# TWO-SIDED METHODS FOR SOLVING INITIAL VALUE PROBLEM FOR NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS

### Abstract

Using the continued fractions and the method of constructing Runge-Kutta methods, numerical methods for solving the Cauchy problem for nonlinear Volterra non-linear integrodifferential equations are proposed. With appropriate values of the parameters, one can obtain an approximation to the exact solution of the first and second order of accuracy. We found a set of parameters for which we obtain two-sided calculation formulas, which at each step of integration allow to obtain the upper and lower approximations of the exact solution.

### References

Baker G. A. Jr., Graves-Morris P. Pade Approximants. Generalizations and applications. Moscow: Mir. 1986. 502 p.

Jones W. B., Tron W. J. Continued fractions. Analytic theory and applications. Moscow: Mir. 1985. 416 p.

Skorobogatko V. Ya. The theory of branching continued fractions and its application in computational mathematics. Moscow: Nauka. 1983. 312 p.

Krylov V. I., Bobkov V. V., Monastyrny P. I. Computational methods. Vol. II. Moscow: Nauka. 1977. 400 p.

Hall G., Watt J. M. Modern numerical methods for solving ordinary differential equations. Moscow: Mir. 1979. 312 P.

Pelekh Y. M., Budz I. S., Kunynets A. V., Fil B. M. Methods of solving the initial problem with estimation of the principal term of the local error. Visnyk of the Lviv University. Series: applied mathematics and computer science. 2019. No. 27. P. 75–88.

Pelekh Y. M., Kunynets A. V., Berehova G. I., Magerovska T. V. Methods solving the initial problem with two-sided estimation of the local error. Physic-mathematical modelling and informational technologies. 2021. No. 33. P. 88–92.

Coroian I. Asupra metodei Runge-Kutta-Fehlberg, pentru ecuatia integrala neliniara de tip Volterra. Stud. Cerc. Math. 1974. Vol. 26. No. 4. P. 505–511.

Dobronets B. S., Shaidurov V. V. Two-sided numerical methods. Novosibirsk: Nauka. 1990. 206 p.