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

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.

Linearization-Discretization process to solve systems of nonlinear Fredholm integral equations in an infinite-dimensional context

“…In this section, we study the convergence of Option A, where we will prove that our approximate solution (12), converges to the exact solution U = (u 1 , u 2 , ..., u N ) ∈ χ dened in (1).…”