ABSTRACT
Laplace’s equation in two dimensions takes the form of the linear, elliptic, and homogeneous partial differential equation
where r2 is the Laplacian operator in the xy plane defined as
: (2.2)
A function that satisfies (2.1) is called harmonic. To compute the solution of (2.1) in a certain solution domain, we require boundary conditions for the unknown function f (Dirichlet boundary condition), its normal derivative (Neumann boundary condition), or a combination thereof (mixed or Robin boundary condition) along overlapping or complementary parts of the boundary. Following the analysis of Chapter 1, we proceed to formulate the solution in terms of an integral equation by working in four stages. First, we discuss Green’s first and second identities and the reciprocal relation. Second, we introduce the Green’s functions of Laplace’s equation in two dimensions. Third, we develop a boundaryintegral representation. Finally, we derive an integral equation that we then solve using the boundary-element method.
