Integral representations for the double-diffusivity system on the half-line
Integral representations for the double-diffusivity system on the half-line
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DOI：
10.1007/s00033-023-02174-8
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发表时间：
2024-04-01
影响因子：
2
通讯作者：
Fokas，Athanassios S.
中科院分区：
文献类型：
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作者：
Chatziafratis，Andreas;Aifantis，Elias C.;Fokas，Athanassios S.
提出了一种显式求解一类耦合发展偏微分方程组半直线上非齐次初边值问题的新方法。所谓的双扩散模型，它是基于一个简单的，但一般的，不均匀的扩散配置，准确地描述了几个重要的物理和机械过程，因此出现在杂项应用，从材料科学，热质传输和固体-流体动力学，石油和化学工程。例如，它出现在纳米技术中，其非均匀版本最近出现在锂离子可充电电池领域。我们的方法是基于扩展的统一变换（也称为Fokas方法），使其可以应用到耦合方程组。首先，我们推导出形式上有效的解表示，然后严格地证明其有效性。这包括重建规定的初始和边界条件，这需要仔细分析公式中出现的各种积分项，证明它们在严格定义的意义上收敛。新的解决方案公式也被用来严格推导解决方案的规则性附近的时空域的边界。特别是，我们证明了一致收敛的解决方案的数据，它的快速衰减在无穷大，以及它的光滑度（和超越）的边界轴，提供一定的数据兼容性条件在四分之一平面角得到满足。作为我们的分析和调查的解决方案及其衍生物的边界行为的重要应用的一个样本，我们都证明了一个新的唯一性定理，并构造了一个“非唯一性反例”。在适定性的框架内，这些补充了前面的“建设性存在”结果。此外，统一变换的优点之一是，它产生定义在复傅立叶平面中的轮廓上的表示，其对于大的值表现出指数衰减。这一重要特性的解决方案，预计将允许一个有效的数值评估，这是设想在未来的数值分析调查。新的公式和本文报道的研究结果也有望在有关非线性对应的适定性问题的研究中找到实用性。此外，我们的严格的方法可以扩展到IBVP的其他重要的数学物理模型，也可能更高的维度和变系数的情况下。
A novel method is presented for explicitly solving inhomogeneous initial-boundary-value problems (IBVPs) on the half-line for a well-known coupled system of evolution partial differential equations. The so-called double-diffusion model, which is based on a simple, yet general, inhomogeneous diffusion configuration, describes accurately several important physical and mechanical processes and thus emerges in miscellaneous applications, ranging from materials science, heat-mass transport and solid–fluid dynamics, to petroleum and chemical engineering. For instance, it appears in nanotechnology and its inhomogeneous version has recently appeared in the area of lithium-ion rechargeable batteries. Our approach is based on the extension of the unified transform (also called the Fokas method), so that it can be applied to systems of coupled equations. First, we derive formally effective solution representations and then justifya posterioritheir validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. The novel solution formulae are also utilized to rigorously deduce the solution’s regularity properties near the boundaries of the spatiotemporal domain. In particular, we prove uniform convergence of the solution to the data, its rapid decay at infinity as well as its smoothness up to (and beyond) the boundary axes, provided certain data compatibility conditions at the quarter-plane corner are satisfied. As a sample of important applications of our analysis and investigation of the boundary behavior of the solution and its derivatives, we both prove a novel uniqueness theorem and construct a ‘non-uniqueness counterexample’. These supplement the preceding ‘constructive existence’ result, within the framework of well-posedness. Moreover, one of the advantages of the unified transform is that it yields representations which are defined on contours in the complex Fourier-plane, which exhibit exponential decay for large values of. This important characteristic of the solutions is expected to allow for an efficient numerical evaluation; this is envisaged in future numerical-analytic investigations. The new formulae and the findings reported herein are also expected to find utility in the study of questions pertaining to well-posedness for nonlinear counterparts too. In addition, our rigorous approach can be extended to IBVPs for other significant models of mathematical physics and potentially also to higher-dimensional and variable-coefficient cases.