ABSTRACT
This chapter presents some examples of ordinary differential equation-boundary value problems (ODE-BVPs) and solves them using a finite difference approach. It introduces the problems involving such banded structures, followed by so-called direct methods to solve the problems. Direct methods are related to Gauss Elimination, but where the banded structure of matrix is exploited to solve the linear problem more efficiently. The chapter discusses iterative methods for solving such problems, where both linear as well as nonlinear equations of the form will be discussed. It expands the linear banded diagonal methods to specific examples of nonlinear equations. Heat conduction in a 2D geometry is a standard problem that leads to the elliptic partial differential equation (PDE). Gauss-Siedel method is the linear equations equivalent of fixed point iteration. The convergence of fixed point iteration for tridiagonal systems can sometimes be slow, or the method may not be robust.
