Boundary value problems (BVPs) play a pivotal role in various scientific and engineering disciplines, providing mathematical models for phenomena ranging from heat conduction to fluid dynamics and beyond. Traditionally, researchers have primarily focused on local BVPs, where the solution at a point depends only on the values of the function and its derivatives at that point. However, in recent years, there has been a growing recognition of the significance of nonlocal BVPs, where the solution involves the integration of the function over an interval rather than evaluation at a single point. This shift in focus arises from the need to model phenomena with nonlocal interactions, such as long-range interactions in materials, spatially distributed processes in biology, and fractional differential equations in physics.
The study of nonlocal BVPs has gained prominence due to its ability to capture the intricate nature of phenomena that exhibit memory effects, long-range interactions, and non-local dependencies. Understanding the mathematical properties, analytical solutions, and computational methods for nonlocal BVPs is essential for advancing our comprehension of diverse physical phenomena.
In this research proposal, we aim to conduct a comprehensive investigation into both local and nonlocal boundary value problems, with the following key objectives described in what follows.
Objectives:
1. Characterization of Local BVPs:
- Review classical methods for solving local BVPs.
- Investigate mathematical properties and solution existence for diverse local BVPs.
- Develop efficient numerical algorithms for local BVPs.
2. Exploration of Nonlocal BVPs:
- Investigate the mathematical foundations of nonlocal BVPs, including fractional calculus and integral equations.
- Examine the existence and uniqueness of solutions for various classes of nonlocal BVPs.
- Propose innovative numerical methods for solving nonlocal BVPs.
3. Integration and Comparative Analysis:
- Develop a unified framework that integrates local and nonlocal BVPs.
- Conduct a comparative analysis to identify scenarios where nonlocal effects significantly impact solutions.
- Assess the computational efficiency and accuracy of existing and proposed methods for both local and nonlocal BVPs.
4. Applications:
- Apply the unified approach to real-world problems in physics, engineering, and biology, emphasizing scenarios with mixed local and nonlocal effects.
- Evaluate the practical significance of considering both local and nonlocal aspects in modeling complex phenomena.
The central purpose of this special issue hosted by Boundary Value Problems is to attract high-level papers written by distinguished scientists all over the world. The topic is modern and very suitable for applications in many fields, including mathematical physics, engineering, Newtonian and non-Newtonian mechanics, etc.
