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NUMERICAL METHODS FOR SOLVING INTEGRAL EQUATIONS WITH SINGULARITIES
Abstract
This study presents a comprehensive examination of numerical methods for solving integral equations with singularities. Integral equations, which arise in various scientific and engineering disciplines, often contain singularities that complicate their analytical and numerical solutions. This research focuses on developing and evaluating robust numerical techniques to address these challenges effectively.
The study begins with an overview of integral equations, highlighting their significance and the types of singularities commonly encountered. It then reviews traditional numerical methods and their limitations when applied to singular integral equations. To overcome these limitations, advanced techniques such as regularization methods, singularity subtraction, and adaptive mesh refinement are introduced and analyzed.
A series of test problems, representing different types of singularities, are used to compare the performance of these numerical methods. The results demonstrate that methods incorporating regularization and adaptive strategies significantly improve the accuracy and convergence of solutions for singular integral equations. The study also explores the computational efficiency of these methods, emphasizing the balance between accuracy and computational cost.
Additionally, practical applications of the developed methods are discussed, including their implementation in solving physical problems in fields such as fluid dynamics, electromagnetics, and quantum mechanics. The research concludes with recommendations for selecting appropriate numerical techniques based on the nature of the singularities and the specific requirements of the problem at hand.
This work contributes to the field by providing a detailed analysis of advanced numerical methods for singular integral equations, offering practical guidance for researchers and practitioners in developing more accurate and efficient solutions. Future research directions are suggested to further enhance these techniques and explore their application to more complex and higher-dimensional integral equations.
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