Iterative methods for solving quadratic Volterra integral equations of the first kind

Background. The paper is devoted to the numerical study of integral equations of the first kind with quadratic nonlinearity, which are part of the generalized Volterra integropower series and describe dynamical systems with one input and one output. Such equations are widely used in the modeling of stationary systems with constant dynamic characteristics during the transfer process. Materials and methods. The proposed iterative numerical methods are based on the preliminary linearization of the integral operator according to the modified Newton-Kantorovich scheme and the use of the regularization parameter to ensure stability to fluctuations in the input data. To solve linear equations ateach iteration, the method of successive approximations is used in combination with the approximation of the exact solution by a polynomial spline constructed for each segment on the zeros of Legendre polynomials. The Gauss compound quadrature formula is used to calculate integrals. Results and conclusions. A number of iterative numerical schemes for solving quadratic Volterra integral equations are proposed. The convergence theorems of the modified Newton-Kantorovich method are formulated. Numerical results confirming the convergence of the methods are presented.