08 / 26 / 2011 - 10:06 — Julia

**ON A WAY FOR CONSTRUCTING HYBRID METHODS**

**Mehdiyeva Galina, doctor of science, prof., Ibrahimov Vagif, doctor of science, prof., Imanova Mehriban, phd. Mirzayev Ramil, qraduate Baku State University**

*There exists a wide class of numerical methods for solving of initial-value problem for ordinary differential equations. These classes may be divided into onestep and multistep ones. For constructing hybrid methods, some relations between one and multistep methods are researched on the base of Runge-Kutta and Adams methods. By means of Runge-Kutta multistep methods and forward jumping methods and contrary ones are constructed. Keywords: Cauchy problem, initial-value problem, hybrid method, implicit Runge-Kutta and forward jumping methods, stability, convergence.*

Consider the following initial-value problem

As is known, Euler first gave the direct numerical method for solving equation (1). This method developed in two directions and as a result appeared the Runge-Kutta and Adams methods (see [1]). The Runge-Kutta methods were explicit, but the Adams methods were both implicit and explicit. By means of Adams concrete methods it was shown that implicit methods are more precise and have a wide area of stability than explicit ones. Therefore, many scientists constructed diagonally-implicit and implicit Runge-Kutta methods (see [2], p.213).

During the development of the Adams methods, multistep methods with constant coefficients appeared. In particular, from these methods one can be obtain both implicit, explicit methods and forward jumping methods (see [3], p.467). Notice that the forward jumping methods were first constructed by Cowell (see [4]). The Cowell type methods were constructed by the famous mathematicians as Laplace and Steklov (see [5]). Taking into account the advantage of one and multistep methods, the specialists suggested to construct new methods on the base of these methods and as the result hybrid methods appeared (see [6], p.19). For improving some properties of numerical methods such as extension of stability area, increase of accuracy and etc. the scientists suggested to use the superposition methods (see [7], p. 403) or two-sided methods (see [9]). Proceeding from what has been said that, determination of some relation between the classic Runge-Kutta methods and Adams methods was a great interest. In [10] passage from the Runge-Kutta explicit method to Adams methods and vice versa was shown by means of investigation of relations between these methods on a concrete example. Here, by means of comparisons of Runge-Kutta diagonal-implicit and implicit methods with multistep methods with constant coefficients, the hybrid methods and also forward jumping type methods are constructed.

Notice that, many papers (see [11],[12]) have been devoted to investigation of problem (1) by means of hybrid methods.

At first consider application of the Runge-Kutta method to numerical solution of problem (1) that in a general form may be written in the form:

get method (4). However, method (5) is a forward jumping method in the case when . Consequently, implicit methods of type (4) and the forward jumping methods have different properties.

By comparison of explicit one and multistep methods we give an example on passing from Adams method to the Runge-Kutta method and to this end consider the central differences method:

**References:**

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8.V. R. Ibrahimov, G. Yu. Mehdiyeva, I. I. Nasirova. On the stability region for the forward jumping methods. Transactions issue methematics and mechanics series of phis.-tech. and mathematical sciences, XX, №4, 2005, 163-170 pp.

9.V. R. Ibrahimov. On a way for consructing two-sided methods. Godishnik na visshite uchebni zavedeniya, Prilozhno math., Sofiya, Bulgariya, 1984, 199-207 p.

10.V. R. Ibrahimov, G. Yu. Mehdiyeva, I. I. Nasirova. On some connections between Runge-Kutta and Adams methods. Transactions issue methematics and mechanics series of phis.-tech. and mathematical sciences, XXV, №7, 2005, 183-190 pp.

11.G. K. Gupta, A polynomial representation of hybrid methods for solving ordinary differential equations, Mathematics of Computation, vol. 33, No. 148, 1979, 1251-1256 pp.

12.J. O. Ehigie, S. A. Okunuga, A. B. Sofoluwe, M. A. Akanbi, On generalized 2-step continuous linear multistep method of hybrid type for the integration of second order ordinary differential equations, Archives of Applied Science Research, Scholar Research Library,2(6), 2010, 362-372 pp.

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

15.V. R. Ibrahimov. One non-linear method to numerical solving of problem Cauchy for ordinary differential equation. Proceeding of 2nd International Conference Differential equations and applications. Russo, Bulgaria, 1981, 310-319 pp.

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