An Iterative Method for Solving the Inverse Problem for an Integral Dynamic Model with a Discontinuous Kernel

The paper addresses an inverse problem for a nonlinear Volterra integral equation of the first kind with a piecewise continuous kernel whose discontinuity curves are the unknown functions. Such models arise in the theory of developing systems, power systems with energy storage, and related applications. We develop an iterative scheme based on the Newton–Kantorovich linearisation of the nonlinear integral operator and obtain explicit recurrent formulas for the discontinuity curve. Both the full Newton-like and a modified (simplified) iterative process are constructed, and their local convergence is proved under natural smoothness and smallness conditions. The performance and accuracy of the method are illustrated by several model problems with known and unknown exact solutions. The algorithm demonstrates rapid convergence and is robust with respect to the choice of the initial approximation.