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

The paper presents a generalized degree method for constructing a sequence of basic solutions to a system of first-order linear differential equations known as the Moisila-Teodorescu systems. To perform this task, the quaternion form of the Moisil-Teodorescu equation is converted to a matrix form. The system is reduced to a form that allows the use of the method of generalized degrees by means of a certain operation called joining. After that, the differentiation operations and the right inverse integration operation are introduced, which are analogs of the differentiation and integration over the complex variable of the solution of the Cauchy-Riemann system. These operations do not deduce 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 provides an algorithm for constructing a sequence of basic solutions of the Moisila-Teodorescu system. Further, the construction of these solutions is given by the method of generalized degrees (OS). Previously, based on the operators $D_1, D_2$, the so-called binary OS operations are constructed with certain formally analogous to the usual numerical powers $X_1^mX_2^nC$ differentiation properties. On their algebraic basis, using the correspondence principle, symmetric OS of the type $\overline Z^mZ^nC$are constructed. The special case $m=0$ gives an infinite sequence of solutions to the Moisil-Theodorek system. The proposed apparatus is largely analogous to the algorithm for constructing complex powers of $z^n$ in the theory of functions of complex variables. Particular examples are given. To facilitate practical calculations, a number of degrees, both binary and symmetric, are given in the applications. The Moisil-Teodorescu system is closely related to the Maxwell system of electromagnetic field equations and to the Dirac system of quantum electrodynamics for particles with mass $m=0$ and coincides with them with a certain identification of the quantities included in it. The proposed work is a direct generalization of the ideas of the American mathematician of European origin L. Bers.