**AUTHORS:**Mehdiyeva Galina, Ibrahimov Vagif, Imanova Mehriban

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**ABSTRACT:**
As is known there are some classes of numerical methods have constructed to solving initial value
problem of the ODE, which fundamentally investigated by the many known scientist. Therefore the specialists
tried to study many scientific and applied problems by using these methods. Here we have defined the direct
way between the initial value problem for the Volterra integro-differential and Ordinary Differential Equations.
By using this way have constructed the multistep methods with constant coefficients, which are applied to
solving initial value problem for the Volterra integro-differential equations and have determined the necessary
and sufficient conditions for its convergence. And also have proven that the constructed here methods are more
accurately than the known, which is illustrated by the application of concrete method to solving of the model
problem.

**KEYWORDS:**
Volterra Integro-Differential Equations, initial value problem of the ODE, multistep methods
(MM), stability and degree for MM

**REFERENCES:**

[1] Volterra V. Theory of functional and of integral and integro-differensial equations, Moskow, Nauka, 1982.

[2] Linz P. Linear Multistep methods for Volterra Integro-Differential equations. Journal of the Association for Computing Machinery, Vol.16, No.2, April, 1969, 295-301.

[3] Feldstein, A., & Sopka, J.R. (1974). Numerical methods for nonlinear Volterra integro differential equations. SIAM J. Numer. Anal. V. 11, 826-846.

[4] Brunner H. Imlicit Runge-Kutta Methods of Optimal order for Volterra integro-differential equation. Mathematics of computation, Volume 42, Number 165, January 1984, 95-109.

[5] Makroglou A. Hybrid methods in the numerical solution of Volterra integro-differential equations. Journal of Numerical Analysis 2, 1982, 21-35

[6] Baker C.T.H. The numerical treatment of integral equations. Oxford; Claerdon, 1977, 1034p.

[7] Verlan A.F, Sizikov V.S. Integral equations: methods, algorithms, programs. Kiev, Naukova Dumka, 1986.

[8] Mehdiyeva G., Ibrahimov V., Imanova M. The application of the hybrid method to solving the Volterra integro-differential equation. World Congress on Engineering 2013, London, U.K., 3-5 July, 2013, 186-190.

[9] Mehdiyeva G., Imanova M., Ibrahimov V. Application of the hybrid method with constant coefficients to solving the integro-differential equations of first order. 9th International conference on mathematical problems in engineering, aerospace and sciences, AIP, Vienna, Austria, 10-14 July 2012, 506-510.

[10] Ibrahimov V., Imanova M. The application of second derivative methods to solving Volterra integro-differential equations. Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2014 (ICNAAM-2014) AIP Conf. Proc. 1648, © 2015 AIP Publishing LLC

[11] Mehdiyeva G.Yu., Imanova M.N., Ibrahimov V.R. On one application of forward jumping methods. Applied Numerical Mathematics, Volume 72, October, 2013, 234-245.

[12] Mehdiyeva G.Yu., Imanova M.N., Ibrahimov V.R. Some refinement of the notion of symmetry for the Volterra integral equations and the construction of symmetrical methods to solve them, Journal of Computational and Applied Mathematics, 306, 2016, 1–9.

[13] Mehdiyeva G., Ibrahimov V., Imanova M. Solving Volterra Integro-Differential Equation by the Second Derivative Methods Applied Mathematics and Information Sciences, Volume 9, No. 5, Sep. 2015, 2521-2527.

[14] Dahlquist G. Convergence and stability in the numerical integration of ordinary differential equations, Math.Scand, 1956, No.4, 33-53.

[15] Henrici P. Discrete variable methods in ordinary differential equation. Wiley, New York, 1962.

[16] Dahlquist G. Stability and error bounds in the numerical integration of ordinary differential equation. Trans. Of the Royal Inst. Of Techn., Stockholm, Sweden, Nr. 130, 1959, 3-87.

[17] Ibrahimov V.R. Ibrahimov V.R. On a relation between order and degree for stable forward jumping formula. Zh. Vychis. Mat. 1990, № 7, 1045-1056.

[18] Hairier E., Norsett S.P., Wanner G. Solving ordinary differential equations. (Russian) М., Mir, 1990.

[19] Butcher J.C. Numerical methods for ordinary differential equations. John Wiley and sons, Ltd, Second Edition, 2008.

[20] Kendall E. Atkinson, Weimin Han Elementary numerical analysis, Third Edition, 2004, 560p.