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We present exact analytical results for the Caputo fractional derivative ofa wide class of elementary functions, including trigonometric andinverse trigonometric, hyperbolic and inverse hyperbolic, Gaussian,quartic Gaussian, Lorentzian, and shifted polynomial functions. Theseresults are especially important for multi-scale physical systems, suchas porous materials, disordered media, and turbulent fluids, in whichtransport is described by fractional partial differential equations. Theexact results for the Caputo fractional derivative are obtained from a singlegeneralized Euler’s integral transform of the generalized hypergeometricfunction with a power-law argument. We present a proof of thegeneralized Euler’s integral transform and directly apply it to theexact evaluation of the Caputo fractional derivative of a broad spectrum offunctions, provided that these functions can be expressed in terms of ageneralized hypergeometric function with a power-law argument. Wedetermine that the Caputo fractional derivative of elementary functions isgiven by the generalized hypergeometric function. Moreover, we show thatin the most general case the final result cannot be reduced toelementary functions, in contrast to both the Liouville-Caputo and Fourier fractionalderivatives. However, we establish that in the infinite limit of theargument of elementary functions, all three definitions of a fractionalderivative - the Coputo, Liouville-Caputo, and Fourier - converge to the same result given by theelementary functions. Finally, we prove the equivalence between Liouville-Caputo and Fourierfractional derivatives.
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This content will become publicly available on January 1, 2026
On the Fractional Dynamics of Kinks in Sine-Gordon Models
In the present work, we explored the dynamics of single kinks, kink–anti-kink pairs and bound states in the prototypical fractional Klein–Gordon example of the sine-Gordon equation. In particular, we modified the order β of the temporal derivative to that of a Caputo fractional type and found that, for 1<β<2, this imposes a dissipative dynamical behavior on the coherent structures. We also examined the variation of a fractional Riesz order α on the spatial derivative. Here, depending on whether this order was below or above the harmonic value α=2, we found, respectively, monotonically attracting kinks, or non-monotonic and potentially attracting or repelling kinks, with a saddle equilibrium separating the two. Finally, we also explored the interplay of the two derivatives, when both Caputo temporal and Riesz spatial derivatives are involved.
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- PAR ID:
- 10616513
- Publisher / Repository:
- MDPI
- Date Published:
- Journal Name:
- Mathematics
- Volume:
- 13
- Issue:
- 2
- ISSN:
- 2227-7390
- Page Range / eLocation ID:
- 220
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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