论文标题
复杂的傅立叶变换域中的波浪分析：一种获得绿色功能的新方法
Wave analysis in the complex Fourier transform domain: A new method to obtain the Green's functions of dispersive linear partial differential equations
论文作者
论文摘要
本文提供了一种新的分析方法，以获得Green的线性分散偏微分方程的功能。在时空和时间的冲动下，Euler-Bernoulli束方程和一维热传导方程（耗散方程）被用作示例。得出了复杂的无限域绿色的绿色功能。提出了一种新方法，以通过复杂的傅立叶变换域中的反射和传输分析从无限域绿色的函数中获得有限域绿色的功能。发现通过这种方法获得的解决方案在短响应时间与传统模态分析获得的解决方案要好得多。此外，通过应用矩阵系列的几何求和公式，一个新的模态扩展解决方案不需要计算每种模式的内部产品，从而在分析上证明波模式二元性并简化了计算。半侵入域病例和耦合域病例也是通过新开发的方法得出的，以显示其有效性和简单性。发现非传播波还具有波速，并且热传导也可以视为传播波
This paper provides a new analytical method to obtain Green's functions of linear dispersive partial differential equations. The Euler-Bernoulli beam equation and the one-dimensional heat conduction equation (dissipation equation) under impulses in space and time are solved as examples. The complex infinite-domain Green's function of the Euler-Bernoulli beam is derived. A new approach is proposed to obtain the finite-domain Green's function from the infinite-domain Green's function by the reflection and transmission analysis in the complex Fourier transform domain. It is found that the solution obtained by this approach converges much better at short response times compared with that obtained by the traditional modal analysis. Besides, by applying the geometric summation formula for matrix series, a new modal expansion solution requiring no calculation of each mode's inner product is derived, which analytically proves the wave-mode duality and simplifies the calculation. The semi-infinite-domain cases and the coupled-domain cases are also derived by the newly developed method to show its validity and simplicity. It is found that the non-propagating waves also possess wave speed, and heat conduction can also be treated as propagating waves
