Difference Methods for Boundary Value Problems in Ordinary Differential Equations

Abstract

A general theory of difference methods for problems of the form \[ \mathcal{N}{\bf y} \equiv {\bf y}' - {\bf f}(t,{\bf y}) = 0,\quad a \leqq t \leqq b,\quad {\bf g}({\bf y}(a),{\bf y}(b)) = 0, \] is developed. On nonuniform nets, $t_0 = a$, $t_j = t_{j - 1} + h_j $, $1 \leqq j \leqq J$, $t_J = b$, schemes of the form \[ \mathcal{N}_h {\bf u}_j \equiv {\bf G}_j ({\bf u}_0 , \cdots ,{\bf u}_J ) = 0,\quad 1 \leqq j \leqq J,\quad {\bf g}({\bf u}_0 ,{\bf u}_J ) = 0 \] are considered. For linear problems with unique solutions, it is shown that the difference scheme is stable and consistent for the boundary value problem if and only if, upon replacing the boundary conditions by an initial condition, the resulting scheme is stable and consistent for the initial value problem. For isolated solutions of the nonlinear problem, it is shown that the difference scheme has a unique solution converging to the exact solution if (i) the linearized difference equations are stable and consistent for the linearized initial value problem, (ii) the linearized difference operator is Lipschitz continuous, (iii) the nonlinear difference equations are consistent with the nonlinear differential equation. Newton’s method is shown to be valid, with quadratic convergence, for computing the numerical solution.