This paper demonstrates an effective and powerful technique, namely fractional He–Laplace method (FHe-LM), to study a nonlinear coupled system of equations with time fractional derivative. The FHe-LM is designed on the basis of Laplace transform to elucidate the solution of nonlinear fractional Hirota–Satsuma coupled KdV and coupled mKdV system but the series coefficients are evaluated in an iterative process with the help of homotopy perturbation method manipulating He’s polynomials. The fractional derivatives are considered in the Caputo sense. The obtained results confirm the suggested approach is extremely convenient and applicable to provide the solution of nonlinear models in the form of a convergent series, without any restriction. Also, graphical representation and the error estimate when compared with the exact solution are presented.

中文翻译：

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非线性 HIROTA-SATSUMA 耦合 KdV 和耦合 mKdV 系统的时间分数导数研究

本文展示了一种有效且强大的技术，即分数 He-Laplace 方法 (FHe-LM)，用于研究具有时间分数导数的非线性耦合方程组。FHe-LM 是在拉普拉斯变换的基础上设计的，用于阐明非线性分数阶 Hirota-Satsuma 耦合 KdV 和耦合 mKdV 系统的解，但在操作 He 多项式的同伦摄动方法的帮助下，在迭代过程中评估级数系数。在 Caputo 意义上考虑分数导数。获得的结果证实，所提出的方法非常方便，适用于以收敛级数的形式提供非线性模型的解，没有任何限制。此外，还提供了与精确解决方案进行比较时的图形表示和误差估计。