Numeric solution of systems of nonlinear Volterra integral equations of the first kind with discontinuous kernels

TL;DR

This paper introduces and analyzes numerical methods for solving nonlinear Volterra integral equations of the first kind with discontinuous kernels, including an iterative approach and polynomial collocation, with proven convergence and demonstrated effectiveness.

Contribution

The paper proposes a novel iterative numerical method and an alternative polynomial collocation technique for these complex integral equations, with theoretical convergence proof.

Findings

The iterative method converges under specified conditions.

The polynomial collocation method provides accurate solutions.

Numerical examples confirm the efficiency of both methods.

Abstract

The systems of nonlinear Volterra integral equations of the first kind with jump discontinuous kernels are studied. The iterative numerical method for such nonlinear systems is proposed. Proposed method employs the modified Newton-Kantorovich iterative process for the integral operators linearization. On each step of the iterative process the linear system of integral equations is obtained and resolved using the discontinuity driven piecewise constant approximation of the exact solution. The convergence theorem is proved. The polynomial collocation technique is also applied to solve such systems as the alternative method. The accuracy of proposed numerical methods is discussed. The model examples are examined in order to demonstrate the efficiency of proposed numerical methods and illustrate the constructed theory.