Explorations of Numerical Techniques and Algorithms for Handling Mathematical Issues and Modelling Real-World Activities
Abstract
Numerical methods and algorithms play a crucial role in solving mathematical problems that are otherwise analytically intractable and in simulating real-world phenomena across various scientific disciplines. These methods encompass a wide range of techniques including finite difference methods (FDM), finite element methods (FEM), spectral methods, and optimization algorithms. Recently, the integration of machine learning has introduced new dimensions to numerical analysis, offering enhanced accuracy and efficiency. This review article explores the development, application, and advancements in numerical methods and algorithms, focusing on their effectiveness, efficiency, and versatility. It delves into methods such as finite difference, finite element, and spectral methods, along with optimization algorithms and machine learning techniques that are increasingly being integrated into numerical analysis. Case studies across physics, engineering, finance, and biology demonstrate the practical applications and ongoing innovations in this field. By highlighting the strengths and limitations of various approaches, the article aims to provide a comprehensive understanding of current trends and future directions in numerical methods and their role in scientific and engineering problem-solving.
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