International Journal of Geometric Methods in Modern Physics

Combined effects of forced and natural convection on the occurrence of flow separation and heat transfer on a vertical paraboloid of revolution under the influence of a magnetic field are studied in this investigation. Inside the boundary layer, free convection is established due to the temperature difference between the fluid and the boundary while forced convection happens due to the presence of an external potential flow. The external potential velocity is chosen from the decelerated nature concerning the stream-wise coordinate due to which the flow separation is expected to occur. The influence of natural convection, the applied magnetic field, and the surface transverse curvature on delaying the flow separation and on the rate of heat transfer in the vicinity of the separation point is the aim of the investigation of this study. Regarding these important physical ingredients, the percentage increase in separation length and the rate of heat transfer are calculated. Moreover, their influence on the flow and heat-transfer phenomena has also been discussed in detail and presented through various graphs and tables. An exact numerical solution is obtained with the aid of an implicit finite difference scheme.

The aim of this paper is to study certain types of metrics such as conformal $\eta$-Ricci soliton and Yamabe soliton in general relativistic spacetime. Here, we have shown the nature of the soliton when the spacetime with semisymmetric energy–momentum tensor admits conformal $\eta$-Ricci soliton, whose potential vector field is torse-forming. We have studied certain curvature conditions on the spacetime that admits conformal $\eta$-Ricci soliton. Also, we have enriched the importance of the Laplace equation on the spacetime admitting conformal $\eta$-Ricci soliton. Next, we have given some applications of physical connection of dust fluid, dark fluid and radiation era on general relativistic spacetime admitting conformal $\eta$-Ricci soliton and Yamabe soliton.

In this paper, Lie symmetry analysis method is applied to time fractional Burgers equation, Korteweg-de Vries equation and generalized reaction-diffusion equation with delays, respectively. The Lie symmetries for fractional partial differential equations with delays (DFPDEs) are obtained, and the group classifications of the equations are established. The obtained group generators are used to reduce the DFPDEs to fractional ordinary differential equations with delays (DFODEs). Some exact solutions constructed for the DFODEs generate group-invariant solutions of the discussed DFPDEs.

We introduce in this paper a tomographic analysis of the properties of a Friedmann–Lemaitre–Robertson–Walker (FLRW) universe with a perfect fluid. We first review previous works where the Schutz’s parametrization in terms of Clebsch potentials was adopted to describe the perfect fluid. This approach allows to introduce a fiducial time in the Wheeler–De Witt equation. We revisit the properties of the perfect fluid and the introduced Clebsch potentials. In particular, we see that thermasy plays an extremely important role in the definition of fiducial time. The definition of a time and the condition $a\ge 0$ for the expansion factor imply that the Hamiltonian operator must be self-adjoint which implies a restriction on the initial conditions for the wave packet. We show that these allow to obtain well-defined tomograms. Tomograms are marginal functions which incorporate all the information contained in the wave function of the universe, but have the properties of classical probability functions. They can be defined for classical distributions on the phase space as well, enabling us to describe quantum and classical states with the same family of functions. The aim of this paper is to compare the difference between classical tomograms where the Hawking and Penrose theorems imply the inevitability of an initial singular state and the well-defined initial quantum states found in literature. Finally the introduction of a time in the Wheeler–DeWitt allows us to consider the evolution of the classical and quantum initial states of the universe which can be accomplished by introducing a transition probability function for tomograms.

In this paper, the generalized quantum states, i.e. positive and normalized linear functionals on $C^{\ast }$-algebras, are studied. First, we study normal states, i.e. states which are represented by density operators, and singular states, i.e. states cannot be represented by density operators. It is given an approach to the resolution of bounded linear functionals into quantum states by applying the Gelfand–Neumark–Segal (GNS) construction, i.e. the fundamental result of Gelfand, Neumark and Segal on the representation theory of $C^{\ast }$-algebras, and theory of projections. Second, it is given an application in quantum information theory. We study covariant cloners, i.e. quantum channels in the Heisenberg and the Schrödinger pictures which are covariant by shifting, and it is shown that the optimal cloners cannot have a singular component. Finally, we discuss on the representation of pure states in the sense of the Gelfand–Pettis integral. We also give physical interpretations and examples in different sections of this work.

We prove a series of van Est type theorems relating the cohomologies of strict Lie 2-groups and strict Lie 2-algebras, and use them to prove the integrability of strict Lie 2-algebras anew.

In this paper, we aim to formulate anisotropic solutions that represent charged self-gravitating static spheres in $f(R)$ gravity. We introduce the effects of anisotropy in the domain of an isotropic solution by including an extra matter source. The spherical configuration is deformed by transforming radial as well as temporal metric functions. As a result of these transformations, the isotropic and anisotropic matter sources can be decoupled from each other. The system of field equations is disintegrated into two arrays such that the first set corresponds to the isotropic source while the second comprises the new gravitational source. The charged Krori–Barua spacetime is employed to specify the first set in the background of $f(R)$ Starobinsky model. Two solutions of the other system are determined by incorporating a barotropic equation of state along with additional constraints on the anisotropic source. The matching conditions are imposed at the hypersurface to determine the values of unknown constants. Different features of the constructed models are inspected by using the mass and radius of Her X-I. One of the constructed solutions agrees with the viability and stability conditions for specific values of charge and the decoupling parameter.

In this paper, Morris Thorne wormholes are considered in the context of $f(R)$ gravity. A shape function is defined as $b(r)=r_0(1-k)+k\frac{r_0^2}{r}$, where $k$ is constant. The equation of state is considered as $p_t=-\rho$ and $f(R)$ function is derived. The wormhole solutions are obtained and energy conditions are examined. Further, the $f(R)$ model is found to be consistent with local gravity tests and stability of cosmological perturbations and late-time de Sitter point. Cosmological evolution is also explored using Friedman-Robertson-Walker (FRW) metric in $f(R)$ gravity.

In the present paper, we study the invariant submanifolds of $f$-Kenmotsu manifolds. Firstly, we show that any invariant submanifold of $f$-Kenmotsu manifold is again $f$-Kenmotsu manifold and minimal. Then, we give some characterizations of totally geodesic submanifolds of $f$-Kenmotsu manifolds. We study 3-dimensional submanifold and prove that a 3-dimensional submanifold of $f$-Kenmotsu manifold is totally geodesic if and only if it is invariant. Also, $\eta$-Ricci soliton is considered on invariant submanifold of $f$-Kenmotsu manifolds. Lastly, the non-trivial examples are constructed to verify some of our obtained results.

In this paper, we study conformal Ricci solitons and conformal gradient Ricci solitons on generalized $(\kappa ,\mu )$-space forms. The conditions for the solitons to be shrinking, steady and expanding are derived in terms of conformal pressure $p$. We show under what conditions a Ricci semi-symmetric generalized $(\kappa ,\mu )$-space form equipped with a conformal Ricci soliton forms an Einstein manifold.

This paper encompassed various physically realistic models of static and spherically symmetric wormholes in extended $f(R)$ gravitational framework by incorporating the features of non-compact matter density distribution of non-commutative geometry. For this assessment, we formulated the $f(R)$ field equations and obtained their exact analytical solutions in presence of cosmologically well-consistent power law type $f(R)=a{R}^n$ corresponding to distinct values of the parameter $n$. We conducted the physical adequacy of such proposed solutions through the graphical manifestation for the necessary existence of wormhole structures. Moreover, we revisited our field equations with a feasible choice of shape function together with reconstructed form of $F(R)$ to provide an again authentication to the existence of wormholes structures. These all implications vigorously depicted that our discussed models are able to expect the required wormhole structures.

Based on the new macroscopic two-velocity difference model, this paper analyzes the linear stability of the new model and studies the nonlinear bifurcation theory. First, the linear stability analysis method is used to study the stability conditions of the shock wave in the model. Then, considering the long wave model in the coarse-grained scale, the reduced perturbation method is used to analyze the characteristics of the traffic flow in the metastable region, and the solitary wave solution of the Korteweg-de Vries (KdV) equation in the metastable region is derived. In addition, by using the bifurcation analysis method, the type, and stability of the equilibrium solution are discussed and the existing conditions of the saddle-node bifurcation are proven. Then, taking the saddle-node bifurcation as the starting point, we draw the density space-time diagram and phase plane diagram of the system. It is proven that the newly proposed model can describe complex traffic phenomena such as stop-and-go and sudden changes in stability, which is of great help to solve traffic congestion.

Korteweg-de Vries (KdV)-type models are frequently seen during the investigations on the optical fibers, cosmic plasmas, planetary oceans and atmospheres. In this paper, for an extended three-coupled KdV system, noncharacteristic movable singular manifold and symbolic computation help us bring about four sets of the auto-Bäcklund transformations with some solitons. All of our results rely on the coefficients in that system.

The action of active diffeomorphisms (diffs) $r\to \rho (r)$ on the Schwarzschild metric leads to metrics which are also static spherically symmetric solutions of the Einstein vacuum field equations. It is shown how in a $limiting$ case it allows to introduce a deformation of the manifold such that $\rho (r=0)=0$, and $\rho (r=0^{+})=2GM$ corresponding, respectively, to the spacelike singularity and horizon of the Schwarzschild metric. In doing so, one ends up with a spherical $void$ surrounding the singularity at $r=0$. In order to explore the “interior” region of this void, we introduce complex radial coordinates whose imaginary components have a direct link to the inverse Hawking temperature, and which furnish a path that provides access to interior region. In addition, we show that the black hole entropy $\frac{A}{4}$ (in Planck units) is equal to the $area$ of a rectangular strip in the $complex$ radial-coordinate plane associated to this path. The gist of the physical interpretation behind this construction is that there is an emergence of thermal dimensions which unfolds as one plunges into the interior void region via the use of complex coordinates, and whose imaginary components capture the span of the thermal dimensions. Namely, the filling of the void leads to an $emergent$ internal/ thermal dimension via the imaginary part ${\beta }_r$ of the complex radial variable $r=r+i{\beta }_r$.

The current study takes into account the evolution of a fractal universe with holographic dark energy through Barrow entropy and dark matter, i.e. without pressure, which interact with one another through mutual interaction. The interaction term for this model is then rebuilt by using the Hubble length as the IR cut-off scale. We represent Barrow holographic dark energy as Nojiri–Odintsov generalized holographic dark energy in fractal universe. The cosmological parameters that change over the course of cosmic history are looked at from the early matter-dominated period through the late time acceleration. The results of the study indicate that the cosmos recently underwent a smooth transition from a decelerated to an accelerated phase of expansion. We also found that the Barrow holographic dark energy equation of state parameter exhibits a rich behavior, lying in the quintessence regime, the phantom regime, or experiencing the phantom-divide crossing during evolution, depending on the values of the coupling term $b^2$ and the Barrow exponent $\mathrm{\Delta }$. It has been reported on the evolution of the model’s Hubble parameter and a comparison with the most recent cosmic chronometer data. The stability of the model has also been examined in order to determine its viability, with the square of sound speed being taken into account.

The Levi-Civita connection usually adopted on the non-commutative three-sphere is generalized to a connection possessing torsion and a non-metricity of the Weyl kind. This leads to geometrical properties that are qualitatively different from the ordinary ones and, in particular, provides an example of non-commutative “teleparallelism”. The comparison is made with the analogous case in classical (i.e. commutative) differential geometry. A general framework is developed, from which the unique connection having a given metric, torsion and non-metricity can be determined.

In this paper, we present solution of Einstein field equations which admit dust distribution and negative energy massless scalar field as a source with cosmological term ($\mathrm{\Lambda }$) in the frame work of Bianchi type II space-time. The models satisfy conservation equation and creation field increases with time satisfying the result as investigated by Hoyle and Narlikar [A new theory of gravitation, Proc. Roy. Soc. A 282 (1964) 191]. The deceleration parameter ($q$) for the model (22) shows a transition from decelerating to accelerating regime with creation. A comparison with other cosmological models shows a transition from decelerating to accelerating regime with and without creation is mentioned. The present model is free from singularity, particle horizon and provides a natural explanation for inflationary scenario and isotropization. These features make the creation field cosmological models theoretically superior to the Big Bang models. Creation field and Einstein field equation are derived using principle of least action and the Lagrangian formulation of variable cosmological term is derived as mentioned by Moffat [Lagrangian formulation of a solution to the cosmological constant problem (1996), arXiv:astro-ph/9608202v1]. The current creation rate with other physical consequences is also discussed. For illustrative purposes, evolutionary behaviors of some cosmological parameters are shown graphically. In the figures, the cosmic time $t$ is given in gravitational units assuming the velocity of light $c=1$. This $t=cT$ and $T=\frac{t}{c}$. Therefore, by multiplying cosmic time $t$ by (light years)⁻¹, we obtain the coordinate time in years.

In this paper, we considered the Tsallis holographic dark energy model in frames of Nojiri–Odintsov gravity with $f(R)=R+\lambda R^2-\sigma \mu /R$. For IR, cutoff event horizon is taken. The cosmological evolution of such universe is investigated for various initial conditions and values of parameters. The dependence of the Hubble parameter $H$ from time in the future has an oscillation. It is shown that for $\mu \ne 0$ appearance of singularities are typical and the time up to these singularities can be relatively small from cosmological viewpoint. The singularity is associated with the zero of second derivative of $f(R)$ on $R$. It is interesting to note that these models can describe observational data from Ia supernovae astrophysics and dependence of the Hubble parameter from redshift $z$ at least not worse than canonical $\mathrm{\Lambda }$CDM model.