Constructive Framework for Analysing Partial Linear Differential Operators on Finite Intervals under Varied Boundary Conditions

DOI: https://doie.org/10.1213/Jbse.2024184013

Avinash Bansidhar Thakare, Dr Sadanand Patil

Keywords:

Partial Differential Operators, Boundary Value Problems, Constructive Analysis, Finite Intervals, Boundary Conditions, Dirichlet Conditions, Neumann Conditions, Mixed Boundary Conditions, Eigenvalue Problems

Abstract:

This article presents a comprehensive framework for analyzing partial linear differential operators on finite intervals, with a particular focus on the impact of various boundary conditions. By investigating boundary value problems (BVPs) associated with Dirichlet, Neumann, and mixed boundary conditions, we elucidate the nuanced relationship between these conditions and the behavior of solutions. Employing constructive techniques such as series solutions, Green’s functions, and variational methods, this study derives explicit and computable solutions, enhancing both theoretical understanding and practical applications. The framework is validated through a series of case studies, including the heat equation, wave equation, and Poisson equation, demonstrating its effectiveness in providing clear insights into operator behavior under differing constraints. The findings reveal that the proposed constructive analysis not only clarifies the role of boundary conditions in determining solution properties but also lays the groundwork for future advancements in operator theory. This work significantly contributes to the field of applied mathematics, offering valuable tools for researchers and practitioners engaged in the study of complex systems across diverse scientific and engineering disciplines.

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