Application of the generalized degree method for constructing solutions of the Moisil-Teodorescu system of differential equations.

The article presents the method of generalized powers (OS) for constructing a sequence of basic solutions for a system of linear differential equations of the first order, known as the Moisil-Teodorescu systems. To accomplish this task, the quaternion form of the Moisil-Teodorescu equation is translated into a matrix form. With the help of a certain operation called joining, the system is reduced to a form that allows the use of the OS method. After that, the operations of differentiation and the right inverse operation of integration are introduced, which are analogues of differentiation and integration with respect to the complex variable of the solution of the Cauchy-Riemann system. These operations do not derive results from the set of solutions of the Moisila-Teodorescu system with given properties in a certain region of four-dimensional space. The possibility of repeated repetition of these operations gives an algorithm for constructing a sequence of basic solutions of the Moisil-Teodorescu system. This system is closely related to Maxwell's system of electromagnetic field equations and Dirac's system of quantum electrodynamics for particles with mass $m=0$ and coincides with them under certain identification of the quantities included in it. Suggested work – this is a direct generalization of the ideas of the American mathematician of European origin L. Bers.

Keywords: generalized Bers degrees, Moisil-Teodorescu system, Cauchy problem, matrix method, boundary conditions.