This study explores the fractional damped generalized regularized long‐wave equation in the sense of Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio fractional derivatives. With the aid of fixed‐point theorem in the Atangana‐Baleanu fractional derivative with Mittag‐Leffler–type kernel, we show the existence and uniqueness of the solution to the damped generalized regularized long‐wave equation. The modified Laplace decomposition method (MLDM) defined in the sense of Caputo, Atangana‐Baleanu, and Caputo‐Fabrizio (in the Riemann sense) operators is used in securing the approximate‐analytical solutions of the nonlinear model. The numerical simulations of the obtained solutions are performed with different suitable values of $\rho$, which is the order of fractional parameter. We have seen the effect of the various parameters and variables on the displacement in figures.

中文翻译：

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带阻尼项的分数阶正则长波方程的分析和数值计算

这项研究探索了Caputo，Atangana-Baleanu和Caputo-Fabrizio分数阶导数意义上的分数阻尼广义正规化长波方程。借助于带有Mittag-Leffler型核的Atangana-Baleanu分数阶导数中的不动点定理，我们证明了阻尼广义正则长波方程解的存在性和唯一性。在Caputo，Atangana-Baleanu和Caputo-Fabrizio（在黎曼意义上）的意义上定义的改进的Laplace分解方法（MLDM）用于确保非线性模型的近似解析解。对获得的解的数值模拟是使用不同的适当值进行的。$\rho$，这是小数参数的顺序。我们已经看到了各种参数和变量对图中位移的影响。