Volterra Integral Equations are a type of integral equation where the unknown function appears under the integral sign and is integrated over a variable limit. They are typically expressed in the form f(t) = g(t) + \int_a^t K(t, s) \phi(s) ds , where f(t) is the unknown function, g(t) is a known function, and K(t, s) is a kernel function that describes the relationship between the variables. These equations are named after the Italian mathematician Vito Volterra, who contributed significantly to the field of functional analysis. Volterra Integral Equations are commonly used in various applications, including physics, engineering, and biology, to model systems where the current state depends on past states, making them essential in understanding dynamic processes.