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Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives

Alzabut Jehad, Houas Mohamed, Abbas Mohamed I.

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20220205

In the current manuscript, we combine the q-fractional integral operator and q-fractional derivative to investigate a coupled hybrid fractional q-differential systems with sequential fractional q-derivatives. The existence and uniqueness of solutions for the proposed system are established by means of Leray-Schauder’s alternative and the Banach contraction principle. Furthermore, the Ulam-Hyers and Ulam-Hyers-Rassias stability results are discussed. Finally, two illustrative examples are given to highlight the theoretical findings.

Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators

Alharthi Nadiyah Hussain, Atangana Abdon, Alkahtani Badr S.

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20220230

Nonlocal operators with different kernels were used here to obtain more general harmonic oscillator models. Power law, exponential decay, and the generalized Mittag-Leffler kernels with Delta-Dirac property have been utilized in this process. The aim of this study was to introduce into the damped harmonic oscillator model nonlocalities associated with these mentioned kernels and see the effect of each one of them when computing the Bode diagram obtained from the Laplace and the Sumudu transform. For each case, we applied both the Laplace and the Sumudu transform to obtain a solution in a complex space. For each case, we obtained the Bode diagram and the phase diagram for different values of fractional orders. We presented a detailed analysis of uniqueness and an exact solution and used numerical approximation to obtain a numerical solution.