Lie Group Techniques Research Articles

Multiple solutions and conserved vectors of a shallow water wave equation arising in fluid mechanics; Lie group analysis

Investigation has shown that shallow water wave model equations constitute a class of hyperbolic partial differential equations describing the movement below a pressure surface in a fluid, and this makes them highly useful to be studied. Therefore, in this paper we analytically investigate a one-dimensional shallow water wave equation that is highly applicable in fluid dynamics. We leverage the robust nature of the Lie group technique to provide symmetries to the equation. Commutator and adjoint representations of symmetries were generated and presented in tabular form. Sequel to that, we perform reductions of the shallow water wave model under study via its point symmetries so that its exact solutions can be found. Thus, to achieve this, various combinations of the symmetries are made. Besides, we examine the travelling wave solutions of the model. The direct integration method is used to obtain elliptic integral functions as well as elliptic solutions in terms of the Weierstrass function. In addition, we utilize standard techniques such as (G′/G)-expansion, power series, and Kudryashov’s approaches to generate more closed-form solutions to the model. Other interesting solutions achieved include hyperbolic, trigonometric, rational, and exponential functions. In search of the physical meaning associated with the results, graphical demonstrations are included, and these are given in 2D, 3D, and density plots. Thereafter, the construction of conserved vectors for the aforementioned model is done with the use of the standard multiplier approach and Ibragimov’s theorem. These techniques give the opportunity to furnish various forms of conserved currents.

Exploring cone-shaped solitons, breather, and lump-forms solutions using the lie symmetry method and unified approach to a coupled breaking soliton model

In this research article, we investigate the coupled breaking soliton (cBS) model using two distinct analytical methods, namely, the Lie symmetry approach and the Unified method. We start by applying the Lie group technique to the cBS model, allowing us to establish infinitesimals, vector fields, commutative and adjoint tables, and an adjoint transformation matrix. Through the utilization of the adjoint transformation matrix, we identify a one-dimensional optimal system of subalgebras. This essential stage allows the cBS model to be reduced into several collections of ordinary differential equations (ODEs) relating to similarity variables resulting from symmetry reduction. By solving these ODE systems under specific parametric constraints, we successfully obtain exact solutions in terms of closed form. Furthermore, the Unified method is employed to address the governing equation, leading us to deduce polynomial and rational function solutions. The dynamic behaviours and characteristics of these such solutions are comprehensively explored through 3-dimensional (3D) plots and contour plots. The graphics show breather solitons, cone-shaped solitons, lump solitons, and patterns of flower petals, periodic solitons, and solitary waves. Additionally, we have connected our mathematical findings with real-world phenomena, which enrich our research work. Furthermore, breathers and lumps arise in many fields of mathematical physics, including fluid dynamics, plasma physics, ocean engineering, nonlinear optics, and physical sciences, as well as several other areas of nonlinear dynamics.

Lie group analysis of flow and heat transfer of a nanofluid in cone–disk systems with Hall current and radiative heat flux

A study of the rheological and heat transport characteristics in cone–disk systems finds relevance in many applications such as viscometry, conical diffusers, and medical devices. Therefore, a three‐dimensional axisymmetric flow with heat transport of a magnetized nanofluid in a cone–disk system subjected to Hall current and thermal radiation effects is investigated. The simplified Navier–Stokes (NS) equations for the cone–disk system given by Sdougos et al. [18] Journal of Fluid Mechanics, 138, 379–404 are solved by using the asymptotic expansion method for the four different models, such as rotating cone with static disk (Model I), rotating disk with static cone (Model II), co‐rotating cone and disk (Model III), and counter‐rotating cone and disk (Model IV). The Khanafer–Vafai–Lightstone (KVL) model along with experimental data‐based properties of 37 nm Al2O3–H2O nanofluid is considered. To obtain the transformations leading to self‐similar equations from the Navier–Stokes (NS) and energy conservation equations, the Lie group technique is used. The self‐similar nonlinear problem is solved numerically to examine the effects of physical parameters. There are critical values of the power exponent at which no heat transport from the disk surface occurs. Nanoparticles significantly enhance heat transport when both the cone and disk rotate in the same or opposite directions. The centrifugal force and thermal radiation improve the heat transport in cone–disk systems.

Scrutinization of Solar Thermal Energy and Variable Thermophysical Properties Effects on Non-Newtonian Nanofluid Flow

Nanofluids generate high values of convection heat transfer coefficients, low specific heat, and density, which improve the solar thermal energy performance by making it work effectively. By utilizing nanotechnology and solar thermal radiation, the modern world is moving in the direction of new technologies. Therefore, this research is communicated to explore the significance of solar thermal energy, variable properties on non-Newtonian nanofluid flow. However, to exemplify the fluid transport features of the Casson nanofluid (CF), the Buongiorno nanofluid model was utilized. Also, the Lie-group technique is used in the framework to develop similarity variables that will be used to reduce the number of independent variables in partial differential equations (PDEs) and is solved numerically by using the weighted residual Galerkin method (WRGM). The graphical findings revealed that when the variable viscosity parameter is increased, the fluid temperature decreases, while the presence of the solar radiation parameter has the opposite impact. Additionally, when the non-Newtonian parameter approaches infinity, the Casson fluid obeys the viscosity law. The report of this study will be of benefit to thermal and chemical engineering for nanotechnology advancement. KEYWORD: Solar Thermal Energy, Nanofluids, Non-Newtonian, weighted residual Galerkin method (WRGM).

Thermal distribution of magneto-tangent hyperbolic flowing fluid over a porous moving sheet: A Lie group analysis

An investigation of magneto-hyperbolic tangent fluid motion through a porous sheet which stretches vertically upward with temperature-reliant thermal conductivity is scrutinized in this study. The current model characterizes thermal radiation and the impact of internal heat source in the heat equation plus velocity and thermal slipperation at the wall. The translation of the transport equations is carried out via the scaling Lie group technique and the resultant equations are numerically tackled via shooting scheme jointly with Fehlberg integration Runge-Kutta scheme. The results are publicized through various graphs to showcase the reactions of the fluid terms on the thermal and velocity fields. From the investigations, it is found that rising values of the material Weissenberg number, slip and suction terms damped the hydrodynamic boundary film whereas the heat field is prompted directly with thermal conductivity.

On metric-invariant relativistic transformations under anisotropy in the context of special relativity

In a context of special relativity under anisotropy, Burde recently presented in this journal a new group of coordinate transformations obtained using Lie group techniques from the principles of relativity and correspondence, which show conformal metric invariance. As a consequence, this author argued that conformal invariance is a necessary condition of any anisotropic transformation satisfying those principles, and rejected the adequacy of any transformation obtained from the Lorentz transformation by a coordinate change, e.g. resynchronization such as the [Formula: see text]-Lorentz transformation. This paper argues against these assertions, and presents, as a counterexample, a relativistic generalization of the Tangherlini transformation obtained by the same Lie group method, that satisfies the previous requirements and enjoys the metric invariance property.

Symmetry reductions, generalized solutions and dynamics of wave profiles for the (2+1)-dimensional system of Broer–Kaup–Kupershmidt (BKK) equations

In this article, a (2+1)-dimensional system of Broer–Kaup–Kupershmidt (BKK) equations, which describes the non-linear, and long gravity waves in a dispersive system, is investigated by applying the method of Lie group of invariance. Applying the Lie group technique, the infinitesimals, vector fields, commutator table, and adjoint table are constructed for the BKK system. Furthermore, a one-dimensional optimal system of subalgebras is determined with the help of the adjoint transformation matrix; thereafter, the BKK system of equations is reduced into many systems of ordinary differential equations (ODEs) with respect to the similarity variables obtained through the symmetry reduction. These systems of ODEs are solved under some parametric constraints to obtain the exact closed form solutions. The obtained results are interpreted physically via graphical representation. Thus, dark–bright solitary waves, multi-peak mixed waves, breather type waves, periodic waves, multi-peakon solitons, and multi-solitons profiles of the obtained solutions are presented to make this research physically significant. A comparison is also presented between the solutions obtained in this article and the solutions reported by Kassem and Rashed in Kassem and Rashed (2019). The solutions obtained in this article are more general, and completely different in view of solutions obtained by Kassem and Rashed. The resulting solutions are found to be useful to understand the dynamics of BKK model and represent the authenticity as well as the effectiveness of the Lie group method. Therefore, the obtained solutions, and their dynamical wave structures are quite significant for understanding the propagation of the long gravity waves in a dispersive system. To obtain the infinitesimal generators and wave profiles of obtained solutions, symbolic computation is performed in the software package MAPLE and MATHEMATICA.

Construction of multiple new analytical soliton solutions and various dynamical behaviors to the nonlinear convection-diffusion-reaction equation with power-law nonlinearity and density-dependent diffusion via Lie symmetry approach together with a couple of integration approaches

By taking advantage of three different computational analytical methods: the Lie symmetry analysis, the generalized Riccati equation mapping approach, and the modified Kudryashov approach, we construct multiple new analytical soliton solutions to the nonlinear convection-diffusion-reaction equation (NCDR) with power-law nonlinearity and density-dependent diffusion. Lie symmetry analysis is one of the powerful techniques that reduce the higher-order partial differential equation into an ordinary differential equation by reduction of independent variables. By the Lie group technique, we obtain one-parameter invariant transformations, determining equations and corresponding vectors for the considered convection-diffusion-reaction equation. By treating the parameters of the governing equation as constants, the applied methods yield a variety of new closed-form solutions, including inverse function solutions, periodic solutions, exponential function solutions, dark solitons, singular solitons, combo bright-singular solitons, and the combine of bright-dark solitons and dark-bright solitons. Moreover, using the Bäcklund transformation of the generalized Riccati equation and modified Kudryashov method, we can construct multiple solitons and other solutions of the considered equation. The obtained new solutions of this work demonstrate that the used approaches are powerful and effective in dealing with nonlinear equations, and that these solutions are required to explain many biological and physical phenomena. Comparing our obtained solutions of this paper with the ones obtained in the literature, we see that our solutions are new and not reported elsewhere. These newly formed soliton solutions will be more beneficial in the various disciplines of ocean engineering, plasma physics, and nonlinear sciences.

Shock wave propagation in a non-ideal rotating medium with azimuthal magnetic field effect using Lie group technique

The present article demonstrates the propagation of a strong cylindrical shock wave in a rotating adiabatic flow of a van der Waals gas in the presence of an azimuthal magnetic field. Using the Lie group technique, self-similar solutions are obtained with an assumption of uniform density in the undisturbed medium. In contrast, the azimuthal component of the fluid velocity and the magnetic field are supposed to vary. The profiles of the flow variables are depicted graphically, followed by a brief discussion on the behavior of the solutions obtained through the graph. The effects of variation in the values of the adiabatic index, Alfvén Mach Number, non-idealness parameter, and the ambient azimuthal velocity exponent on the flow variables are discussed in detail. The computational work is performed by using Mathematica (a software package).

Determination of the symmetry groups of radiation hydrodynamics equations and the compatible equations of state and opacities

The purpose of this study was to find all the symmetry groups of the radiation hydrodynamics equations with no a priori assumptions on the equations of state (EOS) and opacities. As shown in earlier works, the application of the Lie group technique to such a system of equations leads to invariance conditions in the form of linear differential equations, which, up until now, were only partially solved. In this paper, using the same technique and under the same assumptions, but with a simpler formulation, we show that these equations can be entirely solved analytically. This result enables us to list all the one-parameter groups that may be symmetry groups of the system. To be actually so, they must be associated with suitable EOS and opacities whose general expressions are also given. The interesting point is that some of them can be chosen so as to fit realistic data for EOS and opacities. Using this property, we propose a method to design low-scale experiments to simulate radiative processes, which would involve too much energy to have experimented with at their full scale. In addition, we derive the reduced systems associated with the one-parameter symmetry groups found. We show that some classical self-similar problems can be extended to more general EOS and opacities, and we treat in detail the self-similar expansion of a semi-infinite medium submitted to an internal source of energy.

A NEW CONSTRUCTION OF OS OF SUBALGEBRAS AND INVARIANT SOLUTION OF THE BLACK-SCHOLES EQUATION

In this manuscript, the Lie group technique is applied to construct a new OS and invariant solutions of a one-dimensional LA, which describes the symmetries properties of a nonlinear Black-Scholes model. The structure of LA depends on one parameter. We have shown a novel way to construct the so-called OS of subalgebras of the Black-Scholes equation by utilizing the given symmetries. We transform the symmetries of the Black-Scholes equation into a simple ordinary differential equation called the Lie equation, which provides us a way through which to construct a new optimal scheme of subalgebras of the Black-Scholes through applying the concept of LE. The OS which consists of minimal representatives is utilized to develop the invariant solution for the Black-Scholes equation. The fundamental use of the Lie group analysis to the differential equation is the categorization of group invariant solutions of differential equations via OS. Finally, we have utilized the OS to construct the invariant solution of the Black-Scholes equation.

Dynamical reduced basis methods for Hamiltonian systems

We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main factors: the rich geometric structure encoding the physical and stability properties of the dynamics and its local low-rank nature. To address these aspects, we propose a nonlinear structure-preserving model reduction where the reduced phase space evolves in time. In the spirit of dynamical low-rank approximation, the reduced dynamics is obtained by a symplectic projection of the Hamiltonian vector field onto the tangent space of the approximation manifold at each reduced state. A priori error estimates are established in terms of the projection error of the full model solution onto the reduced manifold. For the temporal discretization of the reduced dynamics we employ splitting techniques. The reduced basis satisfies an evolution equation on the manifold of symplectic and orthogonal rectangular matrices having one dimension equal to the size of the full model. We recast the problem on the tangent space of the matrix manifold and develop intrinsic temporal integrators based on Lie group techniques together with explicit Runge–Kutta (RK) schemes. The resulting methods are shown to converge with the order of the RK integrator and their computational complexity depends only linearly on the dimension of the full model, provided the evaluation of the reduced flow velocity has a comparable cost.

Comparison analysis of analytical and lattice Boltzmann methods for simulation of turbulence decay in flows in converging and diverging channels

AbstractThe paper focuses on a study of turbulence decay in flow with streamwise gradient. For the first time, an analytical solution of this problem was obtained based on the k‐ε model of turbulence in one‐dimensional (1D) approximation, as well as on the symmetry properties of the system of differential equations. Lie group technique enabled reducing the problem to a linear differential equation. The analytical solution enabled parametric studies, which are computationally cheap in comparison to CFD based simulations. The lattice Boltzmann method (LBM) in two‐dimensional approximation (2D) was used to validate the analytical results. Large eddy simulation (LES) Smagorinsky approach was used to close the LBM model. Computations revealed that the rate of turbulence decay is significantly different for the cases of positive and negative streamwise pressure gradient. The further comparisons showed that the analytical solution underpredicts the predictions by the numerical methodology, which can be attributed to the simplified problem statement used to derive the closed‐form analytical solution. Comparisons of calculations with experiments revealed that the theoretical models used in the study underpredict the measurements for flows with a positive pressure gradient. Hence it can be concluded that the LBM technique combined with the LES Smagorinsky model requires the further modification.

Subspace controllability of symmetric spin networks

Controlling the dynamics of quantum systems has been a long-standing goal of quantum physics. To understand the controllability of a finite-dimensional quantum system is a fundamental problem in quantum control theory, but can be very challenging in many cases. In particular, results on controllability of open-loop coherently controlled closed quantum systems under symmetry constraints have been elusive. In this paper, we study the controllability of spin systems with some specific symmetry where spins are coupled via Ising interaction and manipulated collectively by a global control field. Based on Lie group and Lie algebra techniques, we find the conditions under which the system is universally controllable in its irreducible subspaces. We believe that the results obtained here will contribute to quantum controlling on symmetric quantum systems.

Holomorphic Automorphism Groups of (m,1)-Circular Domains

A quasi-circular domain of weight (m, p) in $${\mathbb {C}}^2 $$ is in fact a (m, p)-circular domain Cartan (J Math Pures Appl 96:1–114, 1931). In this article, we give isotropic subgroups and holomorphic automorphism groups of bounded (m, 1)-circular domains under linear equivalences by Lie group technique. Adding to Thullen (Math Ann 104:244–259, 1931), Xu (Acta Math Sin 13(3):419–432, 1963), holomorphic automorphisms of bounded (m, p)-circular domains are determined.

On Bounded Positive (m, p)-Circle Domains

Let D be a bounded positive (m, p)-circle domain in ℂ2. The authors prove that if dim(Iso(D)0) = 2, then D is holomorphically equivalent to a Reinhardt domain; if dim(Iso(D)0) = 4, then D is holomorphically equivalent to the unit ball in ℂ2. Moreover, the authors prove the Thullen’s classification on bounded Reinhardt domains in ℂ2 by the Lie group technique.

Symmetries of a reduced fluid-gyrokinetic system

Symmetries of a fluid-gyrokinetic model are investigated using Lie group techniques. Specifically, the nonlinear system constructed by Zocco & Schekochihin (Phys. Plasmas, vol. 18, 2011, 102309), which combines nonlinear fluid equations with a drift-kinetic description of parallel electron dynamics, is studied. Significantly, this model is fully gyrokinetic, allowing for arbitrary $k_{\bot }\unicode[STIX]{x1D70C}_{i}$, where $k_{\bot }$ is the perpendicular wave vector of the fluctuations and $\unicode[STIX]{x1D70C}_{i}$ the ion gyroradius. The model includes integral operators corresponding to gyroaveraging as well as the moment equations relating fluid variables to the kinetic distribution function. A large variety of exact symmetries is uncovered, some of which have unexpected form. Using these results, new nonlinear solutions are constructed, including a helical generalization of the Chapman–Kendall solution for a collapsing current sheet.

An Analysis of Stresses and Applications of Viscoelastic Fluid Flow in a circular Pipe through Porous media via with Oldroyd–B Constitutive Model

In the field of computational fluid dynamics, finding the analytical solutions of partial differential equations governing complex problems presents a challenge to the researchers. Our basic aim is to find the symmetries of Lei- group theory of viscoelastic fluid flows to obtain the results to the PDE’s system including the continuity equation, momentum equations and constitutive equations related with viscoelastic stresses, under suitable initial and boundary conditions.This paper presents the passing hydrodynamics actions of the viscoelastic fluid flow in pipes through porous media related with Darcy-Brinkman model. The research is lectured by means of analytical results and also numerical results of the problems of given differential equations associated by means of initial and boundary conditions, happening in the knowledge of viscoelastic flow in pipes through porous media. The solutions of problem of this paper are obtained adopting the Lie group of transformation theoretic approach and analytical solution is obtained under the symmetries for the equations through Lie group technique.

Projective shape analysis of contours and finite 3D configurations from digital camera images

We analyze infinite dimensional projective shape data collected from digital camera images, focusing on two-sample hypothesis testing for both finite and infinite extrinsic mean configurations. The two sample test methodology is based on a Lie group technique that was derived by Crane and Patrangenaru (J Multivar Anal 102:225–237, 2011) and Qiu et al. (Neighborhood hypothesis testing for mean change on infinite dimensional Lie groups and 3D projective shape analysis of matched contours, 2015). In infinite dimensions, the equality of two extrinsic means is likely to be rejected, thus a neighborhood hypothesis is suitably tested, combining the ideas in these two papers with data analysis methods on Hilbert manifolds in Ellingson et al. (J Multivar Anal 122:317–333, 2013). In this manuscript, we apply these general results to the two sample problem for independent projective shapes of 3D facial configurations and for matched projective shapes of 2D and 3D contours. Digital images of 3D scenes are today at the fingertips of any statistician. Here and in the literature referenced in the in this paper, we provide a methodology for properly analyzing such data, when more pictures of a given scene are available.

The onset of nanofluid flow past a convectively heated shrinking sheet in presence of heat source/sink: A Lie group approach

The influence of uniform heat source or sink of Cu–water and Ag–water nanofluid flow over a convectively heated shrinking sheet has been studied. Lie group technique has been applied to translate highly non-linear governing partial differential equations into ordinary differential equation and then solved numerically by RK-4 scheme with shooting technique. The consequence of relevant parameters on the flow province has been discussed by means of graphical and tabular approach considering both assisting and opposing flow. Our analysis explores that Cu–water achieves high temperature even though Ag has high thermal conductivity with the impact of shrinking ratio parameter. Also temperature and heat transfer rate of Cu–water rise significantly for surface convection parameter.