# Dynamic Programming Method for Best Piecewise Linear Approximation for Vector Field of Nonlinear Boundary Value Problems on the Interval [0, 1]

## TL;DR

This paper introduces a dynamic programming approach for optimally approximating vector fields in nonlinear boundary value problems, ensuring boundary condition adherence and numerical stability, with convergence guarantees under finer discretization.

## Contribution

It presents a novel dynamic programming method for piecewise linear approximation of vector fields in boundary value problems, emphasizing stability and convergence.

## Key findings

- Method guarantees convergence to true solution with finer discretization
- Ensures boundary conditions are maintained during approximation
- Provides a stable computational framework for nonlinear BVPs

## Abstract

An important problem that arises in many engineering applications is the boundary value problem for ordinary differential equations. There have been many computational methods proposed for dealing with this problem. The convergence of the iterative schemes to a true solution, when one such exists, and their numerical stability are the central issues discussed in the literature. In this paper, we discuss a method for approximating the vector field, maintaining the boundary conditions and numerical stability. If a true solution exists, a subsequence of solutions convergent to one such can be produced, by finer discretization of the solution space.