The processes of diffusion and reaction play essential roles in numerous system dynamics. Consequently, the solutions of reaction–diffusion equations have gained much attention because of not only their occurrence in many fields of science but also the existence of important properties and information in the solutions. However, despite the wide range of numerical methods explored for approximating solutions, the adoption of block methods is yet to be investigated. Hence, this article introduces a new two-step third–fourth-derivative block method as a numerical approach to solve the reaction–diffusion equation. In order to ensure improved accuracy, the method introduces the concept of nonlinearity in the solution of the linear model through the presence of higher derivatives. The method obtained accurate solutions for the model at varying values of the dimensionless diffusion parameter and saturation parameter. Furthermore, the solutions are also in good agreement with previous solutions by existing authors.

中文翻译：

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反应扩散方程的非线性求解，采用两步三阶-四阶导数块方法

扩散和反应过程在众多系统动力学中起着至关重要的作用。因此，反应扩散方程的解不仅引起了很多科学领域的关注，而且还因为其中存在重要的性质和信息而备受关注。然而，尽管探索了广泛的数值方法来近似解决方案，但尚未研究采用块方法的问题。因此，本文介绍了一种新的两步三阶-四阶导数块方法，作为求解反应扩散方程的数值方法。为了确保提高的精度，该方法通过存在更高的导数，在线性模型的解中引入了非线性的概念。该方法在无因次扩散参数和饱和度参数的变化值下获得了模型的精确解。此外，这些解决方案也与现有作者先前的解决方案非常吻合。