Direct numerical scheme for all classes of nonlinear Volterra integral equations of the first kind

TL;DR

This paper introduces a direct, efficient numerical scheme using operational vectors for solving all types of nonlinear Volterra integral equations of the first kind, emphasizing low computational cost and high accuracy.

Contribution

The paper develops a novel direct numerical method based on operational vectors for all classes of nonlinear Volterra equations of the first kind, avoiding projection methods.

Findings

The scheme achieves high accuracy in approximating solutions.

It has lower computational cost compared to existing methods.

Error analysis confirms the scheme's efficiency and superiority.

Abstract

This paper presents a direct numerical scheme to approximate the solution of all classes of nonlinear Volterra integral equations of the first kind. This computational method is based on operational matrices and vectors. The operational vector for hybrid block pulse functions and Chebyshev polynomials is constructed. The scheme transforms the integral equation to a matrix equation and solves it with a careful estimate of the error involved. The main characteristic of the scheme is the low cost of setting up the equations without using any projection method which is the consequence of using operational vectors. Simple structure to implement, low computational cost and perfect approximate solutions are the major points of the presented method. Error analysis and comparisons with other existing schemes demonstrate the efficiency and the superiority of our scheme.