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.

Keywords

• third type of Volterra integral equations
• Lagrange polynomial
• modified Lagrange polynomial
• barycentric Lagrange polynomial