R-holomorphic solutions and R-differentiable integrals of multidimensional differential systems

TL;DR

This paper studies the existence and properties of R-holomorphic solutions and integrals in multidimensional differential systems with R-differentiable coefficients, providing theorems, criteria, and methods for their analysis.

Contribution

It introduces new existence theorems, criteria for integrals, and a spectral method for linear systems with R-linear coefficients, advancing the understanding of R-differentiable solutions.

Findings

Proved the existence and uniqueness of R-holomorphic solutions.

Established criteria for R-differentiable first integrals and last multipliers.

Developed a spectral method for linear homogeneous PDE systems.

Abstract

We consider multidimensional differential systems (total differential systems and partial differential systems) with R-differentiable coefficients. We investigate the problem of the existence of R-holomorphic solutions, R-differentiable integrals, and last multipliers. The theorem of existence and uniqueness of R-holomorphic solution is proved. The necessary conditions and criteria for the existence of R-differentiable first integrals, partial integrals, and last multipliers are given. For a completely solvable total differential equation with R-holomorphic right hand side are constructed the classification of R-singular points of solutions and proved sufficient conditions that equation have no movable nonalgebraical R-singular points. The spectral method for building R-differentiable first integrals for linear homogeneous first-order partial differential systems with R-linear coefficients is developed.