Embedded methods with two-sided local error estimation for solving nonlinear integro-differential equations

Abstract

Two-sided numerical methods for solving the Cauchy problem for Volterra's nonlinear integro-differential equations are constructed. With appropriate parameter values, it is possible to obtain an approximation to the exact solution of the first and second order of accuracy. A set of parameters is proposed for which we obtain calculation formulas that at each integration step give the upper and lower approximations to the exact solution. For the approximate solution, we take the half-sum of two-sided approximations, and the modulus of the half-difference gives the error of the method. Calculation formulas are proposed that make it possible to find not only two-sided approximations to the exact solution, but also to calculate the explicit expression of the main term of the local error of the method without additional calculations of the right side of the integro-differential equation.