ABSTRACT
The mathematical modeling of a physical problem often gives rise to ordinary or partial differential equations satisfying a set of initial conditions and/or boundary conditions. It is very essential for one to understand the physical system one is trying to investigate and apply the most suitable approach for analysing the physical system in an efficient manner. Understanding the advantages and limitations of a method is very essential before applying the method. The mathematical models for a large class of physical problems lead to boundary value problems or initial boundary value problems associated with either ordinary differential equations (ODEs) or partial differential equations (PDEs). Methods of separation of variables are applied directly when the differential equation is homogeneous satisfying homogeneous boundary conditions. To the contrary, there is a large class of physical problems which lead to differential equations with a forcing term satisfying a set of initial conditions and/or nonhomogeneous boundary conditions. Such problems are often handled with the help of Green’s function which is used to describe the influence of both nonhomogeneous boundary conditions and forcing terms. Often, it is convenient to provide the integral representation of the boundary value problem (BVP), which incorporates most of the boundary conditions by the suitable application of Green’s identity and Green’s function which is referred as Green’s function technique, named after George Green (1793−1841). In this chapter, emphasis is given to the application of Green’s function to derive the expansion formulae for wave structure interaction problems. Further, the utility of the expansion formulae is illustrated by analysing wave structure interaction problems in specific cases. Before going into the details of the application of Green’s function in wave structure interaction problems, some of the basic characteristics and underlying procedures associated with Green’s function techniques are discussed in brief.
