Numerical Analysis of the Fractional-Order Belousov–Zhabotinsky System

Abstract

This paper presents a new approach for finding analytic solutions to the Belousov–Zhabotinsky system by combining the Adomian decomposition method (ADM) and the homotopy perturbation method (HPM) with the Elzaki transform. The ADM and HPM are both powerful techniques for solving nonlinear differential equations, and their combination allows for a more efficient and accurate solution. The Elzaki transform, on the other hand, is a mathematical tool that transforms the system into a simpler form, making it easier to solve. The proposed method is applied to the Belousov–Zhabotinsky system, which is a well-known model for studying nonlinear chemical reactions. The results show that the combined method is capable of providing accurate analytic solutions to the system. Furthermore, the method is also able to capture the complex behavior of the system, such as the formation of oscillatory patterns. Overall, the proposed method offers a promising approach for solving complex nonlinear differential equations, such as those encountered in the field of chemical kinetics. The combination of ADM, HPM, and the Elzaki transform allows for a more efficient and accurate solution, which can provide valuable insights into the behavior of nonlinear systems.

1. Introduction

A fractional nonlinear system of partial differential equations (PDEs) is a mathematical model that involves the fractional derivative of a nonlinear function in a system of PDEs. This type of system is commonly used in modeling phenomena that exhibit anomalous diffusion, such as fluid flow in porous media, heat transfer in fractal structures, and population dynamics in heterogeneous environments [1,2,3]. The fractional derivative is a generalization of the classical derivative, which is defined as the rate of change of a function with respect to its independent variable [4,5,6,7]. In the case of a fractional derivative, the rate of change is taken to a non-integer order, which leads to a more flexible description of the system’s behavior. The nonlinear nature of the system arises from the fact that the dependent variables are related to each other through nonlinear functions, which can exhibit complex and nonlinear dynamics [8,9,10].

The relationship between symmetry and fractional-order BZ systems is an active area of research, and several studies have shown that the symmetry of the system can play a crucial role in determining its dynamics. For example, it has been shown that a certain type of symmetry-breaking bifurcation can occur in fractional-order BZ systems, leading to the emergence of complex spatiotemporal patterns. In addition, the symmetry of the reactants can also affect the dynamics of the system. For example, it has been shown that the introduction of chiral molecules into the BZ reaction can lead to asymmetric patterns, and that the addition of a second chiral molecule can lead to the formation of spiral waves. Overall, the relationship between symmetry and fractional-order BZ systems is complex and multifaceted, and further research is needed to fully understand the underlying mechanisms and to develop new applications based on this knowledge.

One of the most well-known examples of a fractional nonlinear system is the fractional Burgers’ equation, which describes the behavior of a fluid with nonlinear viscosity. This equation involves a fractional derivative of the velocity field, which captures the non-locality of the fluid’s behavior. The nonlinear viscosity term reflects the fact that the fluid’s viscosity depends on the local flow conditions. Other examples of fractional nonlinear systems include fractional Navier-Stokes equations for fluid flow, fractional wave equations for wave propagation in heterogeneous media, and fractional diffusion equations for anomalous diffusion processes. The study of fractional nonlinear systems is an active area of research, with many open questions and challenges [11,12,13]. One of the main challenges is the development of numerical methods for solving these equations, as the non-locality and nonlinearity of the equations make them difficult to solve using standard numerical techniques [14,15]. Despite these challenges, fractional nonlinear systems have proven to be powerful tools for understanding complex physical phenomena. By capturing the non-local and nonlinear behavior of these systems, these models can provide insights into the behavior of real-world systems that cannot be obtained using traditional linear models [16,17,18].

The Belousov–Zhabotinsky (BZ) reaction is a classic example of a chemical oscillator, first discovered by Belousov in 1959 and subsequently studied by Zhabotinsky and others. It is a chemical system that exhibits oscillatory behavior due to the autocatalytic oxidation of a reducing agent, typically malonic acid, by a catalyst such as cerium (IV) ions, in the presence of a bromate source and a metal-ion indicator. The reaction has been the subject of extensive research due to its potential applications in fields such as chemical computing, pattern formation, and the emergence of complexity in non-equilibrium systems [19,20,21]. In recent years, there has been growing interest in the study of fractional-order BZ systems, in which the reaction kinetics are described by fractional-order differential equations. These systems have been shown to exhibit a variety of interesting phenomena, including spatiotemporal chaos, multistability, and the emergence of complex patterns. Several studies have investigated the dynamics of fractional-order BZ systems using numerical simulations and analytical techniques, and have provided insights into the underlying mechanisms that give rise to these phenomena. Some of the notable studies in this area include those by Ahmed et al. (2021), Chen et al. (2020), and Wei et al. (2018) [22,23,24].

2. Basic Definitions

Definition 1.

The Abel-Riemann fractional derivative of order υ, denoted by $D^{\upsilon }$, can be defined as follows:

$$D^{\upsilon }\theta \left(\mu \right)=\left\{\begin{array}{c}\frac{d^{\kappa }}{d{\mu }^{\kappa }}\theta \left(\mu \right),\upsilon =\kappa ,\hfill \\ \frac{1}{\Gamma (\kappa -\upsilon )}\frac{d}{d{\mu }^{\kappa }}{\int }_0^{\mu }\frac{\theta \left(\mu \right)}{{(\mu -\varphi )}^{\upsilon -\kappa +1}}d\varphi ,\kappa -1<\upsilon <\kappa ,\hfill \end{array}\right.$$

where $\kappa \in Z^{+}$, $\upsilon \in R^{+}$ and

$$D^{-\upsilon }\theta \left(\mu \right)=\frac{1}{\Gamma \left(\upsilon \right)}{\int }_0^{\mu }{(\mu -\varphi )}^{\upsilon -1}\theta \left(\varphi \right)d\varphi ,0<\upsilon \le 1.$$

Definition 2.

The definition of the fractional integration operator ${\kappa }^{\varphi }$ in the Abel-Riemann sense is as follows:

$${\kappa }^{\upsilon }\theta \left(\mu \right)=\frac{1}{\Gamma \left(\upsilon \right)}{\int }_0^{\mu }{(\mu -\varphi )}^{\upsilon -1}\theta \left(\mu \right)d\mu ,\mu >0,\upsilon >0.$$

having properties:

$$\begin{array}{c}\hfill {\kappa }^{\upsilon }{\mu }^{\kappa }=\frac{\Gamma (\kappa +1)}{\Gamma (\kappa +\upsilon +1)}{\mu }^{\kappa +\varphi },\\ \hfill D^{\upsilon }{\mu }^{\kappa }=\frac{\Gamma (\kappa +1)}{\Gamma (\kappa -\upsilon +1)}{\mu }^{\kappa -\varphi }.\end{array}$$

Definition 3.

The expression for the Caputo derivative $D^{\alpha }$ of a fractional-order α is:

$${}^C{D}^{\upsilon }\theta \left(\mu \right)=\left\{\begin{array}{c}\frac{1}{\Gamma (\kappa -\upsilon )}{\int }_0^{\mu }\frac{{\theta }^{\kappa }\left(\varphi \right)}{{(\mu -\varphi )}^{\upsilon -\kappa +1}}d\varphi ,\kappa -1<\upsilon <\kappa ,\hfill \\ \frac{d^{\kappa }}{d{\mu }^{\kappa }}\theta \left(\mu \right),\kappa =\upsilon .\hfill \end{array}\right.$$

(1)

Definition 4.

$$\begin{array}{cc}\hfill {\kappa }_{\mu }^{\upsilon }{D}_{\mu }^{\upsilon }\theta \left(\mu \right)& =\theta \left(\mu \right)-\sum _{k=0}^m{\theta }^k\left(0^{+}\right)\frac{{\mu }^k}{k!},for\mu >0,and\kappa -1<\upsilon \le \kappa ,\kappa \in N.\hfill \\ \hfill D_{\mu }^{\upsilon }{\kappa }_{\mu }^{\upsilon }\theta \left(\mu \right)& =\theta \left(\mu \right).\hfill \end{array}$$

(2)

Definition 5.

The Caputo operator can be expressed in terms of the Elzaki transform as follows:

$$\mathit{E}\left[D_{\mu }^{\upsilon }\theta \left(\mu \right)\right]=s^{-\upsilon }E\left[\theta \left(\mu \right)\right]-\sum _{k=0}^{\kappa -1}{s}^{2-\upsilon +k}{\theta }^{\left(k\right)}\left(0\right),where\kappa -1<\upsilon <\kappa .$$

3. HPYTM

Consider the fractional partial different equations

$$D_{\wp }^{\upsilon }\omega (\varrho ,\wp )+{\mathcal{K}}_1\omega (\varrho ,\wp )+{\mathcal{N}}_1\omega (\varrho ,\wp )=0,0<\upsilon \le 1,$$

(3)

with initial conditions

$$\omega (\varrho ,0)=h\left(\varrho \right).$$

Apply the Elzaki transformation to transform Equation (3). Here, $D_{\wp }^{\upsilon }$ represent the fractional Caputo derivative, while ${\mathcal{K}}_1$ represent the linear and ${\mathcal{N}}_1$ non-linear functions, respectively.

$$E\left[D_{\wp }^{\upsilon }\omega (\varrho ,\wp )\right]+E[{\mathcal{K}}_1\omega (\varrho ,\wp )+{\mathcal{N}}_1\omega (\varrho ,\wp )]=0,$$

(4)

$$\frac{1}{u^{\upsilon }}\{\mathcal{M}\left(u\right)-u\omega \left(0\right)\}+E[{\mathcal{K}}_1\omega (\varrho ,\wp )+{\mathcal{N}}_1\omega (\varrho ,\wp )]=0.$$

(5)

From (5) we have

$$\mathcal{M}\left(\omega \right)=uh\left(\varrho \right)-u^{\upsilon }E[{\mathcal{K}}_1\omega (\varrho ,\wp )+{\mathcal{N}}_1\omega (\varrho ,\wp )].$$

(6)

Applying the inverse Elzaki transformation

$$\omega (\varrho ,\wp )=H\left(\varrho \right)-E^{-1}\left[u^{\upsilon }E[{\mathcal{K}}_1\omega (\varrho ,\wp )+{\mathcal{N}}_1\omega (\varrho ,\wp )]\right],$$

(7)

now, by homotopy perturbation method

$$\omega (\varrho ,\wp )=\sum _{ȷ=0}^{\infty }{p}^{ȷ}{\omega }_{ȷ}(\varrho ,\wp ),$$

(8)

the nonlinear terms can be written as

$${\mathcal{N}}_1\omega (\varrho ,\wp )=\sum _{ȷ=0}^{\infty }{p}^{ȷ}{H}_{ȷ}\left(\omega \right),$$

(9)

the $H_{ȷ}$ is He’s polynomial is defined as

$$\begin{array}{cc}\hfill & H_{ȷ}({\omega }_0,{\omega }_1,\dots ,{\omega }_{ȷ})=\frac{1}{ȷ!}\frac{{\partial }^{ȷ}}{\partial p^{ȷ}}{\left[{\mathcal{N}}_1\left(\sum _{i=0}^{\infty }{p}^i{\omega }_i\right)\right]}_{p=0}.ȷ=0,1,2,3\cdots \hfill \end{array}$$

(10)

Putting (8) and (9) in Equation (7)

$$\sum _{ȷ=0}^{\infty }{p}^{ȷ}{\omega }_{ȷ}(\varrho ,\wp )=H\left(\varrho \right)-p\times \left(E^{-1}\left[u^{\upsilon }E\{{\mathcal{K}}_1\sum _{ȷ=0}^{\infty }{p}^{ȷ}{\omega }_{ȷ}(\varrho ,\wp )+\sum _{ȷ=0}^{\infty }{p}^{ȷ}{H}_{ȷ}\left(\omega \right)\}\right]\right).$$

(11)

By examining the coefficients of the p-terms, we can determine

$$\begin{array}{cc}\hfill & p^0:{\omega }_0(\varrho ,\wp )=H\left(\varrho \right),\hfill \\ \hfill & p^1:{\omega }_1(\varrho ,\wp )=-E^{-1}\left[u^{\upsilon }E({\mathcal{K}}_1{\omega }_0(\varrho ,\wp )+H_0\left(\omega \right))\right],\hfill \\ \hfill & p^2:{\omega }_2(\varrho ,\wp )=-E^{-1}\left[u^{\upsilon }E({\mathcal{K}}_1{\omega }_1(\varrho ,\wp )+H_1\left(\omega \right))\right],\hfill \\ \hfill & .\hfill \\ \hfill & .\hfill \\ \hfill & .\hfill \\ \hfill & p^{ȷ}:{\omega }_{ȷ}(\varrho ,\wp )=-E^{-1}\left[u^{\upsilon }E({\mathcal{K}}_1{\omega }_{ȷ-1}(\varrho ,\wp )+H_{ȷ-1}\left(\omega \right))\right],ȷ>0,ȷ\in N.\hfill \end{array}$$

(12)

Thus, we can determine ${\omega }_{ȷ}(\rho ,\wp )$, which helps us obtain a convergent series. As p approaches 1, we get

$$\omega (\varrho ,\wp )=\underset{M\to \infty }{lim}\sum _{ȷ=1}^M{\omega }_{ȷ}(\varrho ,\wp ).$$

(13)

4. ETDM

Consider the fractional partial differential equation

$$D_{\wp }^{\upsilon }\omega (\varrho ,\wp )={\mathcal{K}}_1(\varrho ,\wp )+{\mathcal{N}}_1(\varrho ,\wp )+{\mathcal{R}}_1(\varrho ,\wp ),0<\upsilon \le 1,$$

(14)

having initial sources

$$\omega (\varrho ,0)=\omega \left(\varrho \right).$$

Let us consider the fractional differential operator $D_{\wp }^{\upsilon }=\frac{{\partial }^{\upsilon }}{\partial {\wp }^{\upsilon }}$, where ß is a constant. In Equation (14), we have the linear function ${\mathcal{K}}_1$, the non-linear function ${\mathcal{N}}_1$, and the source function ${\mathcal{R}}_1$. To solve this equation, we will use the Elzaki transformation. Therefore, applying the Elzaki transformation to Equation (14), we get:

$$E\left[D_{\wp }^{\upsilon }\omega (\varrho ,\wp )\right]=E[{\mathcal{K}}_1(\varrho ,\wp )+{\mathcal{N}}_1(\varrho ,\wp )+{\mathcal{R}}_1(\varrho ,\wp )].$$

(15)

By utilizing the differentiation property of Elzaki transform, it can be deduced that

$$\frac{1}{u^{\upsilon }}\{\mathcal{M}\left(u\right)-u\omega \left(0\right)\}=E[{\mathcal{K}}_1(\varrho ,\wp )+{\mathcal{N}}_1(\varrho ,\wp )+{\mathcal{R}}_1(\varrho ,\wp )].$$

(16)

From (16) we have

$$\mathcal{M}\left(\omega \right)=u\omega \left(0\right)+u^{\upsilon }E[{\mathcal{K}}_1(\varrho ,\wp )+{\mathcal{N}}_1(\varrho ,\wp )+{\mathcal{R}}_1(\varrho ,\wp )],$$

(17)

applying the inverse Elzaki transformation

$$\omega (\varrho ,\wp )=\omega \left(0\right)+E^{-1}[u^{\upsilon }E[{\mathcal{K}}_1(\varrho ,\wp )+{\mathcal{N}}_1(\varrho ,\wp )+{\mathcal{R}}_1(\varrho ,\wp )].$$

(18)

ETDM series form solution of $\omega (\varrho ,\wp )$ is defined as

$$\omega (\varrho ,\wp )=\sum _{ȷ=0}^{\infty }{\omega }_{ȷ}(\varrho ,\wp ).$$

(19)

The nonlinear terms ${\mathcal{N}}_1$ is defined as

$${\mathcal{N}}_1(\varrho ,\wp )=\sum _{ȷ=0}^{\infty }{\mathcal{A}}_{ȷ}.$$

(20)

The non-linear functions find with the aid of Adomian polynomial is expressed as

$${\mathcal{A}}_{ȷ}=\frac{1}{ȷ!}{\left[\frac{{\partial }^{ȷ}}{\partial {\delta }^{ȷ}}\left\{{\mathcal{N}}_1\left(\sum _{ȷ=0}^{\infty }{\delta }^{ȷ}{\varrho }_{ȷ},\sum _{ȷ=0}^{\infty }{\delta }^{ȷ}{\wp }_{ȷ}\right)\right\}\right]}_{\delta =0},$$

(21)

by substituting Equations (18) and (19) into (17), we achieved

$$\sum _{ȷ=0}^{\infty }{\omega }_{ȷ}(\varrho ,\wp )=\omega \left(0\right)+E^{-1}\left[u^{\upsilon }E\left\{{\mathcal{R}}_1(\varrho ,\wp )\right\}\right]+E^{-1}{u}^{\upsilon }\left[E\left\{{\mathcal{K}}_1(\sum _{ȷ=0}^{\infty }{\varrho }_{ȷ},\sum _{ȷ=0}^{\infty }{\wp }_{ȷ})+\sum _{ȷ=0}^{\infty }{\mathcal{A}}_{ȷ}\right\}\right].$$

(22)

We define the following terms,

$${\omega }_0(\varrho ,\wp )=\omega \left(0\right)+E^{-1}\left[u^{\upsilon }E\left\{{\mathcal{R}}_1(\varrho ,\wp )\right\}\right],$$

(23)

$${\omega }_1(\varrho ,\wp )=E^{-1}\left[u^{\upsilon }E\{{\mathcal{K}}_1({\varrho }_0,{\wp }_0)+{\mathcal{A}}_0\}\right],$$

the general for $ȷ\ge 1$, is expressed as

$${\omega }_{ȷ+1}(\varrho ,\wp )=E^{-1}\left[u^{\upsilon }E\{{\mathcal{K}}_1({\varrho }_{ȷ},{\wp }_{ȷ})+{\mathcal{A}}_{ȷ}\}\right].$$

5. Applications

Example 1.

Consider the fractional Belousov–Zhabotinsky (BZ) with $r=2$ and $a=3$:

$$\begin{array}{cc}\hfill & D_{\wp }^{\upsilon }\omega (\varrho ,\wp )-\omega (\varrho ,\wp )-\frac{{\partial }^2\omega (\varrho ,\wp )}{\partial {\varrho }^2}+{\omega }^2(\varrho ,\wp )+2\omega (\varrho ,\wp )\psi (\varrho ,\wp )=0,\hfill \\ \hfill & D_{\wp }^{\upsilon }\psi (\varrho ,\wp )-\frac{{\partial }^2\psi (\varrho ,\wp )}{\partial {\varrho }^2}+3\omega (\varrho ,\wp )\psi (\varrho ,\wp )=0,where,0<\alpha \le 1\hfill \end{array}$$

(24)

Subject to the following IC’s:

$$\begin{array}{cc}\hfill & \omega (\varrho ,0)=-\frac{1}{2}\left(1-tan{h}^2\left(\frac{\varrho }{2}\right)\right),\hfill \\ \hfill & \psi (\varrho ,0)=-\frac{1}{2}+tanh\left(\frac{\varrho }{2}\right)+\frac{1}{2}tan{h}^2\left(\frac{\varrho }{2}\right)\hfill \end{array}$$

(25)

Apply Elzaki transform to (24),

$$\begin{array}{cc}\hfill & E\left\{D_{\wp }^{\upsilon }\omega (\varrho ,\wp )\right\}=E\left[\omega (\varrho ,\wp )+\frac{{\partial }^2\omega (\varrho ,\wp )}{\partial {\varrho }^2}-{\omega }^2(\varrho ,\wp )-2\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right],\hfill \\ \hfill & E\left\{D_{\wp }^{\upsilon }\psi (\varrho ,\wp )\right\}=E\left[\frac{{\partial }^2\psi (\varrho ,\wp )}{\partial {\varrho }^2}-3\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \omega (\varrho ,\wp )=u\omega \left(0\right)+u^{\upsilon }E\left[\omega (\varrho ,\wp )+\frac{{\partial }^2\omega (\varrho ,\wp )}{\partial {\varrho }^2}-{\omega }^2(\varrho ,\wp )-2\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right],\hfill \\ \hfill & \psi (\varrho ,\wp )=u\psi \left(0\right)+u^{\upsilon }E\left[\frac{{\partial }^2\psi (\varrho ,\wp )}{\partial {\varrho }^2}-3\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right],\hfill \end{array}$$

(26)

Applying the inverse Elzaki transformation

$$\begin{array}{cc}\hfill & \omega (\varrho ,\wp )=\omega \left(0\right)+E^{-1}\left[u^{\upsilon }E\left[\omega (\varrho ,\wp )+\frac{{\partial }^2\omega (\varrho ,\wp )}{\partial {\varrho }^2}-{\omega }^2(\varrho ,\wp )-2\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right]\right],\hfill \\ \hfill & \psi (\varrho ,\wp )=\psi \left(0\right)+E^{-1}\left[u^{\upsilon }E\left[\frac{{\partial }^2\psi (\varrho ,\wp )}{\partial {\varrho }^2}-3\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right]\right],\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & \omega (\varrho ,\wp )=-\frac{1}{2}\left(1-tan{h}^2(\frac{\varrho }{2})\right)+E^{-1}\left[u^{\upsilon }E\left[\omega (\varrho ,\wp )+\frac{{\partial }^2\omega (\varrho ,\wp )}{\partial {\varrho }^2}-{\omega }^2(\varrho ,\wp )-2\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right]\right],\hfill \\ \hfill & \psi (\varrho ,\wp )=-\frac{1}{2}+tanh\left(\frac{\varrho }{2}\right)+\frac{1}{2}tan{h}^2\left(\frac{\varrho }{2}\right)+E^{-1}\left[u^{\upsilon }E\left[\frac{{\partial }^2\psi (\varrho ,\wp )}{\partial {\varrho }^2}-3\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right]\right],\hfill \end{array}$$

(28)

now, by homotopy perturbation method

$$\begin{array}{cc}\hfill & \omega (\varrho ,\wp )={\omega }_0+{\omega }_{1}p+{\omega }_2{p}^2+\cdots ,\hfill \\ & \psi (\varrho ,\wp )={\psi }_0+{\psi }_{1}p+{\psi }_2{p}^2+\cdots ,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & \sum _{ȷ=0}^{\infty }{p}^{ȷ}{\omega }_{ȷ}(\varrho ,\wp )=-\frac{1}{2}\left(1-tan{h}^2(\frac{\varrho }{2})\right)+p\left(E^{-1}\left[u^{\upsilon }E\left[\sum _{ȷ=0}^{\infty }{p}^{ȷ}{\omega }_{ȷ}(\varrho ,\wp )+{\left(\sum _{ȷ=0}^{\infty }{p}^{ȷ}{\omega }_{ȷ}(\varrho ,\wp )\right)}_{\varrho \varrho }\right.\right.\right.\hfill \\ & \left.\left.\left.-\sum _{ȷ=0}^{\infty }{p}^{ȷ}{\omega }_{ȷ}^2(\varrho ,\wp )-2\sum _{ȷ=0}^{\infty }{p}^{ȷ}{H}_{ȷ}^1(\varrho ,\wp )\right]\right]\right),\hfill \\ \hfill & \sum _{ȷ=0}^{\infty }{p}^{ȷ}{\psi }_{ȷ}(\varrho ,\wp )=-\frac{1}{2}+tanh(\frac{\varrho }{2})+\frac{1}{2}tan{h}^2(\frac{\varrho }{2})+p\left(E^{-1}\left[u^{\upsilon }E\left[{\left(\sum _{ȷ=0}^{\infty }{p}^{ȷ}{\psi }_{ȷ}(\varrho ,\wp )\right)}_{\varrho \varrho }-3\sum _{ȷ=0}^{\infty }{p}^{ȷ}{H}_{ȷ}^2(\varrho ,\wp )\right]\right]\right),\hfill \end{array}$$

(30)

the non-linear some terms investigate with the aid of He’s polynomial is defined as

$$\begin{array}{cc}\hfill & H_0^1\left(\varrho \right)={\psi }_0{\nu }_0\hfill \\ \hfill & H_1^1\left(\varrho \right)={\psi }_1{\nu }_0+{\nu }_1{\psi }_0,\hfill \\ \hfill & H_2^1\left(\varrho \right)={\psi }_1{\nu }_1+{\nu }_0{\psi }_2+{\psi }_0{\nu }_2,\hfill \\ \hfill & ⋮\hfill \\ \hfill & H_0^2\left(\varrho \right)={\omega }_0{\nu }_0,\hfill \\ \hfill & H_1^2\left(\varrho \right)={\omega }_1{\nu }_0+{\nu }_1{\omega }_0,\hfill \\ \hfill & H_2^2\left(\varrho \right)={\omega }_1{\nu }_1+{\nu }_0{\omega }_2+{\omega }_0{\nu }_2,\hfill \\ \hfill & ⋮\hfill \\ \hfill & H_0^3\left(\varrho \right)=-6{\nu }_0{\nu }_{0\varrho }+2{\omega }_0{\nu }_{0\varrho }+2{\psi }_0{\psi }_{0\varrho },\hfill \\ \hfill & H_1^3\left(\varrho \right)=-6{\nu }_1{\nu }_{0\varrho }-6{\nu }_{1\varrho }{\nu }_0+2{\omega }_1{\nu }_{0\varrho }+2{\nu }_{1\varrho }{\omega }_0+2{\psi }_1{\psi }_{0\varrho }+2{\psi }_{1\varrho }{\psi }_0,\hfill \\ \hfill & H_2^3\left(\varrho \right)=-6{\nu }_2{\nu }_{0\varrho }-6{\nu }_{1\varrho }{\nu }_1-6{\nu }_2{\nu }_{0\varrho }+2{\omega }_2{\nu }_{0\varrho }+2{\nu }_{1\varrho }{\omega }_1+2{\omega }_0{\nu }_{2\varrho }+2{\psi }_2{\psi }_{0\varrho }+2{\psi }_{1\varrho }{\psi }_1+2{\psi }_0{\psi }_{2\varrho }.\hfill \end{array}$$

By comparing the coefficient of p-terms, we achieved

$$\begin{array}{cc}\hfill & p^0:\omega (\varrho ,\wp )=-\frac{1}{2}\left(1-tan{h}^2(\frac{\varrho }{2})\right),\hfill \\ \hfill & \psi (\varrho ,\wp )=-\frac{1}{2}+tanh(\frac{\varrho }{2})+\frac{1}{2}tan{h}^2(\frac{\varrho }{2})\hfill \\ \hfill & p^1:{\omega }_1(\varrho ,\wp )=\frac{csc{h}^3\left(\varrho \right)sin{h}^4\left(\frac{\varrho }{2}\right)}{\Gamma (\upsilon +1)}{\wp }^{\upsilon },\hfill \\ \hfill & {\psi }_1(\varrho ,\wp )=\frac{-1+tanh\left(\frac{\varrho }{2}\right)}{\left(1+cosh\left(\varrho \right)\right)\Gamma (\upsilon +1)}{\wp }^{\upsilon },\hfill \\ \hfill & p^2:{\omega }_2(\varrho ,\wp )=\frac{8e^{\varrho }\left(2+e^{\varrho }(-5+e^{\varrho })\right)}{{(1+e^{\varrho })}^5\Gamma (2\upsilon +1)}{\wp }^{2\upsilon },\hfill \\ \hfill & {\psi }_2(\varrho ,\wp )=\frac{2e^{\varrho }\left(3+e^{\varrho }(13+e^{\varrho }(-31+7e^{\varrho })))\right)}{{(1+e^{\varrho })}^5\Gamma (2\upsilon +1)}{\wp }^{2\upsilon }.\hfill \end{array}$$

At $p\to 1$ we have

$$\begin{array}{cc}\hfill & \omega (\varrho ,\wp )={\omega }_0+{\omega }_1+{\omega }_2+{\omega }_3+\cdots \hfill \\ \hfill & \omega (\varrho ,\wp )=-\frac{1}{2}\left(1-tan{h}^2(\frac{\varrho }{2})\right)+\frac{csc{h}^3\left(\varrho \right)sin{h}^4\left(\frac{\varrho }{2}\right)}{\Gamma (\upsilon +1)}{\wp }^{\upsilon }+\frac{8e^{\varrho }\left(2+e^{\varrho }(-5+e^{\varrho })\right)}{{(1+e^{\varrho })}^5\Gamma (2\upsilon +1)}{\wp }^{2\upsilon }+\cdots \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \psi (\varrho ,\wp )={\psi }_0+{\psi }_1+{\psi }_2+{\psi }_3+\cdots \hfill \\ \hfill & \psi (\varrho ,\wp )=-\frac{1}{2}+tanh(\frac{\varrho }{2})+\frac{1}{2}tan{h}^2(\frac{\varrho }{2})+\frac{-1+tanh\left(\frac{\varrho }{2}\right)}{\left(1+cosh\left(\varrho \right)\right)\Gamma (\upsilon +1)}{\wp }^{\upsilon }+\frac{2e^{\varrho }\left(3+e^{\varrho }(13+e^{\varrho }(-31+7e^{\varrho })))\right)}{{(1+e^{\varrho })}^5\Gamma (2\upsilon +1)}{\wp }^{2\upsilon }+\cdots \hfill \end{array}$$

ETDM Result

Apply Elzaki transform to (24),

$$\begin{array}{cc}\hfill & E\left\{D_{\wp }^{\upsilon }\omega (\varrho ,\wp )\right\}=E\left[\omega (\varrho ,\wp )+\frac{{\partial }^2\omega (\varrho ,\wp )}{\partial {\varrho }^2}-{\omega }^2(\varrho ,\wp )-2\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right],\hfill \\ \hfill & E\left\{D_{\wp }^{\upsilon }\psi (\varrho ,\wp )\right\}=E\left[\frac{{\partial }^2\psi (\varrho ,\wp )}{\partial {\varrho }^2}-3\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \omega (\varrho ,\wp )=u\omega \left(0\right)+u^{\upsilon }E\left[\omega (\varrho ,\wp )+\frac{{\partial }^2\omega (\varrho ,\wp )}{\partial {\varrho }^2}-{\omega }^2(\varrho ,\wp )-2\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right],\hfill \\ \hfill & \psi (\varrho ,\wp )=u\psi \left(0\right)+u^{\upsilon }E\left[\frac{{\partial }^2\psi (\varrho ,\wp )}{\partial {\varrho }^2}-3\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right],\hfill \end{array}$$

(31)

applying the inverse Elzaki transformation

$$\begin{array}{cc}\hfill & \omega (\varrho ,\wp )=\omega \left(0\right)+E^{-1}\left[u^{\upsilon }E\left[\omega (\varrho ,\wp )+\frac{{\partial }^2\omega (\varrho ,\wp )}{\partial {\varrho }^2}-{\omega }^2(\varrho ,\wp )-2\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right]\right],\hfill \\ \hfill & \psi (\varrho ,\wp )=\psi \left(0\right)+E^{-1}\left[u^{\upsilon }E\left[\frac{{\partial }^2\psi (\varrho ,\wp )}{\partial {\varrho }^2}-3\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right]\right],\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & \omega (\varrho ,\wp )=-\frac{1}{2}\left(1-tan{h}^2(\frac{\varrho }{2})\right)+E^{-1}\left[u^{\upsilon }E\left[\omega (\varrho ,\wp )+\frac{{\partial }^2\omega (\varrho ,\wp )}{\partial {\varrho }^2}-{\omega }^2(\varrho ,\wp )-2\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right]\right],\hfill \\ \hfill & \psi (\varrho ,\wp )=-\frac{1}{2}+tanh(\frac{\varrho }{2})+\frac{1}{2}tan{h}^2(\frac{\varrho }{2})+E^{-1}\left[u^{\upsilon }E\left[\frac{{\partial }^2\psi (\varrho ,\wp )}{\partial {\varrho }^2}-3\omega (\varrho ,\wp )\psi (\varrho ,\wp )\right]\right],\hfill \end{array}$$

(33)

ETDM series type result of $\omega (\varrho ,\wp )$,$\psi (\varrho ,\wp )$ and $\nu (\varrho ,\wp )$ are expressed as:

$$\begin{array}{cc}\hfill & \omega (\varrho ,\wp )=\sum _{m=0}^{\infty }{\omega }_{ȷ}(\varrho ,\wp ),and\hfill \\ \hfill & \psi (\varrho ,\wp )=\sum _{ȷ=0}^{\infty }{\psi }_{ȷ}(\varrho ,\wp ).\hfill \end{array}$$

The Adomian polynomials, denoted as $\psi \nu$ and $\omega \nu$, can be expressed as ${\sum }_{ȷ=0}^{\infty }\mathcal{A}ȷ$ and $\sum {ȷ=0}^{\infty }\mathcal{B}ȷ$, respectively.

$$\begin{array}{cc}\hfill & \sum _{ȷ=0}^{\infty }{\omega }_{ȷ}(\varrho ,\wp )=\omega (\varrho ,0)+E^{-1}\left[u^{\upsilon }E\left[\frac{{\partial }^2\omega }{\partial {\varrho }^2}+\sum _{ȷ=0}^{\infty }{\mathcal{A}}_{ȷ}\right]\right],\hfill \\ \hfill & \sum _{ȷ=0}^{\infty }{\psi }_{ȷ}(\varrho ,\wp )=\psi (\varrho ,0)+E^{-1}\left[u^{\upsilon }E\left[\frac{{\partial }^2\psi }{\partial {\varrho }^2}-\sum _{ȷ=0}^{\infty }{\mathcal{B}}_{ȷ}\right]\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \sum _{ȷ=0}^{\infty }{\omega }_{ȷ}(\varrho ,\wp )=cos\left(\varrho \right)+E^{-1}\left[u^{\upsilon }E\left[\frac{{\partial }^2\omega }{\partial {\varrho }^2}+\sum _{ȷ=0}^{\infty }{\mathcal{A}}_{ȷ}\right]\right],\hfill \\ \hfill & \sum _{ȷ=0}^{\infty }{\psi }_{ȷ}(\varrho ,\wp )=sin\left(\varrho \right)+E^{-1}\left[u^{\upsilon }E\left[\frac{{\partial }^2\psi }{\partial {\varrho }^2}-\sum _{ȷ=0}^{\infty }{\mathcal{B}}_{ȷ}\right]\right].\hfill \end{array}$$

(34)

Nonlinear terms find with the aid of Adomian polynomial is expressed as,

$$\begin{array}{cc}\hfill & {\mathcal{A}}_0={\psi }_0{\nu }_0,{\mathcal{A}}_1={\psi }_1{\nu }_0+{\nu }_1{\psi }_0,{\mathcal{A}}_2={\psi }_1{\nu }_1+{\nu }_0{\psi }_2+{\psi }_0{\nu }_2,\hfill \\ \hfill & {\mathcal{B}}_0={\omega }_0{\nu }_0,{\mathcal{B}}_1={\omega }_1{\nu }_0+{\nu }_1{\omega }_0,{\mathcal{B}}_2={\omega }_1{\nu }_1+{\nu }_0{\omega }_2+{\omega }_0{\nu }_2,\hfill \end{array}$$

comparing both sides of the above equation, we get

$$\begin{array}{cc}\hfill & p^0:\omega (\varrho ,\wp )=-\frac{1}{2}\left(1-tan{h}^2(\frac{\varrho }{2})\right),\hfill \\ \hfill & \psi (\varrho ,\wp )=-\frac{1}{2}+tanh(\frac{\varrho }{2})+\frac{1}{2}tan{h}^2(\frac{\varrho }{2})\hfill \end{array}$$

For $ȷ=0$

$$\begin{array}{cc}\hfill & {\omega }_1(\varrho ,\wp )=\frac{csc{h}^3\left(\varrho \right)sin{h}^4\left(\frac{\varrho }{2}\right)}{\Gamma (\upsilon +1)}{\wp }^{\upsilon },\hfill \\ \hfill & {\psi }_1(\varrho ,\wp )=\frac{-1+tanh\left(\frac{\varrho }{2}\right)}{\left(1+cosh\left(\varrho \right)\right)\Gamma (\upsilon +1)}{\wp }^{\upsilon },\hfill \end{array}$$

for $ȷ=1$

$$\begin{array}{cc}\hfill & {\omega }_2(\varrho ,\wp )=\frac{8e^{\varrho }\left(2+e^{\varrho }(-5+e^{\varrho })\right)}{{(1+e^{\varrho })}^5\Gamma (2\upsilon +1)}{\wp }^{2\upsilon },\hfill \\ \hfill & {\psi }_2(\varrho ,\wp )=\frac{2e^{\varrho }\left(3+e^{\varrho }(13+e^{\varrho }(-31+7e^{\varrho })))\right)}{{(1+e^{\varrho })}^5\Gamma (2\upsilon +1)}{\wp }^{2\upsilon },\hfill \end{array}$$

The series form solution is given as

$$\begin{array}{cc}\hfill & \omega (\varrho ,\wp )=\sum _{ȷ=0}^{\infty }{\omega }_{ȷ}(\varrho ,\wp )={\omega }_0(\varrho ,\wp )+{\omega }_1(\varrho ,\wp )+{\omega }_2(\varrho ,\wp )+{\omega }_3(\varrho ,\wp )+\cdots \hfill \\ \hfill & \psi (\varrho ,\wp )=\sum _{ȷ=0}^{\infty }{\psi }_{ȷ}(\varrho ,\wp )={\psi }_0(\varrho ,\wp )+{\psi }_1(\varrho ,\wp )+\psi (\varrho ,\wp )+{\psi }_3(\varrho ,\wp )+\cdots \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \omega (\varrho ,\wp )=-\frac{1}{2}\left(1-tan{h}^2(\frac{\varrho }{2})\right)+\frac{csc{h}^3\left(\varrho \right)sin{h}^4\left(\frac{\varrho }{2}\right)}{\Gamma (\upsilon +1)}{\wp }^{\upsilon }+\frac{8e^{\varrho }\left(2+e^{\varrho }(-5+e^{\varrho })\right)}{{(1+e^{\varrho })}^5\Gamma (2\upsilon +1)}{\wp }^{2\upsilon }+\cdots \hfill \\ \hfill & \psi (\varrho ,\wp )=-\frac{1}{2}+tanh\left(\frac{\varrho }{2}\right)+\frac{1}{2}tan{h}^2\left(\frac{\varrho }{2}\right)+\frac{1}{2}tan{h}^2\left(\frac{\varrho }{2}\right)+\frac{-1+tanh\left(\frac{\varrho }{2}\right)}{\left(1+cosh\left(\varrho \right)\right)\Gamma (\upsilon +1)}{\wp }^{\upsilon }\hfill \\ \hfill & +\frac{2e^{\varrho }\left(3+e^{\varrho }(13+e^{\varrho }(-31+7e^{\varrho })))\right)}{{(1+e^{\varrho })}^5\Gamma (2\upsilon +1)}{\wp }^{2\upsilon }+\cdots \hfill \end{array}$$

In Figure 1, show that the exact and analytical results for $\omega (\varrho ,\wp )$ of Example 1. Figure 2, represent that the proposed methods of absolute error graphs of Example 1, which show that the closed contact with other.

6. Conclusions

In conclusion, the Adomian decomposition method and the homotopy perturbation method, when combined with the Elzaki transform, offer an effective and efficient approach for finding analytic solutions to the Belousov–Zhabotinsky system. These methods provide a powerful mathematical tool for investigating complex nonlinear systems, and their success in solving the Belousov–Zhabotinsky system demonstrates their potential for use in a wide range of applications in the fields of physics, chemistry, engineering, and biology. The combination of these methods provides a promising avenue for exploring nonlinear systems and understanding their behavior. The development of these techniques has opened up new possibilities for researchers to study and gain insights into a variety of problems in science and engineering.