Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30750
On a Way for Constructing Numerical Methods on the Joint of Multistep and Hybrid Methods

Authors: G.Mehdiyeva, V.Ibrahimov, M.Imanova


Taking into account that many problems of natural sciences and engineering are reduced to solving initial-value problem for ordinary differential equations, beginning from Newton, the scientists investigate approximate solution of ordinary differential equations. There are papers of different authors devoted to the solution of initial value problem for ODE. The Euler-s known method that was developed under the guidance of the famous scientists Adams, Runge and Kutta is the most popular one among these methods. Recently the scientists began to construct the methods preserving some properties of Adams and Runge-Kutta methods and called them hybrid methods. The constructions of such methods are investigated from the middle of the XX century. Here we investigate one generalization of multistep and hybrid methods and on their base we construct specific methods of accuracy order p = 5 and p = 6 for k = 1 ( k is the order of the difference method).

Keywords: initial value problem, Multistep and hybrid methods, degree and stability of hybrid methods

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1228


[1] Subbotin M.F. Course of selestial mechanics. Vol.2, M, ONTI, 1937, 404 p. (Russian).
[2] C.S Gear. Hybrid methods for initial value problems in ordinary differential equations. SIAM, J. Numer. Anal. v. 2, 1965, pp. 69-86.
[3] G.K. Gupta. A polynomial representation of hybrid methods for solving ordinary differential equations, Mathematics of comp., volume 33, number 148, 1979, pp.1251-1256.
[4] Butcher J.C. A modified multistep method for the numerical integration of ordinary differential equations. J. Assoc. Comput. Math., v.12, 1965, pp.124-135.
[5] V.R. Ibrahimov. On a nonlinear method for numerical calculation of the Cauchy problem for ordinary differential equation, Diff. equation and applications. Pros. of II International Conference Russe. Bulgarian, 1982, pp. 310-319.
[6] A. Makroglou. Hybrid methods in the numerical solution of Volterra integro-differential equations. Journal of Numerical Analysis 2, 1982, pp.21-35.
[7] G.Mehdiyeva, M.Imanova, V.Ibrahimov Application of the hybrid methods to solving Volterra integro-differential equations World Academy of Science, engineering and Technology, Paris, 2011, 1197- 1201.
[8] G.Dahlquist Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand. 1956, Ôäû4, p.33-53.