The existence and uniqueness of a fractional q-integro differential equation involving the Caputo-Fabrizio fractional derivative and the fractional \(q\)-integral of the Riemann-Liouville type with \(q\)-nonlocal condition

Volume 33, Issue 2, pp 176--188 https://dx.doi.org/10.22436/jmcs.033.02.06

Publication Date: January 08, 2024 Submission Date: September 19, 2023 Revision Date: October 28, 2023 Accteptance Date: November 16, 2023

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Authors

A. A. Ali - Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City , Cairo, Egypt. K. R. Raslan - Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City , Cairo, Egypt. A. A. Ibrahim - October High Institute for Engineering and Technology, Egypt. M. A. Maaty - Basic Science Department, Higher Technological Institute, 10\({^{\text{th}}}\) of Ramadan City, Egypt.

Abstract

The fractional integro-differential equations are presented here, together with the novel definitions of the Caputo and Fabrizio differential operators and the \(q\)-Riemann-Liouville integral operator. In order to determine whether or not a solution does in fact exist, we employ the Schauder fixed point theorem. We discuss how the solution is unique and how it constantly depends on the constant in the nonlocal condition. In addition to this, a numerical solution to the problem will be found by employing a hybrid approach that combines the forward finite difference and trapezoidal approaches. In conclusion, in order to confirm the primary findings, three examples will be provided as illustrations.

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Keywords

• Fractional derivative
• \(q\)-integro-differential equation
• existence and uniqueness of solution
• applications

MSC

•  26Axx
•  34Axx
•  03Exx

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