Boundary-Value Problems and Applications
In 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 Problem Laplace Equation Neumann Problem Continuity Theorem Robin Problem
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