This research focuses on the analysis of a second-order system of partial differential equations linked with the four-variable hypergeometric functions . These specific functions play a critical role in numerous branches of contemporary mathematics and theoretical physics, notably in quantum mechanics and string theory. By treating multidimensional hypergeometric functions within the framework of differential systems, this work explores their structural properties and evaluates solution behavior under specific boundary constraints. The subject of this inquiry involves the function classes  and  originally proposed by Exton. Throughout this investigation, distinct differential operators were established for the aforementioned functions, enabling the derivation of exact analytical expressions for their fundamental solutions. The outcomes highlighted in this paper underscore the significance of multivariate hypergeometric functions in addressing contemporary applied problems and open up new avenues for mathematical modeling. Ultimately, the validity of the computed data is rigorously verified through classical techniques of differential equation theory.