A new operational matrix technique to solve linear boundary value problems

TL;DR

This paper introduces a novel operational matrix method using modified Bernoulli polynomials to efficiently solve linear boundary value problems with high accuracy and simplicity.

Contribution

The paper develops a new orthonormal polynomial set and operational matrix approach for solving linear BVPs, improving simplicity and accuracy over existing methods.

Findings

Method transforms BVP into algebraic system for polynomial solutions.

Achieves high accuracy with simpler computational process.

Validated on four problems with solutions matching exact and numerical results.

Abstract

A new technique is presented to solve a class of linear boundary value problems (BVP). Technique is primarily based on an operational matrix developed from a set of modified Bernoulli polynomials. The new set of polynomials is an orthonormal set obtained with Gram-Schmidt orthogonalization applied to classical Bernoulli polynomials. The presented method changes a given linear BVP into a system of algebraic equations which is solved to find an approximate solution of BVP in form of a polynomial of required degree. The technique is applied to four problems and obtained approximate solutions are graphically compared to available exact and other numerical solutions. The method is simpler than many existing methods and provides a high degree of accuracy.