Pan Ernian
University of Akron, Akron, OH 44325, United States of America.
Rep Prog Phys. 2019 Oct;82(10):106801. doi: 10.1088/1361-6633/ab1877. Epub 2019 Apr 11.
The Green's function (GF) method, which makes use of GFs, is an important and elegant tool for solving a given boundary-value problem for the differential equation from a real engineering or physical field. Under a concentrated source, the solution of a differential equation is called a GF, which is singular at the source location, yet is very fundamental and powerful. When looking at the GFs from different physical and/or engineering fields, i.e. assigning the involved functions to real physical/engineering quantities, the GFs can be scaled and applied to large-scale problems such as those involved in Earth sciences as well as to nano-scale problems associated with quantum nanostructures. GFs are ubiquitous and everywhere: they can describe heat, water pressure, fluid flow potential, electromagnetic (EM) and gravitational potentials, and the surface tension of soap film. In the undergraduate courses Mechanics of Solids and Structural Analysis, a GF is the simple influence line or singular function. Dropping a pebble in the pond, it is the circular ripple traveling on and on. It is the wave generated by a moving ship in the opening ocean or the atom vibrating on a nanoscale sheet induced by the atomic force microscopy. In Earth science, while various GFs have been derived, a comprehensive review is missing. Thus, this article provides a relatively complete review on GFs for geophysics. In section 1, the George Green's potential functions, GF definition, as well as related theorems and basic relations are briefly presented. In section 2, the boundary-value problems for elastic and viscoelastic materials are provided. Section 3 is on the GFs in full- and half-spaces (planes). The GFs of concentrated forces and dislocations in horizontally layered half-spaces (planes) are derived in section 4 in terms of both Cartesian and cylindrical systems of vector functions. The corresponding GFs in a self-gravitating and layered spherical Earth are presented in section 5 in terms of the spherical system of vector functions. The singularity and infinity associated with GFs in layered systems are analyzed in section 6 along with a brief review of various layer matrix methods. Various associated mathematical preliminaries are listed in appendix, along with the three sets of vector function systems. It should be further emphasized that, while this review is targeted at geophysics, most of the GFs and solution methods can be equally applied to other engineering and science fields. Actually, many GFs and solutions methods reviewed in this article are derived by engineers and scientists from allied fields besides geophysics. As such, the updated approaches of constructing and deriving the GFs reviewed here should be very beneficial to any reader.
格林函数（GF）方法利用格林函数，是求解实际工程或物理领域中微分方程给定边值问题的一种重要且精妙的工具。在集中源作用下，微分方程的解被称为格林函数，它在源位置处是奇异的，但却非常基础且强大。当从不同的物理和/或工程领域审视格林函数时，即把所涉及的函数赋予实际的物理/工程量，格林函数可以进行缩放并应用于大规模问题，比如地球科学中涉及的那些问题，以及与量子纳米结构相关的纳米尺度问题。格林函数无处不在：它们可以描述热、水压、流体流动势、电磁（EM）和引力势，以及肥皂膜的表面张力。在本科课程《固体力学》和《结构分析》中，格林函数就是简单的影响线或奇异函数。往池塘里扔一块石子，它就是不断向外传播的圆形涟漪。它是开阔海洋中行驶船只产生的波浪，或者是原子力显微镜在纳米尺度薄片上引起的原子振动。在地球科学领域，虽然已经推导了各种格林函数，但还缺少全面的综述。因此，本文对地球物理学中的格林函数进行了相对完整的综述。在第1节中，简要介绍了乔治·格林的势函数、格林函数定义以及相关定理和基本关系。第2节给出了弹性和粘弹性材料的边值问题。第3节讨论全空间和半空间（平面）中的格林函数。第4节在笛卡尔向量函数系统和柱面向量函数系统中，推导了水平分层半空间（平面）中集中力和位错的格林函数。第5节在球向量函数系统中，给出了自引力分层球形地球中的相应格林函数。第6节分析了分层系统中格林函数的奇异性和无穷性，并简要回顾了各种层矩阵方法。附录中列出了各种相关的数学预备知识，以及三组向量函数系统。需要进一步强调的是，虽然本综述针对地球物理学，但大多数格林函数和求解方法同样可应用于其他工程和科学领域。实际上，本文中综述的许多格林函数和求解方法是由地球物理学以外的相关领域的工程师和科学家推导出来的。因此，这里综述的构建和推导格林函数的更新方法应该对任何读者都非常有益。
