This paper aims to implement an analytical method, known as the Laplace homotopy perturbation transform technique, for the result of fractional-order Whitham–Broer–Kaup equations. The technique is a mixture of the Laplace transformation and homotopy perturbation technique. Fractional derivatives with Mittag-Leffler and exponential laws in sense of Caputo are considered. Moreover, this paper aims to show the Whitham–Broer–Kaup equations with both derivatives to see their difference in a real-world problem. The efficiency of both operators is confirmed by the outcome of the actual results of the Whitham–Broer–Kaup equations. Some problems have been presented to compare the solutions achieved with both fractional-order derivatives.

中文翻译：

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不涉及奇异核算子的时分 Whitham-Broer-Kaup 方程的数值研究

本文旨在实现一种分析方法，称为拉普拉斯同伦微扰变换技术，用于分数阶 Whitham-Broer-Kaup 方程的结果。该技术是拉普拉斯变换和同伦微扰技术的混合体。考虑了具有 Mittag-Leffler 的分数导数和 Caputo 意义上的指数定律。此外，本文旨在展示具有两个导数的 Whitham-Broer-Kaup 方程，以了解它们在实际问题中的差异。Whitham-Broer-Kaup 方程的实际结果证实了这两个算子的效率。已经提出了一些问题来比较两种分数阶导数所获得的解决方案。