RECENT ADVANCES IN INTEGRAL EQUATIONS: NEW CLASSES, ANALYTICAL TECHNIQUES AND INTERDISCIPLINARY APPLICATIONS

Abstract

Integral equations have long been an essential tool for modeling a wide variety of physical, biological, and engineering systems. Over the past few years, the field has witnessed significant advancements, particularly in the development of new integral equation types, analytical techniques, and numerical methods. This review systematically examines key developments from 2020 to 2024, focusing on the introduction of fractional, stochastic, and nonlinear integral equations that address more complex real-world phenomena. We discuss the refinement of traditional analytical methods, such as fixed-point theory and homotopy-based approaches, and highlight cutting-edge numerical solutions including deep learning techniques like Physics-Informed Neural Networks (PINNs). Furthermore, we explore the growing applications of integral equations across diverse fields such as physics, biology, data science, and finance. While substantial progress has been made, we also address current challenges and suggest promising future research directions, including hybrid analytical-numerical approaches and the integration of data-driven methods with traditional integral equation solvers. This review provides a comprehensive synthesis of recent advances and aims to inspire further research and innovation in the field. Keywords: Integral Equations, Fractional Calculus, Stochastic Models, Nonlinear Systems, Analytical Methods, Numerical Solutions, Deep Learning, Physics-Informed Neural Networks, High-Dimensional Problems, Data-Driven Modeling, Applications in Physics and Biology, Inverse Problems, Computational Methods.