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Register a new userUniqueness of the fractional derivative definition
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Richard Herrmann
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For the Riesz fractional derivative besides the well known integral representation two new differential representations are presented, which emphasize the local aspects of a fractional derivative. The consequences for a valid solution of the fractional Schroedinger equation are discussed.
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We prove uniqueness for weak solutions to abstract parabolic equations with the fractional Marchaud or Caputo time derivative. We consider weak solutions in time for divergence form equations when the fractional derivative is transferred to the test function.
Accepting the Komar mass definition of a source with energy-momentum tensor $T_{mu u}$, and using the thermodynamic pressure definition, we find a relaxed energy-momentum conservation law. Thereinafter, we study some cosmological consequences of the obtained energy-momentum conservation law. It has been found out that the dark sectors of cosmos are unifiable into one cosmic fluid in our setup. While this cosmic fluid impels the universe to enter an accelerated expansion phase, it may even show a baryonic behavior by itself during the cosmos evolution. Indeed, in this manner, while $T_{mu u}$ behaves baryonically, some parts of it, namely $T_{mu u}(e)$ which is satisfying the ordinary energy-momentum conservation law, are responsible for the current accelerated expansion.
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