Boundary Value Problems and Initial-Boundary Value Problems

In the earlier chapters, we have analyzed several prototype potential field problems, including potential fluid flows, steady state temperature distribution, electrostatics problems and gravitational potential problems. All of these potential field problems are governed by the Laplace equation. There is no time variable in these problems, and the characterization of individual physical problems is exhibited by the corresponding prescribed boundary conditions. The mathematical problem of finding the solution of a partial differential equation that satisfies the prescribed boundary conditions is called a boundary value problem , of which there are two main types: Dirichlet problems where the boundary values of the solution function are prescribed, and Neumann problems where the values of the normal derivative of the solution function along the boundary are prescribed. In other physical problems, like the heat conduction and wave propagation models, the time variable is also involved in the model. To describe fully the partial differential equations modeling these problems, one needs to prescribe both the associated boundary conditions and the initial conditions. The latter class is called an initial-boundary value problem . This chapter discusses some of the solution methodologies for solving boundary value problems and initial-boundary value problems using complex variables methods. The link between analytic functions and harmonic functions is exhibited by the fact that both the real and imaginary parts of a complex function that is analytic inside a domain satisfy the Laplace equation in the same domain.