Methods of Solving the Initial Value Problem for Nonlinear Integro-Differential Equations with Local Error Estimation

Abstract

One of the modern scientific methods of researching phenomena and pro-cesses is mathematical modeling, which in many cases allows replacing the real process and makes it possible to obtain both a qualitative and a quantitative pic-ture of the process. Since the exact solutions of such models can be found in very individual cases, it is necessary to use approximate methods. In applied mathematics, fractional-rational approximations, which under appropriate con-ditions give a high rate of convergence of algorithms, bilateral and monotonic approximations have become widely used.In this work, using the technique of constructing one-step methods for solv-ing the initial problem for ordinary differential equations and developing the sought solution into a finite continued fraction, a numerical method for solving the Cauchy problem for nonlinear integro-differential equations of the Volterra type is proposed. The values of the parameters at which the nonlinear method of the first and second order of accuracy is obtained are found.Computational formulas are proposed, which at each integration step allow obtaining an upper and lower approximation to the exact solution without additional references to the right-hand side of the integro-differential equation. Calculation formulas, in which the main terms of the local error differ only in sign, form a two-sided method. We take the half-sum of bilateral approximations to the exact solution as the approximate solution at the given integration point, and the absolute value of the half-difference determines the error of the obtained result.The modular nature of the proposed algorithms makes it possible to ob-tain several approximations to the exact solution of the initial problem for the nonlinear integro-differential equation at each point of integration. The comparison of these approximations gives useful information in the matter of choosing the integration step or in assessing the accuracy of the result