Derivation of an approximate formula of the Rabotnov fractional-exponential kernel fractional derivative and applied for numerically solving the blood ethanol concentration system

Ahmed F. S. Aboubakr; Gamal M. Ismail; Mohamed M. Khader; Mahmoud A. E. Abdelrahman; Ahmed M. T. AbdEl-Bar; Mohamed Adel; Ahmed F. S. Aboubakr; Gamal M. Ismail; Mohamed M. Khader; Mahmoud A. E. Abdelrahman; Ahmed M. T. AbdEl-Bar; Mohamed Adel

doi:10.3934/math.20231569

AIMS Mathematics

2023, Volume 8, Issue 12: 30704-30716. doi: 10.3934/math.20231569

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Derivation of an approximate formula of the Rabotnov fractional-exponential kernel fractional derivative and applied for numerically solving the blood ethanol concentration system

1.
Department of Mathematics, College of Science, Taibah University, Medina 41411, Saudi Arabia
2.
Department of Mathematics, University of Fayoum, Fayoum 63514, Egypt
3.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Medina 42210, Saudi Arabia
4.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia
5.
Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt
6.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
7.
Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt
8.
Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

Received: 11 September 2023 Revised: 31 October 2023 Accepted: 02 November 2023 Published: 14 November 2023
MSC : 41A10, 65N12, 65N35

The article aimed to develop an accurate approximation of the fractional derivative with a non-singular kernel (the Rabotnov fractional-exponential formula), and show how to use it to solve numerically the blood ethanol concentration system. This model can be represented by a system of fractional differential equations. First, we created a formula for the fractional derivative of a polynomial function $ t^{p} $ using the Rabotnov exponential kernel. We used the shifted Vieta-Lucas polynomials as basis functions on the spectral collocation method in this work. By solving the specified model, this technique generates a system of algebraic equations. We evaluated the absolute and relative errors to estimate the accuracy and efficiency of the given procedure. The results point to the technique's potential as a tool for numerically treating these models.
- blood ethanol concentration system,
- Rabotnov fractional-exponential,
- Vieta-Lucas polynomials,
- spectral collocation method
Citation: Ahmed F. S. Aboubakr, Gamal M. Ismail, Mohamed M. Khader, Mahmoud A. E. Abdelrahman, Ahmed M. T. AbdEl-Bar, Mohamed Adel. Derivation of an approximate formula of the Rabotnov fractional-exponential kernel fractional derivative and applied for numerically solving the blood ethanol concentration system[J]. AIMS Mathematics, 2023, 8(12): 30704-30716. doi: 10.3934/math.20231569

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Abstract

The article aimed to develop an accurate approximation of the fractional derivative with a non-singular kernel (the Rabotnov fractional-exponential formula), and show how to use it to solve numerically the blood ethanol concentration system. This model can be represented by a system of fractional differential equations. First, we created a formula for the fractional derivative of a polynomial function $ t^{p} $ using the Rabotnov exponential kernel. We used the shifted Vieta-Lucas polynomials as basis functions on the spectral collocation method in this work. By solving the specified model, this technique generates a system of algebraic equations. We evaluated the absolute and relative errors to estimate the accuracy and efficiency of the given procedure. The results point to the technique's potential as a tool for numerically treating these models.

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