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Description
In this study, the Method of Fundamental Solutions (MFS) and the Domain-Decomposition Method (DDM) are integrated with Particle Swarm Optimization (PSO) to develop an efficient and robust framework for solving degenerate boundary problems. The MFS, inherently free from mesh generation and numerical quadrature, is recognized as a promising meshless method. Its implementation requires only field points and source points, which are positioned outside the computational domain. The numerical solution in MFS is expressed as a linear combination of fundamental solutions with unknown coefficients, determined by solving a system of linear algebraic equations that enforce the interior and boundary conditions. To address challenges associated with degenerate boundaries, the DDM discretizes the computational domain. By dividing the domain into smaller subdomains, the DDM enhances numerical stability, facilitates efficient local analysis, and improves solution accuracy. Moreover, the optimal placement of source points, critical to the MFS's performance, is determined using the PSO algorithm. As a modern metaheuristic optimization technique, PSO efficiently explores the computational domain to identify the optimal configuration of source points without requiring additional problem-specific information. Consequently, the integration of MFS, DDM, and PSO ensures superior performance in accuracy, stability, and scalability. The proposed methodology is validated through numerical experiments, demonstrating its effectiveness in solving diverse degenerate boundary problems. Additionally, a systematic analysis of key parameters across the three methods highlights the robustness and adaptability of the combined numerical framework. This study, therefore, provides a powerful and versatile tool for addressing complex boundary problems.
