Purpose

The purpose of this study is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional fisher’s equation. Through our methods, we aim to provide accurate solutions and gain a deeper understanding of the intricate behaviors exhibited by these systems.

Design/methodology/approach

In this study, we use a dual technique that combines the Aboodh residual power series method and the Aboodh transform iteration method, both of which are combined with the Caputo operator.

Findings

We develop exact and efficient solutions by merging these unique methodologies. Our results, presented through illustrative figures and data, demonstrate the efficacy and versatility of the Aboodh methods in tackling such complex mathematical models.

Originality/value

Owing to their fractional derivatives and nonlinear behavior, these equations are crucial in modeling complex processes and confront analytical complications in various scientific and engineering contexts.

中文翻译：

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Aboodh 变换内 Fisher 方程和泡沫排水方程的分数阶视图分析

目的

本研究的目的是求解两个独特但困难的偏微分方程：泡沫排水方程和非线性时间分数费希尔方程。通过我们的方法，我们的目标是提供准确的解决方案，并更深入地了解这些系统所表现出的复杂行为。

设计/方法论/途径

在本研究中，我们使用了将 Aboodh 残差幂级数法和 Aboodh 变换迭代法相结合的双重技术，这两种技术都与 Caputo 算子相结合。

发现

我们通过融合这些独特的方法来开发精确有效的解决方案。我们通过说明性图形和数据呈现的结果证明了 Aboodh 方法在处理此类复杂数学模型方面的有效性和多功能性。

原创性/价值

由于其分数阶导数和非线性行为，这些方程对于复杂过程建模至关重要，并且在各种科学和工程环境中面临分析复杂性。