Fundamental Solution and Integral Equation

Abstract

In the mathematical terminology, a fundamental solution is a singular solution of a linear partial differential equation that is not required to satisfy boundary conditions. A Green’s function, on the other hand, is the singular solution tied to a certain domain geometry and boundary condition. For this reason, a fundamental solution is also called a free space Green’s function. Using the symbolism of a generalized function, known as the Dirac delta function, δ, a fundamental solution is a solution of linear partial differential equation with the Dirac delta as its right hand side. From physical considerations, Dirac delta can be used to approximate a forcing applied to a small region. For example, a fluid mass injected through an injection well, whose radius is small compared to the formation that it is flooding, can be considered as a point source. Similarly, a force acting on a small area can be considered as a point force, and a defect in crystal structure (dislocation) or a local slippage of a geological fault can be approximated as a displacement discontinuity, etc. Hence fundamental solution is not just an abstract mathematical construct; it has its root in physics and can be used to model physical phenomena. The use of fundamental solution to simulate a physical phenomenon and to solve mathematical problems can be traced to George Green (An essay on the application of mathematical analysis to the theories of electricity and magnetism, printed for the author, by T. Wheelhouse, Nottingham, 1828) (see Sect. F.7 for a biography). Green utilized the fundamental solution 1∕r of the Laplace equation to model the electrical and magnetic potential created by concentrated electrostatic and magnetic charges. The potential is a mathematical construct, whose derivative gives the force associated with the field. The use of fundamental solution to solve mathematical problems is closely tied to the integral equations known as Green’s identities. Particularly, the third identify, shown in its original form in the prologue of the chapter, provides the general solution to the boundary value problem known as the Dirichlet problem. In this chapter, we extend the classical work on fundamental solutions and integral equations for potential (satisfying Laplace equation) and elasticity problems to poroelasticity. We shall demonstrate that the several varieties of integral equations derived from different origins and the many different fundamental solutions are intricately related. The presentation begins with the integral equations for general anisotropic poroelastic materials. The fundamental solutions are then derived, but only for the case of isotropy. The presentation follows the work of Cheng and Detournay (Int J Solids Struct 35(34–35):4521–4555, 1998).