Stability and computational results for chemical kinetics reactions in enzyme

Abstract

Kinetic chemical reactions find applications across various fields. In industrial processes, they drive the production of essential materials like fertilizers and pharmaceuticals. In environmental science, they are crucial to understanding pollution dynamics. Additionally, in biochemistry, they underpin vital cellular processes, offering insights into disease mechanisms and drug development. In this work, we present a new advancement of a dynamical system for kinetically controlled chemical reactions and the dependency of its solution on the initial conditions using mathematical techniques for fractional orders. By utilizing this fixed-point approach, we can derive the existence and uniqueness theorem of the proposed model. We further show that the chemical kinetics of the fractional model are stable through the Hyers-Ulam stability condition. We finally run a numerical simulation to verify our conclusions. The manuscript concludes with demonstrative examples.

1 Introduction

Chemical kinetics is a branch of chemistry that focuses on studying the rates of chemical reactions and the factors that influence them. It investigates how fast or slow reactions occur, the mechanisms behind them, and how various conditions affect their rates. Understanding chemical kinetics is essential for a wide range of applications, from industrial processes to environmental chemistry and drug development. In this discussion of chemical kinetics, we delve into the fundamental principles governing the speed of chemical reactions and the intricacies of reaction mechanisms [1, 2]. We explore concepts such as reaction rate laws, activation energy, reaction mechanisms, and factors affecting reaction rates, including temperature, concentration, catalysts, and surface area [3]. Through a synthesis of theoretical principles and practical applications, we aim to unravel the mysteries of enzyme kinetics and their pivotal role in driving biological phenomena. Lahiri et al. [4] highlight the key points of reversible polymerization dynamics in dilute solutions. Further, Rekha et al. [5] described a substrate forming a complex with the immobilized catalyst using a hyperbolic function. Many chemical reactions in organic chemistry, such as nucleophilic substitution reactions, electrophilic addition reactions, and radical reactions, are kinetically controlled. Additionally, reactions in enzymatic catalysis and industrial processes often exhibit kinetic control [6]. Kinetically controlled reactions often involve complex reaction mechanisms with multiple steps and intermediate species. Understanding these mechanisms is crucial for predicting and manipulating reaction rates.

The most essential tool for illuminating the dynamical behavior of many physical systems is the fractional-order derivative. The exploration of fractional differential equations represents a significant and compelling area of inquiry, drawing the interest of numerous researchers. By investigating differential equations featuring a differential operator with a flexible, real, and positive order, we achieve a more realistic understanding of a wide array of natural phenomena. Recent interest in fractional calculus has opened up new possibilities for capturing the detailed dynamics in various fields, including the conservation laws of Pfaff-Birkhoff principles [7], reaction-diffusion models [8, 9], Legendre functions [10], oscillatory systems [11], non-local and non-singular kernels [12], disease spread with saturated incidence rate [13], Hepatitis C [14], vector borne disease [15], SARS-CoV-2 [16], and shale gas production prediction [17]. Additionally, a few models have been studied in a fractional order framework focusing on toxic environments [18, 19]. We also examine the application of fractional differential equations in modeling non-local effects and memory-dependent behaviors. In their study, Singh et al. [20] looked at the biological population model in fractional order using Caputo-Katugampola memory. Baba et al. [21] investigated the dynamics of COVID-19 using the Caputo-Fabrizio fractional order model. Pankavich et al. [22] described the dynamics of the human immunodeficiency of the latent infection. Khan et al. discussed fractional order in the leukemia model [23]. Joseph et al. [24] described a density-dependent mathematical model for strain dissemination among Aedes aegypti mosquitoes. Alkhazzan et al. [25, 26] addressed a stochastic susceptible-vaccinated-infected-recovered model. The well-known Hyers-Ulam stability originated in 1940-1941, when Hyers and Ulam initially proposed it [27]. Rassias later extended and developed the concepts of Hyers-Ulam stability [28]. Additionally, Li et al. motivated the iterative sequential approximate solutions in neural networks [29]. Tassaddiq et al. examined the comparative analysis in the COVID-19 model [30]. Khan et al. [31] studied system of Langevin FDEs for the solution and stability works and given some examples for the illustration of the results.

As far as we know, there is a gap in research investigating the stability analysis of chemical kinetics model with fractional derivative operators. This lack of research highlights the need for further study. To address this gap, this study employs the fractional derivative in the Caputo sense to investigate the stability analysis of chemical kinetics model with fractional derivative operators. The key contributions and aspects of this paper are outlined below:

• 1. Our main focus is on investigating the Hyers-Ulam stability to fill a research gap in the current literature. We specifically examine the stability analysis of chemical kinetics model using the fractional derivative in the Caputo sense.

• 2. We expand on the existence and uniqueness has a unique solution and explore Hyers-Ulam stability results for the given model. To achieve our desired outcomes, we use the well-recognized fixed-point approach.

• 3. The Michaelis-Menten equation, on the other hand, only studies enzyme-catalyzed reactions and enzyme-substrate interactions. By incorporating fractional derivatives, chemical kinetics models can better capture the complex dynamics of reactions occurring in such systems. This helps us understand how useful they are for simulating real-world systems.

Our contributions seek to address this research gap and bring new perspectives to the field of fractional derivatives. We provide valuable theoretical and numerical results for studying nonlinear fractional-order differential equations using fractional derivatives in the Caputo sense.

We organize and structure the remaining part of this paper as follows: In Sect. 2, we present a mathematical model of chemical kinetics and also recall the definitions and associated concepts, along with an outline of a basic approach to computing. Section 3 discusses the Caputo fractional derivative for the proposed model. In Sect. 4, we explore the existence and uniqueness of the solution using fixed-point approach for the proposed model. Section 5 guarantees the Hyers-Ulam stability of chemical kinetics. Numerical simulations and examples supporting our theoretical findings are displayed in Sect. 6. Finally, conclusions are given in Sect. 7.

2 A mathematical formulation for the chemical kinetics

Chemical kinetic reactions are fundamental to understanding the rates and mechanisms by which chemical reactions proceed, influencing diverse applications across industries, environmental science, and biochemistry. In industrial processes, they are indispensable for the production of essential materials like fertilizers, pharmaceuticals, and other chemicals, facilitating optimization and cost reduction. Additionally, in environmental science, they play a crucial role in interpreting the degradation of pollutants in air, water, and soil, aiding in the development of effective pollution control strategies. In biochemistry, kinetic studies provide valuable insights into enzyme catalysis, metabolic pathways, and drug interactions, driving advancements in drug development, personalized medicine, and the understanding of disease mechanisms. Through the lens of chemical kinetics, scientists can optimize processes, mitigate environmental impacts, and unravel complex biochemical phenomena, shaping advancements and innovations across a range of disciplines.

• 1. To the best of our knowledge, this is the first study of this fractional derivative in the Caputo sense applied to our model of chemical kinetics.

• 2. Through this understanding, we can better appreciate the intricate biochemical machinery that sustains life, as shown in Fig. 1.

• 3. Novel fractional masks, which are an amazing step forward in modeling kinetically controlled chemical reactions, give a more accurate picture than integer-order derivatives, mostly because they include memory effects.

In this article, we briefly explore which enzyme has dual binding sites. We look at how the physical forms of the substrate and enzyme interact in a solution that has amounts of ${\mathbb {S}}$, ${\mathbb {E}}$, ${\mathbb {J}_{1}}$, ${\mathbb {J}_{2}}$, and ${\mathbb {P}}$. From the beginning, ${\mathbb {J}_{1}}$ breaks down given a product ${\mathbb {P}}$, and also the constant rates are displayed as $\alpha , \beta , \Psi , \gamma , \rho$ and $\Phi$. Additionally, it can form another dual-bound ${\mathbb {J}_{2}}$ degradation into the products due to a backward reaction. Chemical kinetics involves quantitatively analyzing how enzyme activity changes in response to varying substrate concentrations, reaction conditions, and other factors. The ${\mathbb {J}_{1}}$ and ${\mathbb {J}_{2}}$ values represent the terms presented in the relevant system of reactions. Square brackets indicate the concentration of a substance $[ \ ]$ and ${[\mathbb {S}]=\mathbb {S}}$, ${[\mathbb {E}]=\mathbb {E}}$, ${[\mathbb{S}\mathbb{E}]=\mathbb {J}_{1}}$, ${[\mathbb {S}\mathbb {J}_{2}]=\mathbb {J}_{2}}$ and ${[\mathbb {P}]=\mathbb {P}}$ as follows:

$$\begin{aligned} {{\left\{ \begin{array}{ll} \frac{d\mathbb {S}}{ds}=- \alpha \mathbb {S} \mathbb {E}+(\beta -\gamma \mathbb {S})\mathbb {J}_{1}+\rho \mathbb {J}_{2},\\ \frac{d\mathbb {E}}{ds}=- \alpha \mathbb {S} \mathbb {E}+(\beta +\Psi )\mathbb {J}_{1},\\ \frac{d\mathbb {J}_{1}}{ds}= \alpha \mathbb {S} \mathbb {E}-(\alpha +\gamma \mathbb {S}+\Psi )\mathbb {J}_{1}+(\rho +\Phi )\mathbb {J}_{2},\\ \frac{d\mathbb {J}_{2}}{ds}= \gamma \mathbb {S}\mathbb {J}_{1}-(\rho +\Phi )\mathbb {J}_{2},\\ \frac{d\mathbb {P}}{ds}=\Psi \mathbb {J}_{1}+\Phi \mathbb {J}_{2}.\\ \end{array}\right. }} \end{aligned}$$

(2.1)

Fig. 1

Schematic diagram of chemical kinetics

With the initial condition, the impact of the chemical kinetics model becomes ${\mathbb {S}(0)\ge 0, \mathbb {E}(0)\ge 0, \mathbb {J}_{1}(0)\ge 0, \mathbb {J}_{2}(0)\ge 0}$ and $P(0)\ge 0$.

Fractional calculus, a branch of mathematical analysis dealing with derivatives and integrals of non-integer order, has a rich history dating back to the 17th century. Initially explored by mathematicians like Leibniz and Euler, its formalization came later in the 19th century with pioneers like Riemann and Liouville. However, it wasn’t until the 20th century that fractional calculus gained significant attention, with notable contributions from mathematicians such as Grünwald, Letnikov, and Riemann-Liouville. Today, fractional calculus finds wide-ranging applications across various fields. In physics, it’s used to model phenomena like diffusion, viscoelasticity, and electrical circuits with memory effects [32, 33]. In engineering, it’s applied to control systems, signal processing, and optimization problems. In biology and medicine, it aids in modeling physiological methods and analyzing medical imaging data. The versatility of fractional calculus continues to inspire new research avenues and innovative applications, making it an indispensable tool in modern scientific analysis [34,35,36].

We present some standard definitions that are outstandingly significant to exhibit our essential results in the Caputo fractional derivative.

Definition 1

([37]). The Caputo time-fractional derivative of order n for the function h(s) under integrable differentiation is presented by

$$\begin{aligned} ^{C}D^{n}_{s} h(s)=\frac{1}{\Gamma (\upsilon -n)} \int _{0}^{s} (s-\varphi )^{\upsilon -n-1} h^{\upsilon }(\varphi )d\varphi , \ \ \upsilon -1<n<\upsilon , \ \upsilon \in N. \end{aligned}$$

Particularly, $n \in (0,1)$. Thus, $^{C}D^{n}_{s} h(s) \rightarrow h^{'}(s)$ if $n\rightarrow 1$.

Definition 2

From the above function, as discussed, the integral operator operator of the Caputo fractional operator is displayed as

$$\begin{aligned} ^{C}I^{n}_{s}h(s)=\frac{1}{\Gamma (n)} \int _{0}^{s} h(\varphi )(s-\varphi )^{n-1} d\varphi , \ for \ \ 0<n<1, \ s>0. \end{aligned}$$

3 Chemical kinetics model to fractional derivative

Before presenting the different approaches of our model for chemical kinetics, we will discuss the mathematical form of the model, including the use of a Caputo fractional derivative to convert the chemical kinetics model from an integer to a fractional order. A multiplication factor of $\Upsilon ^{n-1}$ is added to the left-hand side of each equation to address the discrepancy in dimensions between both sides of the equations in system (2.1). This discrepancy arises when integer-order derivatives are substituted with non-integer derivatives in the proposed model. This factor, $\Upsilon$ serves as a time constant that is utilized to adjust for the differences in units [38], following the substitution of the differential operators. Then the new form of the system as follows:

$$\begin{aligned}&\Upsilon ^{n-1} \ \big [ ^{C}D^{n}_{s} \mathbb {S}(s)\big ]=- \alpha \mathbb {S} \mathbb {E}+(\beta -\gamma \mathbb {S})\mathbb {J}_{1}+\rho \mathbb {J}_{2},\nonumber \\&\Upsilon ^{n-1} \ \big [^{C}D^{n}_{s}\mathbb {E}(s)\big ]=- \alpha \mathbb {S} \mathbb {E}+(\beta +\Psi )\mathbb {J}_{1},\nonumber \\&\Upsilon ^{n-1} \ \big [^{C}D^{n}_{s}\mathbb {J}_{1}(s)\big ]= \alpha \mathbb {S}\mathbb {E}-(\alpha +\gamma \mathbb {S}+\Psi )\mathbb {J}_{1}+(\rho +\Phi )\mathbb {J}_{2},\nonumber \\&\Upsilon ^{n-1} \ \big [ ^{C}D^{n}_{s}\mathbb {J}_{2}(s)\big ]= \gamma \mathbb {S}\mathbb {J}_{1}-(\rho +\Phi )\mathbb {J}_{2},\nonumber \\&\Upsilon ^{n-1} \ \big [^{C}D^{n}_{s} \mathbb {P}(s)\big ]=\Psi \mathbb {J}_{1}+\Phi \mathbb {J}_{2}.\end{aligned}$$

(3.1)

We can rewrite it in the following new form as for the above model

$$\begin{aligned} \Upsilon ^{n-1} \ \big [ ^{C}D^{n}_{s}h(s)\big ]&=\omega (s, h(s)), \ 0<s<\infty , \end{aligned}$$

(3.2)

and $h(0)=h_{0}$,

where $h:[0,\infty ) \rightarrow R^{5}$ and $\omega :R^{5}\rightarrow R^{5}$ such that $h(s)=$ $\begin{pmatrix} \mathbb {S}(s)\\ \mathbb {E}(s)\\ \mathbb {J}_{1}(s)\\ \mathbb {J}_{2}(s)\\ \mathbb {P}(s)\\ \end{pmatrix}$. Notice that $\omega (h(s))=$ $\begin{pmatrix} \omega _{1}\\ \omega _{2}\\ \omega _{3}\\ \omega _{4}\\ \omega _{5}\\ \end{pmatrix}$ = $\begin{pmatrix} - \alpha \mathbb {S} \mathbb {E}+(\beta -\gamma \mathbb {S})\mathbb {J}_{1}+\rho \mathbb {J}_{2}\\ - \alpha \mathbb {S} \mathbb {E}+(\beta +\Psi )\mathbb {J}_{1}\\ \alpha \mathbb {S}\mathbb {E}-(\alpha +\gamma \mathbb {S}+\Psi )\mathbb {J}_{1}+(\rho +\Phi )\mathbb {J}_{2}\\ \gamma \mathbb {S}\mathbb {J}_{1}-(\rho +\Phi )\mathbb {J}_{2}\\ \Psi \mathbb {J}_{1}+\Phi \mathbb {J}_{2}\\ \end{pmatrix}$. By making use of the $\omega _{j}, j = 1, 2, 3, 4, 5$ are continuously differentiable functions of ${\mathbb {S}, \mathbb {E}, \mathbb {J}_{1}, \mathbb {J}_{2}}$ and ${\mathbb {P}}$ with respect to time s.

4 Existence and uniqueness results

The existence and uniqueness of our chemical kinectics model solution are shown using the given theorem.

Theorem 4.1

The function $\omega$ in equation (3.2) exhibits Lipschitz continuity with respect to the variable h.

Proof

Consider $\Lambda$ be the line segment in $R^{5}$ connecting the points $h_{1}$ and $h_{2}$. In our case, we have

$$\begin{aligned} \Lambda (h_{1}, h_{2}, \upsilon )=\{h_{1}+\upsilon (h_{2}-h_{1}); \upsilon \in [0,1], h_{1}, h_{2} \in R^{5}\}. \end{aligned}$$

Define the Mean value theorem $\theta \in \Lambda (h_{1},h_{2}, \upsilon )$ and given in the form of

$$\begin{aligned} ||\omega (h_{2})-\omega (h_{1})||_{\infty }= ||\omega ^{'}(\theta , h_{2}-h_{1})||_{\infty }, \end{aligned}$$

where $\omega ^{'}$ $(\theta , h_{2}-h_{1})$ is the directional derivative of $\omega$ defined as

$$\begin{aligned} \begin{aligned} ||\omega ^{'}(\theta , h_{2}-h_{1})||_{\infty }&=\bigg |\bigg |\sum _{j=1}^{5}(\Lambda \omega _{j}(\theta ).(h_{2}-h_{1}))e_{j}\bigg |\bigg |_{\infty }\nonumber \\ {}&=\bigg |\bigg |\sum _{j=1}^{5}\Lambda \omega _{j}(\theta )\bigg |\bigg |.||(h_{2}-h_{1})||_{\infty }. \end{aligned} \end{aligned}$$

Now, we can assure that the $\Lambda \omega _{j}(\theta )$ is the bounded linear operator, and there exists a $\ell >0$, such that for all $\Lambda (h_{1}, h_{2}, \theta )\subseteq R^{5}$,

$$\begin{aligned} \bigg |\bigg |\sum _{j=1}^{5}\Lambda \omega _{j}(\theta )\bigg |\bigg |\le \ell . \end{aligned}$$

Observing the above inequality, we can conclude that

$$\begin{aligned} ||\omega (h_{2})-\omega (h_{1})||_{\infty } \le \ell ||(h_{2}-h_{1})||_{\infty }. \end{aligned}$$

(4.1)

Finally, $\omega$ is Lipschitz continuous in h. $\square$

Theorem 4.2

If $\omega$ satisfied the Lipschitz condition (4.1) then the nonlinear fractional system (3.1) has a unique solution if $\omega (0,h(0))=0$ and provided $\ell \big [E^{*} \frac{1}{\Gamma (n)}\big ]<1$.

Proof

Consider $\omega (0, h(0))=0$. We will now show that the solution h(s) with equation (3.1). Then, we have

$$\begin{aligned} h(s)= ^{C}I^{n}_{s} \omega (s, h(s)). \end{aligned}$$

(4.2)

Suppose that h(s) satisfies system (3.2). Next, applying Definition 2 on the proposed new form model (3.2).

$$\begin{aligned} ^{C}I^{n}_{s} [^{C}D^{n}_{s} h(s)]=^{C}I^{n}_{s} [\omega (s,h(s))]. \end{aligned}$$

Hence, we look for the integral equation

$$\begin{aligned} \omega (s)=\omega (0)+\frac{1}{\Gamma (n)}\int _{0}^{s}(s-\wp )^{n-1} \omega (\wp , h(\wp ))d\wp , \end{aligned}$$

(4.3)

thus $\omega (0,h(0))=0$ and $h(0)=h_{0}$. Then, the desired result follows from the equation (4.2). Here, we will show that the system possesses a unique solution. Suppose that $\aleph =(0,T)$ and $U:C(\aleph , R^{5})\rightarrow C(\aleph , R^{5})$ is presented as

$$\begin{aligned} U[h(s)]=h(0)+\frac{1}{\Gamma (n)}\int _{0}^{s} (s-\wp )^{n-1}\omega (\wp ,h(\wp ))d\wp , \end{aligned}$$

employ equation (4.3), we have

$$\begin{aligned} U[h(s)]= h(s), \end{aligned}$$

(4.4)

and the supremum norm of $\aleph ,||.||_{\aleph }$ becomes $||h(s)||_{\wp }= sup_{s\in \wp }||h(s)||$. Then, $C(\aleph , R^{5})$ with $||.||_{\aleph }$ constructs Banach space. Assume that the function $\iota :C[0,T]\rightarrow C[0,T]$ is written as $\iota \sigma = l$. It is clear that

$$\begin{aligned} l=l(s)=\int _{0}^{s} \ell (s,\wp )h(\wp )d\wp , \end{aligned}$$

where $\ell (s,\wp ):\aleph \times \aleph \rightarrow R$ is the kernel of $\iota$, for a continuous function in the closed region $\Delta = \aleph \times \aleph$ and also bounded. Then, there exists $\ell _{0}\in R$, we have

$$\begin{aligned} |\ell (s,\wp )|\le \ell _{0}, \ \ell (s,\wp )\in \aleph \times \aleph , \end{aligned}$$

and also

$$\begin{aligned} sup_{s,\wp \in \aleph }|\ell (s,\wp )|\le \ell _{0}. \end{aligned}$$

Additionally, we show that the integral operator $\iota$ is bounded. Assume that,

$$\begin{aligned} ||\iota \sigma ||_{\aleph }=\bigg |\bigg |\int _{0}^{s} \ell (s,\wp )h(\wp )d\wp \bigg |\bigg |_{\aleph }. \end{aligned}$$

Hence, $h(\wp )$ ansd $\ell (s,\wp )$ are continuous, then

$$\begin{aligned} \begin{aligned} \bigg |\bigg |\int _{0}^{s} \ell (s,\wp )h(\wp )d\wp \bigg |\bigg |_{\aleph }&=sup_{s\in \aleph }\bigg |\bigg |\int _{0}^{s} \ell (s,\wp )h(\wp )d\wp \bigg |\bigg |_{\aleph }\nonumber \\ {}&\le sup_{s\in \aleph }\int _{0}^{s}||\ell (s,\wp )||||h(\wp )||d\wp . \end{aligned} \end{aligned}$$

Accordingly, it is displayed as

$$\begin{aligned} \bigg |\bigg |\int _{0}^{s} \ell (s,\wp )h(\wp )d\wp \bigg |\bigg |_{\aleph }\le E^{*}||\ell (s,\wp )||_{\aleph }||h(s)||_{\aleph }, \end{aligned}$$

(4.5)

and $h(s)\in C(\aleph , R^{5}), \ell (s,\wp )\in C(\aleph ^{2},R)$ such that

$$\begin{aligned} ||\ell (s, \wp )||_{\aleph }= sup _{s,\wp \in \aleph }|\ell (s,\wp )|, \end{aligned}$$

with equation (4.4), it is immediately computed that

$$\begin{aligned} ||U[h_{1}(s)]-U[h_{2}(s)]||_{\aleph }\le \bigg |\bigg |\frac{1}{\Gamma (n)}\int _{0}^{s}(s-\wp )^{n-1}(\omega (s,h_{1}(s))-\omega (s,h_{2}(s)))d\wp \bigg |\bigg |. \end{aligned}$$

Regarding to the equation (4.1) along the consequence of equation (4.5), we get

$$\begin{aligned} ||U[h_{1}(s)]-U[h_{2}(s)]||_{\aleph }\le \ell \bigg [E^{*}\frac{1}{\Gamma (n)}\bigg ]||h_{1}(s)-h_{2}(s)||_{\aleph }, \end{aligned}$$

then U will be contraction if $\ell \bigg [E^{*}\frac{1}{\Gamma (n)}\bigg ]<1$. $\square$

By applying the fixed point theorem, which implies that the given system (3.1) has a unique solution.

5 Hyers-Ulam stability

In the following theorem, we prove the Hyers-Ulam stability, which also improves a result on the generalized Hyers-Ulam stability using the fractional derivative method. Consider the function space ${H = C(\mathcal {L}, \mathbb {R}^{5})}$, which consists of continuous functions on the interval $\mathcal {L} = [0, u]$ to ${\mathbb {R}^{5}}$ with the norm $||h|| = \sup _{s \in \mathcal {L}} |h(s)|$. We introduce the following inequalities:

Definition 3

Considering the well particularity of system (3.1), the results are further interesting is given by

$$\begin{aligned} {\left\{ \begin{array}{ll} ^{C}D^{n}_{s}h(s)=\omega (s,h(s)),\\ h(0)=h_{0}, \end{array}\right. } \end{aligned}$$

(5.1)

is Hyers-Ulam stable, if there exists a non-negative constant $\mathcal {K}>0$ given for all $\Theta >0$ for each of the solution $\hbar (s)$ of (5.1) satisfying the following inequality

$$\begin{aligned} ||^{C}D^{n}_{s}\hbar (s)-\omega (s,\hbar (s))||\le \Theta , s\in \mathcal {L}. \end{aligned}$$

(5.2)

There exists a unique solution h within the set H of the given new form (5.1), such that

$$\begin{aligned} ||\hbar (s)-h(s)||\le \mathcal {K}\Theta , s\in \mathcal {L}. \end{aligned}$$

Definition 4

([39]). The new form of the model (5.1) is said to be “generalised Hyers-Ulam stable” if there exists a continuous function $\delta :R^{+}\rightarrow R^{+}$ with initial condition $\delta (0)=0$ such that for any alternative solution $\hbar$ in the set H for the inequality (5.1), there exists a unique solution h within the same set H, satisfies the following conditions is presented as

$$\begin{aligned} ||\hbar (s)-h(s)||\le \delta \Theta , s \in \mathcal {L}. \end{aligned}$$

Remark 1

A function $\hbar \in H$ is the solution of the inequality (5.2) iff there exists a function $\Im \in H$ such that

• 1. $||\Im (s)||\le \Theta , s\in \mathcal {L}$.

• 2. $^{C}D^{n}\hbar (s)=\omega (s,\hbar (s))+\Im (s), \ s \in \mathcal {L}$.

Lemma 1

Le $\hbar \in H$ be a solution of the inequality (5.2), and a solution $\hbar$ satisfying the following inequality

$$\begin{aligned} \bigg |\hbar (s)-\bigg [\hbar _{0}+\frac{1}{\Gamma (n)}\int _{0}^{s}(s-\wp )^{n-1}\omega (\wp ,\hbar (\wp ))d\wp \bigg ]\bigg |\le \sigma \Theta . \end{aligned}$$

Proof

Then by Remark 5.3, the solution of the equation $^{C}D^{n} \hbar (s)= \omega (s,\hbar (s))+\Im (s)$, $s \in \mathcal {L}$. With the help of Caputo integral, now we estimate

$$\begin{aligned} \hbar (s)=\hbar _{0}+\frac{1}{\Gamma (n)}\int _{0}^{s}(s-\wp )^{n-1}\omega (\wp ,\hbar (\wp ))d\wp +\frac{1}{\Gamma (n)} \int _{0}^{s}(s-\wp )^{n-1} \Im (\wp )d\wp . \end{aligned}$$

The improved expansion technique and then taking the norm on both sides, according to Remark 5.3, we have

$$\begin{aligned} \begin{aligned}&\bigg |\hbar (s)-\bigg [\hbar _{0}+\frac{1}{\Gamma (n)}\int _{0}^{s}(s-\wp )^{n-1}\omega (\wp ,\hbar (\wp ))d\wp \bigg ]\bigg |\\&\quad \le \frac{1}{\Gamma (n)}\int _{0}^{s}(s-\wp )^{s-1}|\Im (\wp )|d\wp \\ {}&\le \bigg (\frac{u^{n}}{\Gamma (n+1)}\bigg )\Theta \\&\quad \le \sigma \Theta . \end{aligned} \end{aligned}$$

$\square$

Theorem 5.1

For all $h \in H$ and the Lipschitz mapping $\omega :\mathcal {L}\times R^{5}\rightarrow R^{5}$ with the Lipschitz constant $\Bbbk >0$ with $1-\sigma \Bbbk >0$, where $\sigma =\frac{u^{n}}{\Gamma (n+1)}$, the model (5.1) is generalised Hyers-Ulam stable.

Proof

Let $\hbar \in H$ be a solution of the inequality (5.2) $h\in H$ be the unique solution of (5.1). Notice that $\Theta >0$ for $s\in \mathcal {L}$, with the benefit of Lemma 1, we obtain that

$$\begin{aligned} \begin{aligned} ||\hbar (s)-h(s)||&= sup_{s \in \mathcal {L}}\bigg |\hbar _{0}+\frac{1}{\Gamma (n)}\int _{0}^{s}(s-\wp )^{n-1} \omega (\wp , \hbar (\wp ))d\wp \\&\qquad +\frac{1}{\Gamma (n)}\int _{0}^{s}(s-\wp )^{n-1}\Im (\wp )d\wp \\&\qquad -\bigg [\hbar _{0}+\frac{1}{\Gamma (n)}\int _{0}^{s}(s-\wp )^{n-1} \omega (\wp , \hbar (\wp ))d\wp \bigg ]\bigg |,\\ {}&\le sup_{s \in \mathcal {L}}|\hbar _{0}-h_{0}|+sup_{s \in \mathcal {L}}\bigg (|\Im (s)|\bigg [\frac{1}{\Gamma (n)}\int _{0}^{s}(s-\wp )^{n-1}d\wp \bigg ]\bigg )\\&\qquad + sup_{s \in \mathcal {L}} \frac{1}{\Gamma (n)} \int _{0}^{s}(s-\wp )^{n-1}|\omega (s,\hbar (s))-\omega (s,h(s))|d\wp , \\&\le \sigma \Theta +\frac{\Bbbk ||\hbar -h||}{\Gamma (n)}sup_{s\in \mathcal {L}}\int _{0}^{s}(s-\wp )^{n-1}d\wp \\&\le \sigma \Theta + \bigg [\frac{u^{n}}{\Gamma (n+1)}\bigg ]\Bbbk ||\hbar -h||\\&\le \sigma \Theta +\sigma \Bbbk ||\hbar (s)-h(s)||. \end{aligned} \end{aligned}$$

Then we get

$$\begin{aligned} ||\hbar -h||\le \ell \Theta , \end{aligned}$$

choose, $\ell = \frac{\sigma }{1-\sigma \Bbbk }$. Therefore, system $G(\Theta )=\Bbbk \Theta$ and also $G(0)=0$. This shows that the both Hyers-Ulam and Hyers-Ulam generalised stability of the nonlinear dynamical system (5.1). $\square$

6 Numerical scheme

According to model (3.1), the numerical resolution of the adopted fractional model by the Adams-Bashforth method and the numerical results demonstrate the technique’s applicability. First, by employing the model (3.1), it becomes an iterative scheme. In this section, we utilize graphs created through numerical simulation. We employed a fractional differential equation nonlinear system to mathematically analyze the chemical kinetics model. In comparison to ordinary derivatives, we discover a greater degree of flexibility in the fractional chemical kinetics model. Through numerical simulation, we obtained the results for various values of n.

$$\begin{aligned} D^{n} h(s) = \omega (s, h(s)). \end{aligned}$$

(6.1)

Furthermore, it is presumed that the Voltera integral of the above equation (6.1) is displayed as

$$\begin{aligned} h(s)= \sum _{i=0}^{(n)-1} \frac{h_{0}^{(i)} s^{i}}{i!}+\frac{1}{\Gamma (n)}\int _{0}^{s} (s-\upsilon )^{n-1}\omega (\upsilon ,h(\upsilon ))d \upsilon . \end{aligned}$$

Again integrating the above equation, we gain as

$$\begin{aligned} h(s_{y+1})= & {} \sum _{i=0}^{(n)-1} \frac{h_{0}^{(i)} s^{i}}{i!}\\{} & {} +\frac{q^{n}}{\Gamma (n+2)}\omega (s_{y+1}, h^{\upsilon }(s_{y+1}))+\frac{q^{n}}{\Gamma (n+2)}\sum _{x=0}^{b}c_{x,y+1}\omega (s_{x},h(s_{x})). \end{aligned}$$

Observe that $q=\frac{T}{R}, s_{y}= yq, y= 0,1,2,..R\in z^{+}$. By making using the above scheme for the suggested novel model, we get the following solution

$$\begin{aligned} { \begin{aligned} \mathbb {S}(s_{r+1})=&\mathbb {S}(s_{0})+\frac{q^{n}}{\Gamma (n+2)}\{- \alpha \mathbb {S} \mathbb {E}+(\beta -\gamma \mathbb {S})\mathbb {J}_{1}+\rho \mathbb {J}_{2}\}\\&\quad +\frac{q^{n}}{\Gamma (n+2)}\sum _{p=0}^{r} c_{p,r+1}\{- \alpha \mathbb {S} \mathbb {E}+(\beta -\gamma \mathbb {S})\mathbb {J}_{1}+\rho \mathbb {J}_{2}\},\\ \mathbb {E}(s_{r+1})&=\mathbb {E}(s_{0})+\frac{q^{n}}{\Gamma (n+2)}\{- \alpha \mathbb {S} \mathbb {E}+(\beta +\Psi )\mathbb {J}_{1}\}\\&\quad +\frac{q^{n}}{\Gamma (n+2)}\sum _{p=0}^{r} c_{p,r+1}\{- \alpha \mathbb {S} \mathbb {E}+(\beta +\Psi )\mathbb {J}_{1}\},\\ \mathbb {J}_{1}(s_{r+1})=&\mathbb {J}_{1}(s_{0})+\frac{q^{n}}{\Gamma (n+2)}\{\alpha \mathbb {S}\mathbb {E}-(\alpha +\gamma \mathbb {S}+\Psi )\mathbb {J}_{1}+(\rho +\Phi )\mathbb {J}_{2}\}\\&\quad +\frac{q^{n}}{\Gamma (n+2)}\sum _{p=0}^{r} c_{p,r+1}\{\alpha \mathbb {S}\mathbb {E}-(\alpha +\gamma \mathbb {S}+\Psi )\mathbb {J}_{1}+(\rho +\Phi )\mathbb {J}_{2}\},\\ \mathbb {J}_{2}(s_{r+1})&=\mathbb {J}_{2}(s_{0})+\frac{q^{n}}{\Gamma (n+2)}\{\gamma \mathbb {S}\mathbb {J}_{1}-(\rho +\Phi )\mathbb {J}_{2}\}\\&\quad +\frac{q^{n}}{\Gamma (n+2)}\sum _{p=0}^{r} c_{p,r+1}\{\gamma \mathbb {S}\mathbb {J}_{1}-(\rho +\Phi )\mathbb {J}_{2}\},\\ \mathbb {P}(s_{r+1})&=\mathbb {P}(s_{0})+\frac{q^{n}}{\Gamma (n+2)}\{\Psi \mathbb {J}_{1}+\Phi \mathbb {J}_{2}\}+\frac{q^{n}}{\Gamma (n+2)}\sum _{p=0}^{r} c_{p,r+1}\{\Psi \mathbb {J}_{1}+\Phi \mathbb {J}_{2}\}. \end{aligned}} \end{aligned}$$

(6.2)

Fig. 2

Behaviour of the substrate ${(\mathbb {S})}$ for various values of parameter n with $\alpha = 0.2, \beta = 0.03, \gamma = 0.03, \rho = 0.02, \Psi = 0.12$ and $\Phi = 0.03$

Fig. 3

Behaviour of the enzyme ${(\mathbb {E})}$ for various values of parameter n with $\alpha = 0.2, \beta = 0.03, \gamma = 0.03, \rho = 0.02, \Psi = 0.12$ and $\Phi = 0.03$

Fig. 4

Behaviour of the enzyme-substrate ${\mathbb {J}_{1}}$ for various values of parameter nwith $\alpha = 0.2, \beta = 0.03, \gamma = 0.03, \rho = 0.02, \Psi = 0.12$ and $\Phi = 0.03$

Fig. 5

Behaviour of the enzyme-substrate ${\mathbb {J}_{2}}$ for various values of parameter n with $\alpha = 0.2, \beta = 0.03, \gamma = 0.03, \rho = 0.02, \Psi = 0.12$ and $\Phi = 0.03$

Fig. 6

Behaviour of the Product ${(\mathbb {P})}$ for various values of parameter n with $\alpha = 0.2, \beta = 0.03, \gamma = 0.03, \rho = 0.02, \Psi = 0.12$ and $\Phi = 0.03$

Fig. 7

Expressing the results of reaction rate in Michaelis-Menten equation at 0.6

Fig. 8

Expressing the results of reaction rate in Michaelis-Menten equation at 0.7

Fig. 9

Expressing the results of reaction rate in Michaelis-Menten equation at 0.8

Fig. 10

Expressing the results of reaction rate in Michaelis-Menten equation at 0.9

Fig. 11

Energy changes in a reaction

6.1 Results and discussion

Using MATLAB for graphical representation, we simulate the numerical schemes using equation (6.2). The initial data used are ${\mathbb {S} = 0.7}$, ${\mathbb {E} = 0.3}$, ${\mathbb {J}_{1} = \mathbb {J}_{2} = 0}$, and ${\mathbb {P} = 0}$. Figures 2 and 3 illustrates how varying fractional orders n impact the concentration of various species. As n decreases, the substrate concentration ${\mathbb {S}}$ decreases, and the rate at which enzyme-substrate complexes dissociate to form enzyme-substrate complexes increases. This reduction delay suggests that ${\mathbb {S}}$ needs more time to decrease in size. Likewise, the enzyme concentration ${\mathbb {E}}$ exhibits an increase for approximately 20 units of time with smaller n values. This observation implies that the reformation of ${\mathbb {E}}$ slows down with smaller values of n in the overall reaction.

Fig. 12

Benefits of chemical reaction with other catalyst

From the beginning, in Fig. 4, ${\mathbb {J}_{1}}$ decreases for 5 units of time and then increases after a certain period. This behavior aligns with the decrease in n, which leads to reduced concentrations of both ${\mathbb {S}}$ and ${\mathbb {E}}$. Clearly, the interaction between ${\mathbb {S}}$ and ${\mathbb {E}}$ results in a reduced formation of ${\mathbb {J}_{1}}$. Similarly, the concentration ${\mathbb {J}{2}}$ mirrors the behavior observed in ${\mathbb {J}_{1}}$ and also depends on the fractional parameter of order n. The decrease in n leads to reduced formation of ${\mathbb {J}_{1}}$, which slows down the overall reaction rate. Consequently, the formation of ${\mathbb {J}_{2}}$ diminishes as it is formed from the reaction between ${\mathbb {S}}$ and ${\mathbb {J}_{1}}$, relating only to ${\mathbb {J}_{1}}$. Likewise, the consumption of ${\mathbb {J}_{2}}$ rapidly falls due to the overall reduction in reaction kinetics, requiring more time to diminish as shown 5. Ultimately, in Fig. 6, the concentration profile of ${\mathbb {P}}$ increases as the fractional parameter of order n decreases. This decline in ${\mathbb {P}}$ formation is attributed to the transformation of ${\mathbb {J}_{1}}$ and ${\mathbb {J}_{2}}$. As previously noted, the formation of ${\mathbb {J}_{1}}$ and ${\mathbb {J}_{2}}$ decelerates with decreasing values of n, and ${\mathbb {J}_{2}}$ itself takes longer to dissipate. Consequently, both ${\mathbb {J}_{1}}$ and ${\mathbb {J}_{2}}$ require more time to generate ${\mathbb {P}}$ due to insufficient quantities. According to the law of mass action, chemical reactions are proportional to the concentration interaction between ${\mathbb {J}_{1}}$ and ${\mathbb {J}_{2}}$. In comparison, the parameters in the Michaelis-Menten equation [2] include the maximum reaction rate (Vmax) and the Michaelis constant (Km), which are expressed in Figs. 7, 8, 9 and 10. These various parameters characterize the enzyme’s efficiency and substrate affinity in a novel study of fractional Caputo chemical kinetics.

6.2 Application and example

The application and example below concentrate on the impact of chemical kinetics. Enzymes catalyze specific reactions by binding to substrates (reactant molecules) and facilitating their transformation into products, as shown in Fig. 11. They are biological catalysts that increase the rate of chemical reactions by lowering the activation energy required for the reaction to occur.

• 1. Iron rusting is a very slow reaction, and determining its rate is difficult..

• 2. Digestion is the breaking down of complex food materials into simpler forms.

• 3. Reaction kinetics can describe the rate at which silver chloride precipitates from solution. This involves studying how quickly the reactants (silver ions and chloride ions) combine to form the product (AgCl).

For instance given in third order reaction with $K= \frac{1}{2t}\bigg [\frac{x(2a-x)}{a^{2}(a-x)^{2}}\bigg ]$ represents the expression for the rate constant.

• 1. Reaction of nitric oxide with oxygen or chlorine/bromine $2No (g)+o_{2} (g) \rightarrow 2No_{2} (g)$ This reaction is crucial in atmospheric chemistry and combustion processes. It is part of the mechanism for nitrogen dioxide formation, which contributes to air pollution and tropospheric ozone formation.

• 2. Oxidation of ferrous sulphate in water.

• 3. $2Fecl_{3} (aq)+ H_{2} (g) \rightarrow N_{2}o (g)+H_{2}o (g)$

• 4. The reaction between $C_{6}H_{5}COCl$ and alcohol in ether solution.

Many biological processes exhibit chemical kinetics involving two enzymes. The enzymatic pathway involved in the metabolism of drugs or toxins in the human body, specifically in the liver, is a real-life application. For a simplified example, consider drug metabolism: Enzymes in the liver, particularly the Cytochrome P450 enzymes (CYP450), are responsible for metabolizing many drugs. Multiple enzymes often work sequentially or in parallel to convert the drug into its metabolites, which the body then excretes. For instance, one enzyme may first act on a drug molecule to convert it into an intermediate compound, which another enzyme then further metabolizes to produce the final metabolite. In photosynthesis, multiple enzymes are involved in the conversion of carbon dioxide and water into glucose and oxygen. Enzymes such as RuBisCO (ribulose-1,5-bisphosphate carboxylase/oxygenase) and various others within the Calvin cycle work together to catalyze different steps of the process.

Likewise, enzymes catalyze chemical reactions at rates that are astounding relative to uncatalyzed chemistry under the same reaction conditions. Some intriguing topics deserve further investigation using the rate of reaction in chemical kinetics with fractional operators in the model (3.1). This approach investigates the speed at which reactants transform into products and the mechanisms behind these transformations. Key concepts in chemical kinetics include reaction rates, rate laws, reaction mechanisms, and reaction kinetics. The substrate moves from the Michaelis complex to the product by changing the way electrons are distributed in the substrate. This is possible because enzymes are biocatalysts, as shown in Fig. 12. Enzymes alter the electronic structure through protonation, proton abstraction, electron transfer, and interaction with Lewis acids and bases, among other processes. Additionally, this knowledge may be useful in real-life pharmaceutical applications such as chiral syntheses, drug modification, bioconjugation, and derivatization, as well as further protein engineering and computational biology.

7 Conclusion

In this research article, we have successfully obtained comparable results to a fractional model for kinetically controlled chemical reactions, which play a crucial role as catalysts in various reactions occurring within living organisms. We use fixed-point theory to briefly establish the existence and uniqueness of solutions, as well as demonstrate Hyers-Ulam stability. Using the fractional Adams-Bashforth approach for the iterative solution of the model, we have graphed the numerical results to justify the importance of the fractional-order derivative. Finally, the application of chemical kinetics is fundamental to understanding and manipulating chemical reactions, playing a crucial role in various fields such as industrial processes, environmental studies, and pharmaceutical development. Through the study of reaction rates, mechanisms, and influencing factors, chemical kinetics provides valuable insights into reaction pathways, reaction efficiencies, and product yields.