Boundary-Value Problems and Applications

Abstract

AbstractIn the preceding chapters, we have treated the initial-value and initial boundary-value problems. In this chapter, we shall be concerned with boundary-value problems. Mathematically, a boundary-value problem is finding a function which satisfies a given partial differential equation and particular boundary conditions. Physically speaking, the problem is independent of time, involving only space coordinates. Just as initial-value problems are associated with hyperbolic partial differential equations, boundary-value problems are associated with partial differential equations of elliptic type. In marked contrast to initial-value problems, boundary-value problems are considerably more difficult to solve. This is due to the physical requirement that solutions must hold in the large unlike the case of initial-value problems, where solutions in the small, say over a short interval of time, may still be of physical interest.KeywordsDirichlet ProblemLaplace EquationNeumann ProblemContinuity TheoremRobin ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.