Damped Burger’s equation describes the characteristics of one-dimensional nonlinear shock waves in the presence of damping effects and is significant in fluid dynamics, plasma physics, and other fields. Due to the potential applications of this equation, thus the objective of this investigation is to solve and analyze the time fractional form of this equation using methods with precise efficiency, high accuracy, ease of application and calculation, and flexibility in dealing with more complicated equations, which are called the Aboodh residual power series method and the Aboodh transform iteration method (ATIM) within the Caputo operator framework. Also, this study intends to further our understanding of the dynamic characteristics of solutions to the Damped Burger’s equation and to assess the effectiveness of the proposed methods in addressing nonlinear fractional partial differential equations. The two proposed methods are highly effective mathematical techniques for studying more complicated nonlinear differential equations. They can produce precise approximate solutions for intricate evolution equations beyond the specific examined equation. In addition to the proposed methods, the fractional derivatives are processed using the Caputo operator. The Caputo operator enhances the representation of fractional derivatives by providing a more accurate portrayal of the underlying physical processes. Based on the proposed two approaches, a set of approximations to damped Burger’s equation are derived. These approximations are discussed graphically and numerically by presenting a set of two- and three-dimensional graphs. In addition, these approximations are analyzed numerically in several tables, including the absolute error for each approximate solution compared to the exact solution for the integer case. Furthermore, the effect of the fractional parameter on the behavior of the derived approximations is examined and discussed.

中文翻译：

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使用 Aboodh 剩余幂级数和 Aboodh 变换迭代方法近似流体和等离子体中的分数非线性阻尼 Burger 型方程

阻尼伯格方程描述了存在阻尼效应时一维非线性冲击波的特征，在流体动力学、等离子体物理等领域具有重要意义。由于该方程的潜在应用，因此本研究的目的是使用高效、准确度高、易于应用和计算以及灵活处理更复杂方程的方法求解和分析该方程的时间分数形式，在 Caputo 算子框架内称为 Aboodh 剩余幂级数法和 Aboodh 变换迭代法 (ATIM)。此外，本研究旨在进一步了解阻尼伯格方程解的动态特性，并评估所提出的方法在解决非线性分数阶偏微分方程方面的有效性。所提出的两种方法是研究更复杂的非线性微分方程的高效数学技术。它们可以为超出特定检查方程的复杂演化方程提供精确的近似解。除了所提出的方法之外，还使用 ​​Caputo 算子处理分数阶导数。 Caputo 算子通过提供对底层物理过程的更准确描述来增强分数导数的表示。基于所提出的两种方法，导出了一组阻尼伯格方程的近似值。通过呈现一组二维和三维图，以图形和数字方式讨论这些近似值。此外，这些近似值在几个表中进行了数值分析，包括每个近似解与整数情况的精确解相比的绝对误差。此外，还检查和讨论了分数参数对导出的近似值的行为的影响。