Solving mixed Volterra–Fredholm integro-differential equations: A Rishi transform-based approach

This paper presents a novel analytical method for solving mixed Volterra–Fredholm integro-differential equations (MVFIDEs) using the recently introduced Rishi transform. These equations frequently arise in modeling complex phenomena in physics, engineering, and applied sciences, where both memory (Volterra) and global interaction (Fredholm) effects are present. We focus on deriving exact solutions without the need for iterative or purely numerical schemes. The proposed technique simplifies MVFIDEs by transforming them into algebraic equations using the operational rules and properties of the Rishi transform. Some analytical examples are presented to illustrate the complete procedure for determining the solution. In particular, we consider MVFIDEs that apply the Rishi Transform to reduce the integro-differential equation into an algebraic form involving the transform parameters ϑ and φ.