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Generalized Hamilton-Jacobi Formulation of Damped Harmonic System Using Fractional Derivatives |
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PP: 13-24 |
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doi:10.18576/pfda/09S102
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Author(s) |
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Yazen M. Alawaideh,
Yasser M. Aboel-Magd,
Bashar M. Al-khamiseh,
Ramiz Assaf,
Mohammad Kanan,
Samer Alawideh,
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Abstract |
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| In this paper, we establish a fractional Hamilton-Jacobi formulation in terms of fractional derivatives for a damped harmonic system. A fractional Riemann–Liouville fractional derivative operator is defined, and a fractional Hamilton–Jacobi function is established using this formulation. The fractional Hamilton-Jacobi function for fractional damped harmonic systems is obtained using the Hamilton-Jacobi theory. We used this new formulation to solve the Hamilton-Jacobi partial differential equation for a fractional damped harmonic system after introducing the Hamilton- Jacobi formulation using fractional derivatives. The motion equations are then defined in terms of Poisson brackets, and the Heisenberg equations are expressed in terms of commutators, the quantum counterpart of the Poisson bracket. Finally, we look at an example to help explain the findings. According to the findings of this study, fractional calculus, owing to the order of the fractional derivative and the fractional operator, allows for more flexible models than classical calculus. This property is critical for developing a novel formulation of the Hamilton-Jacobi formulation of the Damped Harmonic System equation generalized using the Riemann–Liouville derivative.
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