An iterative approximate method of solving boundary value problems using dual Bernstein polynomials

TL;DR

This paper introduces an efficient iterative method for solving boundary value problems by approximating solutions with dual Bernstein polynomials, applicable to linear and nonlinear, as well as higher order differential equations.

Contribution

It presents a novel approach combining least squares approximation with dual Bernstein polynomials for high-efficiency boundary value problem solutions.

Findings

Effective for linear and nonlinear differential equations

Applicable to second and higher order boundary value problems

Demonstrated versatility through illustrative examples

Abstract

In this paper, we present a new iterative approximate method of solving boundary value problems. The idea is to compute approximate polynomial solutions in the Bernstein form using least squares approximation combined with some properties of dual Bernstein polynomials which guarantee high efficiency of our approach. The method can deal with both linear and nonlinear differential equations. Moreover, not only second order differential equations can be solved but also higher order differential equations. Illustrative examples confirm the versatility of our method.