Higher Dimensional Systems of Differential Equations Obtained by Iterative Use of Complex Methods

TL;DR

This paper introduces an iterative complex splitting method to derive higher-dimensional systems of ODEs and PDEs, expanding the class of systems accessible beyond previous single-split approaches.

Contribution

It develops a novel iterative complex splitting technique to generate three and four dimensional differential systems, surpassing limitations of earlier methods.

Findings

Successfully derived three and four dimensional ODE systems

Generated new classes of four ODE systems distinct from single-split methods

Provided illustrative examples demonstrating the method's applicability

Abstract

Systems of two ordinary and partial differential equations (ODEs and PDEs) had been obtained from a scalar complex ODE by splitting it into its real and imaginary parts. The procedure was also carried out to obtain a four dimensional system by splitting a complex system of two ODEs into its real and imaginary parts. Systems of three ODEs had not been accessible by these methods. In this paper the complex splitting is used iteratively to obtain three and four dimensional systems of ODEs and four dimensional systems of PDEs for four functions of two and four variables. The new systems of four ODEs are distinct from the class obtained by the single split of a two dimensional system. Illustrative examples are provided.