In this article we implement an Operational matrix of fractional integration for Legendre polynomials. We proposed an algorithm to obtain an approximation solution for fractional differential equations, described in Riemann-Liouville sense, based on shifted Legendre polynomials. This method was applied to solve linear multiorder fractional differential equation with initial conditions, and the exact solutions obtained for some illustrated examples. Numerical results reveal that this method gives ideal approximation for linear multi-order fractional differential equations.

In this paper, the functional Volterra integral equations of the Hammerstein type are studied. First, some conditions that ensure the existence and uniqueness of the solutions to these equations within the space of square-integrable functions are established and then the Euler Operational matrix of integration is constructed and applied within the collocation method for approximating the solutions. This approach transforms the integral equation into a set of nonlinear algebraic equations, which can be efficiently solved by employing standard numerical methods like Newton's method or Picard iteration. One significant advantage of this method lies in its ability to avoid the need for direct integration to discretize the integral operator. Error estimates are provided and two illustrative examples are included to demonstrate the method’s effectiveness and practical applicability.

In this paper, we propose a method to approximate the solution of a linear Fredholm integro-differential equation by using the Chebyshev wavelet of the first kind as basis. For this purpose, we introduce the first Chebyshev Operational matrix of integration. Chebyshev wavelet approximating method is then utilized to reduce the integro-differential equation to a system of algebraic equations. Illustrative examples are included to demonstrate the advantages and applicability of the technique.

Purpose: In this paper, we construct the Operational matrix of fractional integration of arbitrary order for Laguerre polynomials.Methods: We introduce some necessary definitions and give some relevant properties of Laguerre polynomials. The fractional integration is described in the Riemann-Liouville sense. We develop a direct solution technique for solving the integrated forms of fractional differential equations (FDEs) on the half line using the Laguerre tau method based on Operational matrix of fractional integration in the Riemann-Liouville sense.Results: In order to show the fundamental importance of the Laguerre Operational matrix, we apply it together with the spectral Laguerre tau method for the numerical solution of general linear multi-term FDEs on a semi-infinite interval.Conclusions: The results obtained by the present methods reveal that the present method is very effective and convenient for linear FDEs. Illustrative examples are included to demonstrate the validity and applicability of the new technique for linear muti-term FDEs on a semi-infinite interval.

Russian Federation is recognized as the main inheritor of Soviet Union on December 31, 1991 which was able to collect eleven newly independent Republics in the post-Soviet. Russian policy toward Central Eurasian region indicates that, the country is trying to preserve surrounding area at the same time to be effective in the process and policies of these countries. Therefore multilateral approach, formation and strengthening of regional organization are important approaches of Russia for aligning of central Eurasian countries with its goals, interests and policiesIn this study, based on the importance of multilateralism and integration of views on Russia's foreign policy, the collect efforts by Russian to create integration in the central Eurasia will be discussed in the three sets: integration of politics, security and economics and more important than, that success or failure of Russia in this  field will be evaluated. Thus the main question of this research is "What are the challenges and limitation of Russian Federation in creating regional integration in central Eurasian? "And the finding of this research shows that economic restrictions in Russia in responding to material and technological needs of the region, some important challenges  facing the central  Eurasian region and barriers to trans-regional, such as the presence and intervention of trans-regional actor are the most important Russian challenges to create integration in central Eurasia.