Fractional Order Integrals for the Sustainable Development Model

Abstract

Sustainable development is now engaged in many leading countries. The notion is connected with the activity of countries, their economies, social and political institutions. One of the criteria of sustainability is the economic sphere, the study of which will allow you to control and predict its activity. It is known that the description of continuous processes that arise in economic problems, using definite integrals. In the case when the studied continuous development is carried out in small jumps, when it is the description of the use of fractional integral and differential equations. However, their exact solution is possible only in special cases. Therefore, the construction of approximate methods for solving fractional integral and differential equations is of great importance. In this paper, we give an approximate solution of fractional integral equations of the first kind, where the fractional integral is constructed using the Weyl fractional integral. It turns out that the constructed operator of fractional integration is symmetrical and positively definite. These properties are built by the operator enables to introduce the scalar product on the basis of the specified operator and the corresponding energy space. Next, using the same methodology that is used in the finite element method, approximate solution is constructed, constructed a rough scheme of the method and rationale of the proposed method to construct the energy space. The integral equation of the standard form is given as a specific example, as well as an example solution to illustrate the effectiveness of the proposed method.