On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type

Khalid K. Ali; K. R. Raslan; Amira Abd-Elall Ibrahim; Mohamed S. Mohamed; Khalid K. Ali; K. R. Raslan; Amira Abd-Elall Ibrahim; Mohamed S. Mohamed

doi:10.3934/math.2023925

AIMS Mathematics

2023, Volume 8, Issue 8: 18206-18222. doi: 10.3934/math.2023925

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Research article

On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type

1.
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt
2.
October High Institute for Engineering and Technology
3.
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 12 March 2023 Revised: 08 May 2023 Accepted: 14 May 2023 Published: 26 May 2023
MSC : 26Axx, 34Axx, 45Jxx, 45Lxx

The major objective of this scheme is to investigate both the existence and the uniqueness of a solution to an integro-differential equation of the second order that contains the Caputo-Fabrizio fractional derivative and integral, as well as the q-integral of the Riemann-Liouville type. The equation in question is known as the integro-differential equation of the Caputo-Fabrizio fractional derivative and integral. This equation has not been studied before and has great importance in life applications. An investigation is being done into the solution's continued reliance. The Schauder fixed-point theorem is what is used to demonstrate that there is a solution to the equation that is being looked at. In addition, we are able to derive a numerical solution to the problem that has been stated by combining the Simpson's approach with the cubic-b spline method and the finite difference method with the trapezoidal method. We will be making use of the definitions of the fractional derivative and integral provided by Caputo-Fabrizio, as well as the definition of the q-integral of the Riemann-Liouville type. The integral portion of the problem will be handled using trapezoidal and Simpson's methods, while the derivative portion will be solved using cubic-b spline and finite difference methods. After that, the issue will be recast as a series of equations requiring algebraic thinking. By working through this problem together, we are able to find the answer. In conclusion, we present two numerical examples and contrast the outcomes of those examples with the exact solutions to those problems.
- q-integro-differential equation,
- fractional Caputo-Fabrizio,
- existence and uniqueness of solution,
- numerical solutions
Citation: Khalid K. Ali, K. R. Raslan, Amira Abd-Elall Ibrahim, Mohamed S. Mohamed. On study the fractional Caputo-Fabrizio integro differential equation including the fractional q-integral of the Riemann-Liouville type[J]. AIMS Mathematics, 2023, 8(8): 18206-18222. doi: 10.3934/math.2023925

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Abstract

The major objective of this scheme is to investigate both the existence and the uniqueness of a solution to an integro-differential equation of the second order that contains the Caputo-Fabrizio fractional derivative and integral, as well as the q-integral of the Riemann-Liouville type. The equation in question is known as the integro-differential equation of the Caputo-Fabrizio fractional derivative and integral. This equation has not been studied before and has great importance in life applications. An investigation is being done into the solution's continued reliance. The Schauder fixed-point theorem is what is used to demonstrate that there is a solution to the equation that is being looked at. In addition, we are able to derive a numerical solution to the problem that has been stated by combining the Simpson's approach with the cubic-b spline method and the finite difference method with the trapezoidal method. We will be making use of the definitions of the fractional derivative and integral provided by Caputo-Fabrizio, as well as the definition of the q-integral of the Riemann-Liouville type. The integral portion of the problem will be handled using trapezoidal and Simpson's methods, while the derivative portion will be solved using cubic-b spline and finite difference methods. After that, the issue will be recast as a series of equations requiring algebraic thinking. By working through this problem together, we are able to find the answer. In conclusion, we present two numerical examples and contrast the outcomes of those examples with the exact solutions to those problems.

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