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Numerical resolution of some BVP using Bernstein polynomials

Gianluca Argentini

TL;DR

This work addresses the numerical solution of boundary value problems by a Bernstein polynomial based method that avoids finite differences, finite elements, matrices, and Newton iterations. It reformulates differential equations in a parametric Bézier framework, initializes with a pivot-driven mid-point control, and iteratively refines the solution via a collocation-like error control. The method is demonstrated on linear and nonlinear BVPs, including a linear ODE, a nonlinear equation with a quadratic term, and a growth-led nonlinear problem, producing high-degree Bézier representations that closely track benchmark solutions and reveal qualitative behavior such as blow-up. Technical notes document a Mathematica implementation and provide access to notebooks, highlighting practical viability for Bernstein-based BVP solvers.

Abstract

In this work we present a method, based on the use of Bernstein polynomials, for the numerical resolution of some boundary values problems. The computations have not need of particular approximations of derivatives, such as finite differences, or particular techniques, such as finite elements. Also, the method doesn't require the use of matrices, as in resolution of linear algebraic systems, nor the use of like-Newton algorithms, as in resolution of non linear sets of equations. An initial equation is resolved only once, then the method is based on iterated evaluations of appropriate polynomials.

Numerical resolution of some BVP using Bernstein polynomials

TL;DR

This work addresses the numerical solution of boundary value problems by a Bernstein polynomial based method that avoids finite differences, finite elements, matrices, and Newton iterations. It reformulates differential equations in a parametric Bézier framework, initializes with a pivot-driven mid-point control, and iteratively refines the solution via a collocation-like error control. The method is demonstrated on linear and nonlinear BVPs, including a linear ODE, a nonlinear equation with a quadratic term, and a growth-led nonlinear problem, producing high-degree Bézier representations that closely track benchmark solutions and reveal qualitative behavior such as blow-up. Technical notes document a Mathematica implementation and provide access to notebooks, highlighting practical viability for Bernstein-based BVP solvers.

Abstract

In this work we present a method, based on the use of Bernstein polynomials, for the numerical resolution of some boundary values problems. The computations have not need of particular approximations of derivatives, such as finite differences, or particular techniques, such as finite elements. Also, the method doesn't require the use of matrices, as in resolution of linear algebraic systems, nor the use of like-Newton algorithms, as in resolution of non linear sets of equations. An initial equation is resolved only once, then the method is based on iterated evaluations of appropriate polynomials.

Paper Structure

This paper contains 6 sections, 13 equations, 3 figures.

Table of Contents

  1. Basic concepts
  2. The algorithm of Bernstein polynomials
  3. A BVP with a linear differential equation
  4. A BVP with a non linear differential equation
  5. A more subtle BVP
  6. Technical notes

Figures (3)

  • Figure 1: The continuous line is the graphic of the exact solution, while dots are the cartesian pairs of the numerical solution as in previous table.
  • Figure 2: The continuous line is the graphic of the exact solution, while dots are the cartesian pairs of the numerical solution as in previous table.
  • Figure 3: The continuous line is the graphic obtained by the Mathematica function NDSolve, while dots are the cartesian pairs of the solution provided by the method of Bernstein polynomials.