Спосіб раціональної модифікації ітераційних алгоритмів чисельного розв'язання нелінійних інтегральних рівнянь - Протасов, Сергій; Федорчук, Володимир (Kamianets-Podilskyi National Ivan Ohiienko University, 2021)
Iterative methods for solving integral equations are a powerful tool for theoretical research and practical calculations. The peculiarity of iterative methods lies in the simplicity of computational algorithms that is essential in the process of computer realization. The disadvantages of this class of methods underlie in the problem of convergence, namely, the iterative process should be convergent, and the convergence rate should be high, which is inherent in the numerical solution of nonlinear integral equations. The article discusses the use of a combination of the Newton-Kantorovich method and quadratic formulas, that allows to obtain a high-precision numerical algorithm for solving nonlinear integral equations as Fredholm equation of the second kind. The results of test example solution are provided, which testify to the effectiveness and high accuracy of the method. The possibility of using the algorithm of solving nonlinear integral equations based on the method of sequential approximation during interpolation of the nucleus by cubic spline is analyzed. The disadvantage of these methods in computer implementation is the task of choosing the "best" initial approximation, which in turn accelerates the convergence of the method and thereby reduces the accumulation of error. The considered method of modernization of iterative algorithms of numerical solution in nonlinear integral equations allows to determine the "better" initial approximation, which makes it possible to increase the convergence of the iterative process in the initial method. The results of computational experiments in the solution of the Fredholm integral equations of the second kind confirm the use effectiveness of the modernized algorithm based on the method of simple iterations with preliminary optimization of the initial approximation.