Chapter 3 - Pinning It Down: Boundary Value Problems

Abstract

This chapter discusses boundary value problems. For a single first order equation y' = f(x,y), data at one point completely determines the solution so that if conditions at more than one point are given, either higher order equations or systems of equations must be treated. The second order equation requires two additional conditions, and the simplest possibility is to prescribe the solution values at the end-points. If the function f is nonlinear, the boundary value problem is nonlinear. Diagonal dominance is important for a number of reasons, one of which is that it is an approach to showing nonsingularity of the matrix. Diagonal dominance by itself is not sufficient. Neumann boundary conditions also lead to tridiagonal systems of equations, but periodic boundary conditions introduce matrix elements outside of the tridiagonal part.