Numerical Solutions of Volterra Integral Equations of Third Kind and Its Convergence Analysis

Abstract

The current work suggests a method for the numerical solution of the third type of Volterra integral equations (VIEs), based on Lagrange polynomial, modified Lagrange polynomial, and barycentric Lagrange polynomial approximations. To do this, the interpolation of the unknown function is considered in terms of the above polynomials with unknown coefficients. By substituting this approximation into the considered equation, a system of linear algebraic equations is obtained. Then, we demonstrate the method’s convergence and error estimations. The proposed approaches retain the possible singularity of the solution. To the best of the authors’ knowledge, the singularity case has not been addressed by researchers yet. To illustrate the applicability, effectiveness, and correctness of new methods for the proposed integral equation, examples with both types of kernels, symmetric as well as non-symmetric, are provided at the end.

An Iterative Approach to Solve Volterra Nonlinear Integral Equations

“…Solution The exact solution of equation ( 12) is φ (τ ) = 2τ . To get the solution by ADM, we apply Aboodh transform to equation (12) to get…”

“…The development of integral equations was further facilitated by mathematical physics models of diffraction issues, astrophysics, quantum mechanics scattering, conformal mapping, and water waves [4,5,7,9,26]. Moreover, the study of nonlinear IDEs has appeared in many fields of science, because of the great number of applications that could describe, such as chemical kinetics, queuing theory, and others [12,21,27,28,34]. Thus researchers have developed many techniques to handle these problems such as He's homotopy perturbation method [37], variation iteration method [14], least square method [8], decomposition method [13] and others.…”

Precision and efficiency of an interpolation approach to weakly singular integral equations

“…To address the singularity of the solution, they transform the kernel of the two types of discussed equations; for the first kind of Fredholm integral equations, the parametrization of the kernel has been done, while for the second kind of Volterra integral equations, the interpolation of the kernel has been done twice, respectively. To solve second-kind weakly singular Volterra integral equations (WSVIEs), various academic papers have been published (Bhat and Mishra, 2022; He et al , 2022; Hou et al , 2019; Ramadan et al , 2020). We examine some of these methods.…”

On the solvability of a class of nonlinear functional integral equations involving Erdélyi–Kober fractional operator

“…The practical applications of nonlinear integral equations have recently received a lot of attention from studies that incorporate the same in distinct areas of knowledge that include mathematical modeling of real-world problems in various branches of science, like chemistry, physics, electrical networks, control of the dynamic system, optics, biological science, signal processing, and acoustic scattering [1][2][3][4][5][6][7]. Precisely, the recent development on these equations is focused on their solutions by using the measure of noncompactness technique [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”