Application of orthonormal Bernoulli polynomials for approximate solution of some Volterra integral equations

TL;DR

This paper introduces a novel numerical method using orthonormal Bernoulli polynomials to efficiently solve linear Volterra integral equations with high accuracy, demonstrated through multiple examples.

Contribution

The paper develops a new approach employing orthonormal Bernoulli polynomials and operational matrices for solving Volterra integral equations, enhancing solution accuracy and efficiency.

Findings

High accuracy of numerical solutions demonstrated

Effective for non-singular Volterra equations

Method outperforms traditional approaches

Abstract

In this work, a new approach has been developed to obtain numerical solution of linear Volterra type integral equations by obtaining asymptotic approximation to solutions. Using the classical Bernoulli polynomials, a set of orthonormal polynomials have been derived, and these orthonormal polynomials have been used to form an operational matrix of integration which is has been implemented to find numerical or exact solution of non-singular Volterra integral equations. Two linear Volterra integral and two convolution integral equations of second kind have been solved to demonstrate the effectiveness of present method. Obtained approximate solutions have been compared with the exact solutions for numerical values. High degree of accuracy of numerical solutions has established the credibility of the present method.