Barenblatt-Sobolev-Galpern伪抛物型方程界面问题的积分表示
Integral Representations for Interface Problems for the Barenblatt-Sobolev-Galpern Pseudoparabolic Equation
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中文总结 AI 辅助
该研究针对Barenblatt-Sobolev-Galpern伪抛物型方程的初边值及界面问题，采用统一变换法推导了复傅里叶平面围道积分形式的新型积分表示，其结果可用于解的定性性质探究及相关非线性、相变等问题研究。
中文摘要 AI 辅助
我们针对定义在实直线、半直线和有限区间上的Barenblatt-Zheltov-Kochina Sobolev-Galpern型伪抛物型方程，其完全非齐次初边值问题及界面问题的解，获得了以复傅里叶平面上的围道积分形式表达的新型积分表示。该数学物理领域的基础偏微分方程（PDE）广泛出现于连续介质力学、热力学、化学工程、固态电子学、半导体器件、电池研究及纳米技术等多种自然现象与应用科学中。本研究全程采用一种称为统一变换法（Unified Transform Method）的现代方法，由于该PDE存在高阶混合导数项，且所考虑的问题具有一般性，因此面临特殊挑战。边界与界面条件看似非标准，但实则由PDE本身的结构自然决定。我们的显式解析公式可直接用于后续探究解的定性性质，如渐近行为、时空动力学、正则性及适定性。本研究预计也将对非线性对应问题的研究，以及根据能量平衡定律动态演化的相变现象和自由边界问题的研究具有实用价值。
英文摘要
We obtain novel integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous initial-boundary-value as well as interface problems for the Barenblatt-Zheltov-Kochina pseudo-parabolic equation of Sobolev-Galpern type formulated on the real line, half-line and finite interval. This fundamental partial differential equation (PDE) of mathematical physics emerges in a wide variety of natural phenomena and applied sciences including continuum mechanics, thermodynamics, chemical engineering, solid-state electronics, semi-conductor devices, battery research, and nanotechnology. A suitable implementation of a modern methodology, known as Unified Transform Method, is in force throughout this study, with particular challenges arising due to the higher-order mixed-derivative term of the PDE and the generality of the problems under consideration altogether. The boundary and interface conditions appear to be non-standard but are naturally dictated by the structure of the PDE itself. Our explicit analytical formulae directly lend themselves to future explorations of the solutions' qualitative properties such as asymptotic behavior, spatio-temporal dynamics, regularity and well-posedness. This work is expected to be of utility also in the investigation of nonlinear counterparts as well as towards the study of phase-transition phenomena and free-boundary problems, where the interface evolves dynamically according to energy balance laws.
