We consider a fractional‐order nerve impulse model which is known as FitzHugh–Nagumo (F–N) model in this paper. Knowing the solutions of this model allows the management of the nerve impulses process. Especially, considering this model as fractional‐order ensures to be able to analyze in detail because of the memory effect. In this context, first, we use an analytical solution and with the aim of this solution, we obtain numerical solutions by using two numerical schemes. Then, we demonstrate the walking wave‐type solutions of the stated problem. These solutions include complex trigonometric functions, complex hyperbolic functions, and algebraic functions. In addition, the linear stability analysis is performed and the absolute error is occurred by comparing the numerical results with the analytical result. All of the results are depicted by tables and figures. This paper not only points out the exact and numerical solutions of the model but also compares the differences and the similarities of the stated solution methods. Therefore, the results of this paper are important and useful for either neuroscientists and physicists or mathematicians and engineers.

中文翻译：

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分数阶神经冲动模型的解析和数值方法

在本文中，我们考虑了分数阶神经冲动模型，称为FitzHugh–Nagumo（F–N）模型。了解该模型的解决方案可以管理神经冲动过程。特别是，由于记忆效应，将此模型视为分数阶可确保能够进行详细分析。在这种情况下，首先，我们使用解析解，并且以该解决方案为目标，我们通过使用两个数值方案来获得数值解。然后，我们演示了所述问题的行波型解决方案。这些解决方案包括复杂的三角函数，复杂的双曲函数和代数函数。另外，进行线性稳定性分析，并且通过将数值结果与分析结果进行比较来产生绝对误差。所有结果均以表格和数字表示。本文不仅指出了模型的精确解和数值解，而且还比较了所述解法的异同。因此，本文的结果对于神经科学家和物理学家或数学家和工程师都非常重要和有用。