This paper is mainly concerned with introducing a numerical method for solving initial–boundary value problems with integer and fractional order time derivatives. The method is based on discretizing the considered problems with respect to spatial and temporal domains. With the help of finite difference methods, we transformed the studied problem into a set of fractional differential equations. Then, we implemented the fractional Adams method to solve this set in order to provide approximate solutions to the main problem. This combination results in an algorithm that can efficiently and accurately solve a general class of integer and fractional order initial–boundary value problems, such that it does not need to solve large systems of linear equations. In addition, we discussed the stability of the proposed scheme. Three illustrative examples are numerically solved to reveal the effectiveness and validity of the proposed technique.

中文翻译：

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具有整数和分数阶时间导数的 PDE 的初始-边界值问题的数值离散化


本文主要关注介绍一种数值方法，用于解决整数和分数阶时间导数的初始边界值问题。该方法基于对空间和时间域方面所考虑的问题进行离散化。在有限差分方法的帮助下，我们将研究的问题转化为一组分数阶微分方程。然后，我们实现了分数阶 Adams 方法来求解这组，以便为主要问题提供近似解。这种组合产生的算法可以高效、准确地求解一类一般的整数和分数阶初始边界值问题，因此不需要求解大型线性方程组。此外，我们还讨论了拟议方案的稳定性。对三个说明性示例进行了数值求解，以揭示所提出技术的有效性和有效性。