Numerical Methods of Solving the Initial Value Problem for Ordinary Differential Equations with Evaluation of the Main Member of the Local Error

One of the modern scientific methods for investigating phenomena and processes is mathematical modeling. A mathematical modeling is the effective method of study of economic processes, in many important cases allows to replace the real process, and also gives an opportunity to get both quality and quantitative picture of the designed problems. Since the exact solutions of the investigated models can be got only in very partial cases, then it is necessary to use numerical methods. In the design of economic problems there is a necessity not only to find the numeral solutions of such models but also research of the estimations of their local and global errors. Continuous fractions are widely used in computational mathematics. They make it possible to obtain monotonic and two-sided approximations, have a weak sensitivity to rounding errors, and also correctly reflect the basic properties of the problems studied. A research object is initial value problem for ordinary differential equations. The purpose of the study is to develop methods and algorithms to build computational methods for the numerical solution of the Cauchy problem for ordinary differential equations. Formulas of the Runge-Kutta type of the fourth order of accuracy of the solution of the initial problem for ordinary differential equations based on continued fractions are derived. New two-sided numerical methods of the third order of accuracy are offered which at each node point make it possible to obtain not only upper and lower approximations to an exact solution, but also without additional computational costs give information about the magnitude of the leading term of the local error of the approximate solution. Two-sided formulas at each step of integration use less number of calculations of the right-hand side of the differential equation than the known bilateral methods of Runge-Kutta type. The proposed formulas, using only four calculations of the right-hand side of the differential equation, allow at each step to obtain a method of fourth-order accuracy method and two bilateral formulas of the third order of accuracy.