A new technique to solve linear integro-differential equations (IDEs) with modified Bernoulli polynomials

TL;DR

This paper introduces a novel method using modified Bernoulli polynomials and operational matrices to efficiently approximate solutions of linear integro-differential equations, demonstrating high accuracy and simplicity.

Contribution

The paper presents a new technique that simplifies and improves the approximation of linear IDEs using modified Bernoulli polynomials and operational matrices.

Findings

High accuracy in numerical solutions compared to exact solutions

Method reduces computational complexity and length

Error of approximation can be minimized to desired level

Abstract

In this work, a new technique has been presented to find approximate solution of linear integro-differential equations. The method is based on modified orthonormal Bernoulli polynomials and an operational matrix thereof. The method converts a given integro-differential equation into a set of algebraic equations with unknown coefficients, which is easily obtained with help of the known functions appearing in the equation, modified Bernoulli polynomials and operational matrix. Approximate solution is obtained in form of a polynomial of required degree. The method is also applied to three well known integro-differential equations to demonstrated the accuracy and efficacy of the method. Numerical results of approximate solution are plotted to compared with available exact solutions. Considerably small error of approximation is observed through numerical comparison, which is further reducible to a required level of significance. Method is comparatively simpler and shorter than many existing methods.