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2 2023

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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.

Hardy–Stein identity for pure-jump Dirichlet forms

Gutowski Michał

Bulletin of the Polish Academy of Sciences. Mathematics

2023

Vol. 71, no 1

65--84

We prove the Lp variant of the Hardy–Stein identity for Sobolev–Bregman forms associated with pure-jump Dirichlet forms with killing, under rather mild assumptions. Along the way, we obtain a general result in terms of the p-form defined in a more abstract way.