Purpose

Whereas, the classical Green-Naghdi Type II (GN-II) model struggles to accurately represent the thermo-mechanical behavior of thermoelectric MHD due to its inability to account for the memory effect. A new mathematical model of the GN-II theory incorporates a fractional order of heat transport to address this issue.

Design/methodology/approach

The employment of the matrix exponential method, which forms the basis of the state-space approach in contemporary theory, is central to this strategy. The resulting formulation, together with the Laplace transform techniques, is applied to a variety of problems. Solutions to a thermal shock problem and to a problem of a layer media both without heat sources are obtained. Also, a problem with the distribution of heat sources is considered. The numerical technique is used to achieve the Laplace transform inversion.

Findings

According to the numerical results and its graphs, the influences of the fractional order parameters, figure-of-merit factor, thermoelectric power and Peltier coefficient on the behavior of the field quantities are investigated in the new theory.

Originality/value

The new modeling of thermoelectric MHD has advanced significantly as a result of this work, providing a more thorough and precise tool for forecasting the behavior of these materials under a range of thermal and magnetic conditions.

中文翻译：

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用于分析热电 MHD 的改进 Green-Naghdi 分数阶模型

 目的

然而，经典的 Green-Naghdi II 型 (GN-II) 模型由于无法解释记忆效应，因此难以准确表示热电 MHD 的热机械行为。 GN-II 理论的新数学模型结合了热传输的分数阶来解决这个问题。

设计/方法论/途径

矩阵指数方法的使用是该策略的核心，它构成了当代理论中状态空间方法的基础。由此产生的公式与拉普拉斯变换技术一起应用于各种问题。获得了热冲击问题和没有热源的层介质问题的解决方案。此外，还考虑了热源分布的问题。利用数值技术实现拉普拉斯变换反演。

 发现

根据数值结果及其图表，新理论研究了分数阶参数、品质因数、热电势和珀耳帖系数对场量行为的影响。

 原创性/价值

由于这项工作，热电 MHD 的新模型取得了显着进展，为预测这些材料在一系列热和磁条件下的行为提供了更全面、更精确的工具。