Numerical solution of differential-algebraic equations of arbitrary index with Riemann-Liouville derivative

In the article, linear systems of ordinary differential equations of fractional order  are investigated. In contrast to previously known results, the authors consider the case when the matrix before the fractional differentiation operation is degenerate. Problems in such a formulation are called differential-algebraic equations of fractional order. The fundamental differences of such systems from the classical problems of fractional differentiation and integration are emphasized, namely, the systems under consideration can have an infinite number of solutions, or a solution of the original problem depends on the high fractional derivative of the right-hand side. Corresponding examples are given. The authors pass to a different, equivalent formulation of the problem, namely, they rewrite it in the form of a system of linear integral equations of the Abel type (with a weak singularity). This technique allows one to use the apparatus of regular matrix bundles to investigate the existence and uniqueness of the original problem. Using this result, the authors give sufficient conditions for the existence of a unique solution to the class of problems under consideration. Further, an algorithm for the numerical solution of such equations is proposed. The method is based on the product integration method and the quadrature formula of right rectangles. Calculations and graphs of the errors of the proposed method for various fractional differentiation exponents and various indices of the initial matrix bundles are presented.