Analyzing Higher-Order Boundary Value Problems within the Framework of Time Scales

Abstract

This research focuses on higher-order boundary value problems (BVPs) for delay differential equations, a topic of growing significance in control theory, physics andapplied mathematics. While lower-order BVPs with delays have been widely explored, higher-order cases remain comparatively underexamined. This study aims to bridge that gap by presenting a clear first-order system representation and developing multi-point higher-order BVPs for ordinary differential equations. The paper explores the intricacies of two-point and multi-point boundary conditions, emphasizing the role of distinct boundary constraints and the periodic reformulation of systems to enhance computational efficiency. It critically investigates the fundamental issues of existence and uniqueness of solutions, acknowledging the theoretical challenges these problems present. In addition, the study provides a concise review of numerical approaches such as collocation methods, shooting techniques andfinite difference schemes. As the analysis unfolds, it becomes evident that higher-order delayed BVPs pose complex theoretical and computational challenges, offering fertile ground for further research and refinement of numerical strategies. Ultimately, this work contributes to a deeper understanding of boundary value problems on time scales and lays a strong foundation for future advancements in mathematical modeling, computational methods anddynamic system analysis.