DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH BOUNDARY TERMS

Abstract

In the referenced study, a differential equation (DE) exhibiting a hybrid structure is examined. The principal objective of this manuscript is to determine the feasibility of substituting the given supplementary boundary conditions with alternative equivalent conditions. This is achieved through the establishment and proof of four theorems, providing a rigorous foundation for the proposed substitutions. Incipiently, the existing (3) conditions are considered in a nonhomogeneous context. Subsequently, new conditions, denoted as (7), are introduced. These newly formulated conditions are demonstrated to be equivalent to the original ones, ensuring the unique solvability of the hybrid-structured system labeled as (1). The system under consideration is characterized as hybrid due to the presence of both unknown  and algebraic components. This dual nature necessitates a nuanced approach to boundary condition formulation and analysis. The methodology employed in this study underscores the importance of flexibility in boundary condition specification, particularly in complex or hybrid configurations. By establishing the equivalence of different boundary conditions, article provides valuable insights into the solvability and analysis of such frameworks. Furthermore, the study meticulously details prior research in this domain, delineating the specific conditions and configurations previously explored. This comprehensive review situates the current manuscript within the broader context of hybrid DE analysis.

Key words: boundary, hybrid, dissipative, DAE, BVP.