In the particular configuration of the scalar field k-essence in the Wheeler–DeWitt quantum equation, for some age in the Bianchi type I anisotropic cosmological model, a fractional differential equation for the scalar field arises naturally. The order of the fractional differential equation is $\beta =\frac{2\alpha }{2\alpha -1}$. This fractional equation belongs to different intervals depending on the value of the barotropic parameter; when ${\omega }_X\in [0,1]$, the order belongs to the interval $1\le \beta \le 2$, and when ${\omega }_X\in [-1,0)$, the order belongs to the interval $0<\beta \le 1$. In the quantum scheme, we introduce the factor ordering problem in the variables $(\mathsf{\Omega },\varphi )$ and its corresponding momenta $({\mathsf{\Pi }}_{\mathsf{\Omega }},{\mathsf{\Pi }}_{\varphi })$, obtaining a linear fractional differential equation with variable coefficients in the scalar field equation, then the solution is found using a fractional power series expansion. The corresponding quantum solutions are also given. We found the classical solution in the usual gauge N obtained in the Hamiltonian formalism and without a gauge. In the last case, the general solution is presented in a transformed time $T\left(\tau \right)$; however, in the dust era we found a closed solution in the gauge time $\tau$.

In this work, we will explore the effects of non-commutativity in fractional classical and quantum schemes using the flat Friedmann–Robertson–Walker (FRW) cosmological model coupled to a scalar field in the K-essence formalism. In previous work, we have obtained the commutative solutions in both regimes in the fractional framework. Here, we introduce non-commutative variables, considering that all minisuperspace variables $q_{nc}^i$ do not commute, so the symplectic structure was modified. In the quantum regime, the probability density presents a new structure in the scalar field corresponding to the value of the non-commutative parameter, in the sense that this probability density undergoes a shift back to the direction of the scale factor, causing classical evolution to arise earlier than in the commutative world.

Equations of ideal magnetohydrodynamics that describe stationary flows of an inviscid ideally electrically conducting fluid are considered. Classes of exact solutions of these equations are...

We construct an effective four-dimensional model by compactifying a ten-dimensional theory of gravity coupled with a real scalar dilaton field on a time-dependent torus. This approach is applied to anisotropic cosmological Bianchi type I model for which we study the classical coupling of the anisotropic scale factors with the two real scalar moduli produced by the compactification process. Under this approach, we present an isotropization mechanism for the Bianchi I cosmological model through the analysis of the ratio between the anisotropic parameters and the volume of the Universe which in general keeps constant or runs into zero for late times. We also find that the presence of extra dimensions in this model can accelerate the isotropization process depending on the momenta moduli values. Finally, we present some solutions to the corresponding Wheeler-DeWitt (WDW) equation in the context of standard quantum cosmology.