Exploring the dynamics of coupled systems: fractional q-integro differential equations with infinite delay

Abstract

The objective of this study is to demonstrate the existence and uniqueness of a solution to a coupled system of infinite-delay fractional q integro-differential equations. The equation that will be the focus of this paper's analysis is noteworthy and innovative since it combines fractional differentiation with fractional q integration and has numerous applications to real-world occurrences. The existence of the solution is established using the Schauder fixed-point theorem. The uniqueness of a solution will be evaluated using Banach's fixed-point theorem. The definitions of the Riemann fractional q integral and the Caputo fractional derivative are used to arrive at a numerical solution to the coupled system. Then, the merge of the trapezoidal finite difference methodology is integrated. The finite difference approach is used for the first and second derivative components, whereas the trapezoidal method is used for the integral components. As a result, a set of straightforward algebraic equations will be produced, from which we can deduce the solution. Some applications are provided to illustrate our main conclusions.

Keywords

• Fractional derivative
• coupled system of q-integro-differential equation
• existence and uniqueness of solution
• applications