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

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