# Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Lemma**

**1.**

- 1.
- ${D}_{t}^{\alpha}{t}^{p}=\frac{\mathsf{\Gamma}(\alpha +1)}{\mathsf{\Gamma}(p-\alpha +1)}{t}^{p-\alpha}$
- 2.
- ${D}_{t}^{\alpha}\lambda =0$
- 3.
- ${D}_{t}^{\alpha}{I}_{t}^{\alpha}R(\zeta ,t)=R(\zeta ,t)$
- 4.
- ${I}_{t}^{\alpha}{D}_{t}^{\alpha}R(\zeta ,t)=R(\alpha ,t)-{\sum}_{i=0}^{n-1}{\partial}^{i}R(\zeta ,0)\frac{{t}^{i}}{i!}$

## 3. General Procedure of OAFM

**Step 1**: To solve Equation (3), we will use an approximate solution that has two components, such as:$$R(\xi ,\tau )={R}_{0}(\xi ,\tau )+{R}_{1}(\xi ,\tau ,{C}_{i}),\phantom{\rule{1.em}{0ex}}i=1,2,3,\dots ,p$$**Step 2**: To determine the zero and first-order solutions, we substitute Equation (5) into Equation (3), which results in:$$\frac{{\partial}^{\alpha}{R}_{0}(\xi ,\tau )}{\partial {\tau}^{\alpha}}+\frac{{\partial}^{\alpha}{R}_{1}(\xi ,\tau )}{\partial {\tau}^{\alpha}}+g(\xi ,\tau )+N\left[\frac{{\partial}^{\alpha}{R}_{0}(\xi ,\tau )}{\partial {\tau}^{\alpha}}+\frac{{\partial}^{\alpha}{R}_{1}(\xi ,\tau ,{C}_{i})}{\partial {\tau}^{\alpha}}\right]=0$$**Step 3**: For the purpose of determining the first approximation ${R}_{0}(\xi ,\tau )$ based on the linear equation.$$\frac{{\partial}^{\alpha}{R}_{0}(\xi ,\tau )}{\partial {\tau}^{\alpha}}+g(\xi ,\tau )=0$$

**Step 4**: The nonlinear term seen in expanding form (6) is,$$N\left[\frac{{\partial}^{\rho}{R}_{0}(\xi ,\tau )}{\partial {\tau}^{\alpha}}+\frac{\partial {R}_{1}^{\alpha}(\xi ,\tau ,{C}_{i})}{\partial {t}^{\alpha}}\right]=N\left[{R}_{0}(\xi ,\tau )\right]+\sum _{k=1}^{\infty}\frac{{R}_{1}^{k}}{k!}{N}^{\left(k\right)}\left[{R}_{0}(\xi ,\tau )\right]$$**Step 5**: To quickly solve Equation (9) and accelerate the convergence of the first-order approximations $R(\xi ,\tau )$, we propose the following alternative equation:$$\frac{{\partial}^{\alpha}{R}_{1}(\xi ,\tau ,{C}_{i})}{\partial {\tau}^{\alpha}}={A}_{1}\left[{R}_{0}(\xi ,\tau )\right]N\left[{R}_{0}(\xi ,\tau )\right]+{A}_{2}\left[{R}_{0}(\xi ,\tau ),{C}_{j}\right].$$**Step 6**: We find a first-order solutions ${R}_{1}(\xi ,\tau )$ by using the inverse operator after putting the Auxiliary function into Equation (10).**Step 7**: Several methods are used to find the numerical values of the convergence control parameters ${C}_{i}$, including least squares, Galerkin’s, Ritz, and collocation. To eliminate mistakes, we employ the least squares approach.$$J({C}_{i},{C}_{j})={\int}_{0}^{t}{\int}_{\mathsf{\Omega}}{\Re}^{2}(y,t;{C}_{i},{C}_{j})d\xi d\tau .$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \Re (\xi ,\tau ,{C}_{i},{C}_{j})=\frac{\partial R(\xi ,\tau ,{C}_{i},{C}_{j})}{\partial \tau}+g(\xi ,\tau )+N\left[R(\xi ,\tau ,{C}_{i},{C}_{j})\right],\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& i=1,2,\dots ,s,\phantom{\rule{0.277778em}{0ex}}j=S+1,S+2,\dots ,p\hfill \end{array}$$

## 4. Applications

#### 4.1. Problem 1

#### 4.2. Problem 2

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The 2D-plots of $R(\xi ,\tau )$ and $T(\xi ,\tau )$ using the OAFM solution where $\lambda =1.5$ and $\gamma =2.5$.

**Figure 2.**The 3D-plots of $R(\xi ,\tau )$ and $T(\xi ,\tau )$ using OAFM solution where the values $\lambda =1.5$ and $\gamma =2.5$.

**Figure 3.**The 2D-plots of $R(\xi ,\tau )$,$T(\xi ,\tau )$ using OAFM solution at different values of fractional order.

**Figure 4.**The 3D-plots of $R(\xi ,\tau )$,$T(\xi ,\tau )$ using OAFM solution at different values of fractional order.

**Table 1.**A comparison of the $R(\xi ,\tau )$ OAFM solution and exact solution and their corresponding absolute error at fractional order $\alpha =1$ where $\lambda =2$ and $\gamma =3$ for Problem 1.

$\mathit{\xi}$ | $\mathit{R}(\mathit{\xi},\mathit{\tau})$ OAFM | $\mathit{R}(\mathit{\xi},\mathit{\tau})$ Exact | Abs.error |
---|---|---|---|

0.1 | −0.4987 | −0.4987 | 5.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.2 | −0.495 | −0.4950 | 3.3$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.3 | −0.4889 | −0.4889 | 1.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.4 | −0.4805 | −0.4805 | 2.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

0.5 | −0.470 | −0.4700 | 2.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.6 | −0.4576 | −0.4575 | 3.7$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.7 | −0.4434 | −0.4434 | 5.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.8 | −0.4279 | −0.4278 | 6.6$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.9 | −0.411 | −0.411 | 7.8$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

1 | −0.3933 | −0.3932 | 8.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

**Table 2.**Comparison of the $T(\xi ,\tau )$ OAFM solution and exact solution and their corresponding absolute error at fractional order $\alpha =1$ for Problem 1.

$\mathit{\xi}$ | $\mathit{T}(\mathit{\xi},\mathit{\tau})$ OAFM | $\mathit{T}(\mathit{\xi},\mathit{\tau})$ Exact | Abs.error |
---|---|---|---|

0.1 | −0.4487 | −0.4487 | 4.9$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.2 | −0.3953 | −0.3953 | 8.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.3 | −0.34 | −0.3399 | 1.0 $\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

0.4 | −0.2831 | −0.283 | 1.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

0.5 | −0.2251 | −0.225 | 1.2$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

0.6 | −0.1663 | −0.1661 | 1.2 $\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

0.7 | −0.1071 | −0.1069 | 1.2 $\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

0.8 | −0.0479 | −0.0478 | 1.2 $\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

0.9 | 0.0109 | 0.01102 | 1.2 $\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

1 | 0.06889 | 0.06901 | 1.1$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

**Table 3.**A comparison of the OAFM solution and exact solution and their corresponding absolute error at fractional order $\alpha =2$ for Problem 2.

$\mathit{\xi}$ | $\mathit{R}(\mathit{\xi},\mathit{\tau})$ OAFM | $\mathit{R}(\mathit{\xi},\mathit{\tau})$ Exact | Abs.error |
---|---|---|---|

0.1 | 0.01 | 0.01 | 4.06$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

0.15 | 0.0225 | 0.0225 | 2.72$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

0.2 | 0.04001 | 0.04 | 7.12$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

0.25 | 0.06251 | 0.0625 | 1.30$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.3 | 0.09002 | 0.09 | 1.90$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.35 | 0.12252 | 0.1225 | 2.10$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.4 | 0.16002 | 0.16 | 1.70$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

0.45 | 0.2025 | 0.2025 | 2.74$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

0.5 | 0.24998 | 0.25 | 2.20$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ |

**Table 4.**A comparison of the OAFM solution and exact solution and their corresponding absolute error at fractional order $\alpha =2$ for Problem 2.

$\mathit{\xi}$ | $\mathit{T}(\mathit{\xi},\mathit{\tau})$ OAFM | $\mathit{T}(\mathit{\xi},\mathit{\tau})$ Exact | Abs.error |
---|---|---|---|

0.10 | 0.005 | 0.005 | 2.000$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

0.15 | 0.01125 | 0.01125 | 2.0001$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

0.20 | 0.0200 | 0.0200 | 2.0025$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

0.25 | 0.03125 | 0.03125 | 2.0242$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

0.30 | 0.0450 | 0.04500 | 2.1515$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

0.35 | 0.06125 | 0.06125 | 2.7125$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

0.40 | 0.0800 | 0.0800 | 4.7134$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ |

0.45 | 0.10125 | 0.10125 | 1.0770$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

0.50 | 0.1250 | 0.12500 | 2.6870$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

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**MDPI and ACS Style**

Alshehry, A.S.; Yasmin, H.; Ahmad, M.W.; Khan, A.; Shah, R.
Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator. *Axioms* **2023**, *12*, 825.
https://doi.org/10.3390/axioms12090825

**AMA Style**

Alshehry AS, Yasmin H, Ahmad MW, Khan A, Shah R.
Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator. *Axioms*. 2023; 12(9):825.
https://doi.org/10.3390/axioms12090825

**Chicago/Turabian Style**

Alshehry, Azzh Saad, Humaira Yasmin, Muhammad Wakeel Ahmad, Asfandyar Khan, and Rasool Shah.
2023. "Optimal Auxiliary Function Method for Analyzing Nonlinear System of Belousov–Zhabotinsky Equation with Caputo Operator" *Axioms* 12, no. 9: 825.
https://doi.org/10.3390/axioms12090825