A new operational matrix based on Bernoulli polynomials

TL;DR

This paper introduces a novel operational matrix based on Bernoulli polynomials for solving differential and integral equations, demonstrating its simplicity, efficiency, and accuracy through numerical examples.

Contribution

The paper develops a new operational matrix using Bernoulli polynomials and applies it to transform differential equations into algebraic systems, enhancing computational methods.

Findings

Method is computationally simple

Demonstrates high accuracy in numerical examples

Applicable to various differential and integral equations

Abstract

In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized to transform the differential equation to a matrix equation which corresponds to a system of algebraic equations with unknown Bernoulli coefficients. This method can be used for many problems such as differential equations, integral equations and so on. Numerical examples show the method is computationally simple and also illustrate the efficiency and accuracy of the method.