A new approach to find an approximate solution of linear initial value problems

TL;DR

This paper introduces a novel method using Bernoulli polynomials and operational matrices to efficiently approximate solutions of linear initial value problems, demonstrating high accuracy across multiple test cases.

Contribution

It presents a new approach combining orthonormal Bernoulli polynomials and operational matrices for solving linear initial value problems.

Findings

High accuracy in approximate solutions

Effective for different problem types

Compared favorably with existing solutions

Abstract

This work investigates a new approach to find closed form analytical approximate solution of linear initial value problems. Classical Bernoulli polynomials have been used to derive a finite set of orthonormal polynomials and a finite operational matrix to simplify derivatives of dependent variable. These orthonormal polynomials together with the operational matrix of relevant order provides a good approximation to the solution of a linear initial value problem. Depending upon the nature of a problem, a series form approximation or numerical approximation can be obtained. The technique has been demonstrated through three problems. Approximate solutions have been compared with available exact or other numerical solutions. High degree of accuracy has been noted in numerical values of solutions for considered problems.