Exploring the Dynamics of Fractional q-Integro-Differential Equations with Infinite Time Delays: A Study in Mathematical Analysis

A novel equation that combines fractional calculus and integral operations is investigated in this study. The unique properties of the equation and its potential applications to various real-world phenomena have not been previously explored. The existence and uniqueness of a solution to this equation are the primary objectives of this research. To achieve this, the Schauder fixed point theorem and Banach’s fixed point theorem are utilized. The role of these theorems is crucial in demonstrating the existence and uniqueness of the solution. In addition to theoretical analysis, numerical methods are developed, employing the Caputo fractional derivative and the Riemann fractional q integral. By combining the finite difference approach with the trapezoidal method, the equation is transformed into a system of algebraic equations that can be efficiently solved. This numerical approach allows the acquisition of a reliable numerical solution to the problem at hand. Practical examples are presented to underscore the significance of the findings and to showcase the diverse applications of this equation. These examples serve as compelling evidence for the profound impact of this study on real-world phenomena. Through rigorous mathematical reasoning and innovative numerical techniques, the mysteries surrounding this intriguing equation are uncovered. Consequently, this exploration not only contributes to a deeper understanding of fractional calculus but also sheds light on its practical implications in a variety of real-world scenarios.