On solutions of fractional differential equations for the mechanical oscillations by using the Laplace transform

Changdev P. Jadhav; Tanisha B. Dale; Vaijanath L. Chinchane; Asha B. Nale; Sabri T. M. Thabet; Imed Kedim; Miguel Vivas-Cortez; Changdev P. Jadhav; Tanisha B. Dale; Vaijanath L. Chinchane; Asha B. Nale; Sabri T. M. Thabet; Imed Kedim; Miguel Vivas-Cortez

doi:10.3934/math.20241562

AIMS Mathematics

2024, Volume 9, Issue 11: 32629-32645. doi: 10.3934/math.20241562

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On solutions of fractional differential equations for the mechanical oscillations by using the Laplace transform

1.
Department of Mathematics, Deogiri Institute of Engineering and Management Studies Chhatrapati Sambhaji nagar-431005, India
2.
Department of Mathematics, Rajashri Shahu Science, Commerce and Arts Mahavidyalaya, Pathri, India
3.
Department of Mathematics, MGM University, Chhatrapati Sambhaji nagar-431 003, India
4.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602105, Tamil Nadu, India
5.
Department of Mathematics, Radfan University College, University of Lahej, Lahej, Yemen
6.
Department of Mathematics, College of Science, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02814, Republic of Korea
7.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
8.
Faculty of Exact and Natural Sciences, School of Physical Sciences and Mathematics, Pontifical Catholic University of Ecuador, Sede Quito, Ecuador

Received: 26 August 2024 Revised: 30 October 2024 Accepted: 04 November 2024 Published: 19 November 2024
MSC : 26A33, 34A08, 44A10

In this article, we employ the Laplace transform (LT) method to study fractional differential equations with the problem of displacement of motion of mass for free oscillations, damped oscillations, damped forced oscillations, and forced oscillations (without damping). These problems are solved by using the Caputo and Atangana-Baleanu (AB) fractional derivatives, which are useful fractional derivative operators consist of a non-singular kernel and are efficient in solving non-local problems. The mathematical modelling for the displacement of motion of mass is presented in fractional form. Moreover, some examples are solved.
- fractional derivative,
- Laplace transform,
- fractional differential equations,
- oscillations
Citation: Changdev P. Jadhav, Tanisha B. Dale, Vaijanath L. Chinchane, Asha B. Nale, Sabri T. M. Thabet, Imed Kedim, Miguel Vivas-Cortez. On solutions of fractional differential equations for the mechanical oscillations by using the Laplace transform[J]. AIMS Mathematics, 2024, 9(11): 32629-32645. doi: 10.3934/math.20241562

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Abstract

In this article, we employ the Laplace transform (LT) method to study fractional differential equations with the problem of displacement of motion of mass for free oscillations, damped oscillations, damped forced oscillations, and forced oscillations (without damping). These problems are solved by using the Caputo and Atangana-Baleanu (AB) fractional derivatives, which are useful fractional derivative operators consist of a non-singular kernel and are efficient in solving non-local problems. The mathematical modelling for the displacement of motion of mass is presented in fractional form. Moreover, some examples are solved.

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