An Overview of Volterra Integral Equations: Techniques and Practical Applications

Volterra integral equations (VIEs) represent a powerful tool in mathematical modeling across various disciplines, owing to their ability to capture dynamic processes with memory effects. An extensive discussion of the techniques and uses of Volterra integral equations in current research is given in this review study. The discussion begins with an introduction to the theoretical framework of VIEs, highlighting their formulation and fundamental properties. Subsequently, it explores numerical methods tailored for solving VIE’s encompassing both classical techniques and recent advancements in computational approaches. Volterra integral equations play a fundamental role in mathematical modeling, particularly in physics, engineering, biology, and economics. These equations, categorized into first- and second-kind forms, integral equations with variable limits in which the unknown function appears beneath the integral sign. In addition to quadrature-based numerical techniques like the trapezoidal and Runge-Kutta methods, analytical and numerical approaches for solving Volterra integral equations include the Laplace transform, A domian decomposition, and variational iteration methods. . The practical applications of Volterra integral equations span diverse fields, including population dynamics, viscoelastic materials, financial modeling, and infectious disease modeling. Their ability to represent hereditary and memory-dependent processes makes them indispensable in describing real-world systems. This paper provides an overview of solution techniques and their effectiveness in various applications, emphasizing their role in contemporary computational and applied mathematics.