_{\aleph + 1}^{p} - m\frac{{u_{\aleph + 1}^{p} v_{\aleph + 1}^{p} }}{{1 + \left( {u_{\aleph + 1}^{p} } \right)^{2} }}} \right\} \\ & \quad + \frac{{h^{{\alpha_{2} }} }}{{{{\varGamma }}\left( {\alpha_{2} + 2} \right)}}\mathop \sum \limits_{d = 0}^{\aleph } a_{d,\aleph + 1} \left\{ {m\left( u \right)_{d} - m\frac{{\left( u \right)_{d} \left( v \right)_{d} }}{{1 + \left( {\left( u \right)_{d} } \right)^{2} }}} \right\}, \\ \end{aligned} $$

where

$$ u_{\aleph + 1}^{p} = u\left( 0 \right) + \frac{1}{{{{\varGamma }}\left( {\alpha_{1} } \right)}}\mathop \sum \limits_{d = 0}^{\aleph } b_{d,\aleph + 1} \left\{ {l - \left( u \right)_{d} - 4\frac{{\left( u \right)_{d} \left( v \right)_{d} }}{{1 + \left( {\left( u \right)_{d} } \right)^{2} }}} \right\}, $$

$$ v_{\aleph + 1}^{p} = v\left( 0 \right) + \frac{1}{{{{\varGamma }}\left( {\alpha_{2} } \right)}}\mathop \sum \limits_{d = 0}^{\aleph } b_{d,\aleph + 1} \left\{ {m\left( u \right)_{d} - m\frac{{\left( u \right)_{d} \left( v \right)_{d} }}{{1 + \left( {\left( u \right)_{d} } \right)^{2} }}} \right\}, $$

with

$$ a_{d,\aleph + 1} = \left\{ {\begin{array}{*{20}l} {\aleph^{{\alpha_{i} + 1}} - \left( {\aleph - \alpha_{i} } \right)(\aleph + 1)^{{\alpha_{i} }} ,} \hfill & {d = 0} \hfill \\ {(\aleph - d + 2)^{{\alpha_{i} + 1}} + (\aleph - d)^{{\alpha_{i} + 1}} - 2(\aleph - d + 1)^{{\alpha_{i} + 1}} } \hfill &\quad {1 \le d \le \aleph } \hfill \\ {1,} \hfill & {d = \aleph + 1} \hfill \\ \end{array} } \right. $$

and

$$ b_{d,\aleph + 1} = \frac{{h^{{\alpha_{i} }} }}{{\alpha_{i} }}\left( {\left( {\aleph - d + 1} \right)^{{\alpha_{i} }} - \left( {\aleph - d} \right)^{{\alpha_{i} }} } \right), 0 \le d \le \aleph $$

with $i = 1,2,3$.

Case 1

Let us take $\alpha_{1} = \frac{1}{2}$, $\alpha_{2} = \frac{1}{4}$ $\left( {p = 4} \right)$, $l = 1$, $m = 12$ and $\left( {u_{0} , v_{0} } \right) = \left( {1,1} \right)$. In this case we have from (20)

$$ \lambda^{3} + \frac{30}{13}\lambda^{2} + \frac{61}{13}\lambda + \frac{150}{13} = 0, $$

So by Routh–Hurwitz stability condition $\left( {n = 3} \right)$ is satisfied, because $a_{1} = \frac{30}{13}$, $a_{2} = \frac{61}{13}$ and $a_{3} = \frac{150}{13}$.

Also $a_{1} a_{2} = \left( {\frac{30}{13}} \right)\left( {\frac{61}{13}} \right) > \frac{150}{13} = a_{3}$. Thus, $E_{0} = \left( {0.2,1.04} \right)$ is local asymptotically stable as shown in the Fig. 2.

Fig. 2

Simulation of system (15), for $\alpha_{1} = \frac{1}{2}$, $\alpha_{2} = \frac{1}{4}$, $l = 1$ and $m = 12$

Case 2

Let us take $\alpha_{1} = \frac{1}{2}$, $\alpha_{2} = \frac{1}{2}$ $\left( {p = 2} \right)$, $l = 10$, $m = 8$ and $\left( {u_{0} , v_{0} } \right) = \left( {1,1} \right)$. In this case we have from (20)

$$ \lambda^{2} + \frac{33}{5}\lambda + 16 = 0. $$

In this respect, the eigenvalues from characteristic equation are $\lambda_{1} = - 3.3 - 2.26053i$, $\lambda_{2} = - 3.3 + 2.26053i$.

Since both the eigenvalues are negative.

Also if we use the Routh–Hurwitz stability condition $\left( {n = 2} \right)$, it is satisfied, because $a_{1} = \frac{33}{5} > 0$ and $a_{2} = 16 > 0$.

Thus $E_{0} = \left( {2,5} \right)$ is local asymptotically stable as shown in Fig. 3.

Fig. 3

Simulation of system (15), for $\alpha_{1} = \frac{1}{2}$, $\alpha_{2} = \frac{1}{2}$, $l = 10$ and $m = 8$

Case 3

Let us take $\alpha_{1} = \frac{1}{4}$, $\alpha_{2} = \frac{1}{6}$ $\left( {p = 12} \right)$, $l = 5$, $m = 2$ and $\left( {u_{0} , v_{0} } \right) = \left( {1,1} \right)$. In this case we have from (20)

$$ \lambda^{5} + \lambda^{3} + \lambda^{2} + 5 = 0. $$

In this respect, the eigenvalues from characteristic equation are $\lambda_{1} = - 1.34365$, $\lambda_{2} = -\, 0.277824 - 1.38736i$, $\lambda_{3} = - 0.277824 + 1.38736i$, $\lambda_{4} = 0.94965 - 0.97824i$, $\lambda_{5} = 0.94965 + 0.97824i$.

It is clear that $\left| {{ \arg }\left( {\lambda_{1} , \lambda_{2} ,\lambda_{3} , \lambda_{4} ,\lambda_{5} } \right)} \right| > \frac{\gamma \pi }{2} = \frac{\pi }{24}$. On the other hand

$$ \left| {{ \arg }\left( {\lambda_{1} } \right)} \right| = \left| {\tan^{ - 1} \left( {\frac{0}{ - 1.34365}} \right)} \right| = \pi > \frac{\pi }{24}, $$

$$ \left| {{ \arg }\left( {\lambda_{2} } \right)} \right| = \left| {\tan^{ - 1} \left( {\frac{ - 1.38736}{ - 0.277824}} \right)} \right| = 1.373157 > \frac{\pi }{24}, $$

$$ \left| {{ \arg }\left( {\lambda_{3} } \right)} \right| = \left| {\tan^{ - 1} \left( {\frac{1.38736}{ - 0.277824}} \right)} \right| = 1.373157 > \frac{\pi }{24}, $$

$$ \left| {{ \arg }\left( {\lambda_{4} } \right)} \right| = \left| {\tan^{ - 1} \left( {\frac{ - 0.97824}{0.94965}} \right)} \right| = 0.80023 > \frac{\pi }{24}, $$

$$ \left| {{ \arg }\left( {\lambda_{5} } \right)} \right| = \left| {\tan^{ - 1} \left( {\frac{0.97824}{0.94965}} \right)} \right| = 0.80023 > \frac{\pi }{24}. $$

Thus, $E_{0} = \left( {1,2} \right)$ is local asymptotically stable as shown in Fig. 4.

Fig. 4

Simulation of system (15), for $\alpha_{1} = \frac{1}{4}$, $\alpha_{2} = \frac{1}{6}$, $l = 5$ and $m = 2$

Case 4

Let us take $\alpha_{1} = 1$, $\alpha_{2} = 1$ $\left( {p = 1} \right)$, $l = 25$, $m = 1$ and $\left( {u_{0} , v_{0} } \right) = \left( {1,1} \right)$. In this case we have from (20)

$$ \lambda^{2} - \frac{5}{2}\lambda + \frac{25}{26} = 0. $$

In this respect, the eigenvalues from characteristic equation are $\lambda_{1} = 0.474783$, $\lambda_{2} = 2.02522$

Since both the eigenvalues are positive.

Also if we use the Routh–Hurwitz stability condition $\left( {n = 2} \right)$, it is not satisfied, because $a_{1} = - \frac{5}{2} < 0$ and $a_{2} = \frac{25}{26} > 0$.

Thus $E_{0} = \left( {5,26} \right)$ is unstable as shown in Fig. 5.

Fig. 5

Simulation of system (15), for $\alpha_{1} = 1$, $\alpha_{2} = 1$, $l = 25$ and $m = 1$

4.1 Numerical simulation

In this subsection, we apply the discretization process represented in El-Sayed et al. (2014) and Agarwal et al. (2013) for FOLECR. Over here we have used the piece wise constant arguments (PWCA) method to see the behavior of the system (15) by varying the parameters and order of the system. We can discretize (15) with PWCA method as following.

$$ \left\{ {\begin{array}{*{20}l} {D^{{\alpha_{1} }} u\left( \tau \right) = l - u\left( {\left[ {\frac{\tau }{s}} \right]s} \right) - \frac{{4u\left( {\left[ {\frac{\tau }{s}} \right]s} \right)v\left( {\left[ {\frac{\tau }{s}} \right]s} \right)}}{{1 + u\left( {\left[ {\frac{\tau }{s}} \right]s} \right)^{2} }}} \hfill \\ {D^{{\alpha_{2} }} v\left( \tau \right) = m u\left( {\left[ {\frac{\tau }{s}} \right]s} \right)\left( {1 - \frac{{v\left( {\left[ {\frac{\tau }{s}} \right]s} \right)}}{{1 + u\left( {\left[ {\frac{\tau }{s}} \right]s} \right)^{2} }}} \right)} \hfill \\ \end{array} .} \right. $$

(22)

First, let $\tau \in \left[ {0,s} \right)$, i.e., $\frac{\tau }{s} \in \left[ {0,1} \right)$. Thus, we obtain

$$ \left\{ {\begin{array}{*{20}l} {D^{{\alpha_{1} }} u\left( \tau \right) = l - u\left( 0 \right) - \frac{4u\left( 0 \right)v\left( 0 \right)}{{1 + u\left( 0 \right)^{2} }}} \hfill \\ {D^{{\alpha_{2} }} v\left( \tau \right) = m u\left( 0 \right)\left( {1 - \frac{v\left( 0 \right)}{{1 + u\left( 0 \right)^{2} }}} \right)} \hfill \\ \end{array} } \right. $$

(23)

and the solution of (23) is reduced to

$$ \left\{ {\begin{array}{*{20}c} {u_{1} \left( \tau \right) = u\left( 0 \right) + J^{{\alpha_{1} }} \left( {l - u\left( 0 \right) - \frac{4u\left( 0 \right)v\left( 0 \right)}{{1 + u\left( 0 \right)^{2} }}} \right)} \\ {v_{1} \left( \tau \right) = v\left( 0 \right) + J^{{\alpha_{2} }} \left( {m u\left( 0 \right) - \frac{mu\left( 0 \right)v\left( 0 \right)}{{1 + u\left( 0 \right)^{2} }}} \right)} \\ \end{array} } \right. $$

$$ \left\{ {\begin{array}{*{20}c} {u_{1} \left( \tau \right) = u\left( 0 \right) + \frac{{\tau^{{\alpha_{1} }} }}{{\alpha_{1} {{\varGamma }}\left( {\alpha_{1} } \right)}}\left( {l - u\left( 0 \right) - \frac{4u\left( 0 \right)v\left( 0 \right)}{{1 + u\left( 0 \right)^{2} }}} \right)} \\ {v_{1} \left( \tau \right) = v\left( 0 \right) + \frac{{\tau^{{\alpha_{2} }} }}{{\alpha_{2} {{\varGamma }}\left( {\alpha_{2} } \right)}}\left( {m u\left( 0 \right) - \frac{mu\left( 0 \right)v\left( 0 \right)}{{1 + u\left( 0 \right)^{2} }}} \right)} \\ \end{array} } \right. $$

(24)

second, let $\tau \in \left[ {s,2s} \right)$, i.e., $\frac{\tau }{s} \in \left[ {1,2} \right)$. Thus, we obtain

$$ \left\{ {\begin{array}{*{20}c} {D^{{\alpha_{1} }} u\left( \tau \right) = l - u_{1} - \frac{{4u_{1} v_{1} }}{{1 + u_{1}^{2} }}} \\ {D^{{\alpha_{2} }} v\left( \tau \right) = m u_{1} \left( {1 - \frac{{v_{1} }}{{1 + u_{1}^{2} }}} \right)} \\ \end{array} } \right. $$

(25)

which has the following solution

$$ \left\{ {\begin{array}{*{20}l} {u_{2} \left( \tau \right) = u_{1} \left( s \right) + J_{s}^{{\alpha_{1} }} \left( {l - u_{1} \left( s \right) - \frac{{4u_{1} \left( s \right)v_{1} \left( s \right)}}{{1 + u_{1} \left( s \right)^{2} }}} \right)} \hfill \\ {v_{2} \left( \tau \right) = v_{1} \left( s \right) + J_{s}^{{\alpha_{2} }} \left( {mu_{1} \left( s \right)\left( {1 - \frac{{v_{1} \left( s \right)}}{{1 + u_{1} \left( s \right)^{2} }}} \right)} \right)} \hfill \\ \end{array} } \right.. $$

$$ \left\{ {\begin{array}{*{20}l} {u_{2} \left( \tau \right) = u_{1} \left( s \right) + \frac{{\left( {\tau - s} \right)^{{\alpha_{1} }} }}{{\alpha_{1} {{\varGamma }}\left( {\alpha_{1} } \right)}}\left( {l - u_{1} \left( s \right) - \frac{{4u_{1} \left( s \right)v_{1} \left( s \right)}}{{1 + u_{1} \left( s \right)^{2} }}} \right)} \hfill \\ {v_{2} \left( \tau \right) = v_{1} \left( s \right) + \frac{{(\tau - s)^{{\alpha_{2} }} }}{{\alpha_{2} {{\varGamma }}\left( {\alpha_{2} } \right)}}\left( {m u_{1} \left( s \right)\left( {1 - \frac{{v_{1} \left( s \right)}}{{1 + u_{1} \left( s \right)^{2} }}} \right)} \right)} \hfill \\ \end{array} } \right., $$

(26)

where $J_{s}^{{\alpha_{i} }} = \frac{1}{{{{\varGamma }}\left( {\alpha_{i} } \right)}}\mathop \int \limits_{s}^{t} \left( {\tau - p} \right)^{{\alpha_{i} - 1}} {\text{d}}p$, $0 < \alpha_{i} \le 1$ and $i = 1,2$. Thus, after repeating the discretization process $n$ times, we obtain the discretized FOLECR as following

$$ \left\{ {\begin{array}{*{20}l} {u_{n + 1} \left( \tau \right) = u_{n} \left( {ns} \right) + \frac{{\left( {\tau - ns} \right)^{{\alpha_{1} }} }}{{\alpha_{1} {{\varGamma }}\left( {\alpha_{1} } \right)}}\left( {l - u_{n} \left( {ns} \right) - \frac{{4u_{n} \left( {ns} \right)v_{n} \left( {ns} \right)}}{{1 + u_{n} \left( {ns} \right)^{2} }}} \right)} \hfill \\ {v_{n + 1} \left( \tau \right) = v_{n} \left( {ns} \right) + \frac{{(\tau - ns)^{{\alpha_{2} }} }}{{\alpha_{2} {{\varGamma }}\left( {\alpha_{2} } \right)}}\left( {m u_{n} \left( {ns} \right)\left( {1 - \frac{{v_{n} \left( {ns} \right)}}{{1 + u_{n} \left( {ns} \right)^{2} }}} \right)} \right),} \hfill \\ \end{array} } \right. $$

(27)

where $\tau \in \left[ {ns, \left( {n + 1} \right)s} \right)$. For $\tau \to \left( {n + 1} \right)s$, the system (27) is reduced to

$$ \left\{ {\begin{array}{*{20}l} {u_{n + 1} = u_{n} + \frac{{s^{{\alpha_{1} }} }}{{\alpha_{1} {{\varGamma }}\left( {\alpha_{1} } \right)}}\left( {l - u_{n} - \frac{{4u_{n} v_{n} }}{{1 + u_{n}^{2} }}} \right)} \hfill \\ {v_{n + 1} = v_{n} + \frac{{s^{{\alpha_{2} }} }}{{\alpha_{2} {{\varGamma }}\left( {\alpha_{2} } \right)}}\left( {m u_{n} \left( {1 - \frac{{v_{n} }}{{1 + u_{n}^{2} }}} \right)} \right).} \hfill \\ \end{array} } \right. $$

(28)

It should be notice that if $\alpha_{i} \to 1$, $\left( {i = 1,2} \right)$ in (28), we obtain the corresponding Euler discretization of discretized FOLECR with commensurate order. It is different from predictor corrector method. The obtained result is a two dimensional discrete system. When we use cases (1–4) stated above, then by using PWCA method, we have obtained (Figs. 6, 7, 8, 9).

Fig. 6

Simulation of system (15), for $\alpha_{1} = \frac{1}{2}$, $\alpha_{2} = \frac{1}{4}$, $l = 1$ and $m = 12$

Fig. 7

Simulation of system (15), for $\alpha_{1} = \frac{1}{2}$, $\alpha_{2} = \frac{1}{2}$, $l = 10$ and $m = 8$

Fig. 8

Simulation of system (15), for $\alpha_{1} = \frac{1}{4}$, $\alpha_{2} = \frac{1}{6}$, $l = 5$ and $m = 2$

Fig. 9

Simulation of system (15), for $\alpha_{1} = 1$, $\alpha_{2} = 1$, $l = 25$ and $m = 1$

5 Conclusion

In this article, we propose a nonlinear mathematical model to study the dynamics of fractional order Lengyel–Epstein chemical reaction model being the oscillating chemical reactions. The model consists on modeling the interaction among chlorine dioxide, iodine, and malonic acid. When simulating the model with the given algorithms, we have observed that the methods are converging to equilibrium point, but through different paths for different values of $\alpha$. The values are very close to each other to true equilibrium points. The time consumed by the simulations using PWCA (average time 23 s) is far less than the Adams Bashforth-Moulton (PECE) method (average time 955 s). So we can say that PCWA is superior than PECE by time wise. It is found that the effect of fractional order on trajectory shape and generation mechanism is small, because the type of the equilibrium points that remains unchanged with the variation of fractional order. However, power law stability $t^{ - \alpha }$ in fractional order system will make the period become longer. At $\alpha = 1$, the stability behavior of system (15) will be similar to the nonlinear system of ordinary differential equations discussed in (Chicone 2010) with similar results.