Fractional calculus in beam deflection: Analyzing nonlinear systems with Caputo and conformable derivatives

Abdelkader Lamamri; Iqbal Jebril; Zoubir Dahmani; Ahmed Anber; Mahdi Rakah; Shawkat Alkhazaleh; Abdelkader Lamamri; Iqbal Jebril; Zoubir Dahmani; Ahmed Anber; Mahdi Rakah; Shawkat Alkhazaleh

doi:10.3934/math.20241050

AIMS Mathematics

2024, Volume 9, Issue 8: 21609-21627. doi: 10.3934/math.20241050

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Research article

Fractional calculus in beam deflection: Analyzing nonlinear systems with Caputo and conformable derivatives

1.
Laboratory of LAMDA-RO, University of Blida 1, Algeria
2.
Department of Mathematics, Al-Zaytoonah University of Jordan, Amman, 11733, Jordan
3.
University of Oran USTO, Oran, Algeria
4.
University of Alger 1, Algeria
5.
Department of Mathematics, Faculty of Science and Information Technology, Jadara University, Jordan

Received: 07 June 2024 Revised: 24 June 2024 Accepted: 01 July 2024 Published: 05 July 2024
MSC : 30C45, 39B72, 39B82

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems.
- existence of solution,
- beam deflection,
- Caputo derivative,
- conformable fractional derivative,
- Than method,
- traveling waves,
- differential system
Citation: Abdelkader Lamamri, Iqbal Jebril, Zoubir Dahmani, Ahmed Anber, Mahdi Rakah, Shawkat Alkhazaleh. Fractional calculus in beam deflection: Analyzing nonlinear systems with Caputo and conformable derivatives[J]. AIMS Mathematics, 2024, 9(8): 21609-21627. doi: 10.3934/math.20241050

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Abstract

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems.

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