ABSTRACT
We will discuss mathematical models involving potential fields and related Laplace’s, Pois-son’s, and other equations, which are encountered in different flow fields in continuum mechanics and physics. There are different methods to solve boundary value problems involving these equations, namely, analytic methods including Green’s function, conformal mapping method, and numerical approximations using finite and boundary elements. In this chapter we will confine to the following two-dimensional equations: Laplace’s, Poisson’s, Helmholtz, biharmonic, and membrane equations, and provide their solutions in different domains in the (x, y)-plane, including related Green’s functions; some examples are also provided. Conformal mapping methods will be discussed in subsequent chapters.
