A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transformation method to obtain approximate solutions for the time-fractional Gray-Scott model. This method gives a more realistic series of solutions that converge rapidly to the exact solution. We can ensure convergence by solving the series resultant. We study the convergence analysis of fractional homotopy analysis transformation method by determining the interval of convergence employing the -curves and the average residual error. We also test the accuracy and the efficiency of this method by comparing our results numerically with the exact solution. Moreover, the effect of the fractionally obtained derivatives on the reaction-diffusion is analyzed. The fractional homotopy analysis transformation method algorithm can be easily applied for singular and nonsingular fractional derivative with partial differential equations, where a few terms of series solution are good enough to give an accurate solution.

中文翻译：

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时间分数阶Gray-Scott某些新模型的数值解

反应扩散系统可以用 Gray-Scott 模型表示。在本研究中，我们讨论了具有 Liouville-Caputo、Caputo-Fabrizio-Caputo 和 Atangana-Baleanu-Caputo 分数阶导数的一维时间分数 Gray-Scott 模型。我们利用分数同伦分析变换方法来获得时间分数格雷斯科特模型的近似解。这种方法提供了一系列更现实的解决方案，这些解决方案快速收敛到精确解决方案。我们可以通过求解级数结果来确保收敛。我们通过确定收敛区间来研究分数同伦分析变换方法的收敛性分析——曲线和平均残差。我们还通过将我们的结果与精确解进行数值比较来测试该方法的准确性和效率。此外，分析了部分获得的衍生物对反应扩散的影响。分数同伦分析变换方法算法可以很容易地应用于具有偏微分方程的奇异和非奇异分数阶导数，其中几项级数解足以给出准确的解。