Lie Group Method Research Articles

Lie symmetry analysis, traveling wave solutions and conservation laws of a Zabolotskaya-Khokholov dynamical model in plasma physics

This article analyzes the analytic and solitary wave solutions of the one-dimensional Zabolotskaya-Khokholov (ZK) dynamical model which provides information about the propagation of sound beam or confined wave beam in nonlinear media and studies of beam deformation. By the Lie symmetry analysis method, we acquire the vector fields, commutation relations, optimal system, reduction, and analytic solutions to the specified equation by exerting the Lie group method. Moreover, the solitary wave solutions of the ZK model are procured by exerting the new auxiliary equation method (NAEM). The behavior of the acquired outcomes for several cases is exhibited graphically through two and three-dimensional dynamical wave profiles. Furthermore, the conservation laws of the ZK model are acquired by Ibragimov’s new conservation theorem.

Signature of conservation laws and solitary wave solution with different dynamics in Thomas–Fermi plasma: Lie theory

We propose a Lie group method to discuss the modified KP equation appearing in Thomas–Fermi (TM) plasma, characterised by cold and hot electrons. The Lie method facilitates the identification of similarity reductions, infinitesimal symmetries, group-invariant solutions, and novel analytical solutions for nonlinear models. The similarity reduction method is carried out to transform the nonlinear partial differential equations (NLPDE) into the nonlinear ordinary differential equations (NLODE). This study focuses on solitary wave profiles due to their usefulness in various engineering applications, including monitoring public transportation systems, managing coastlines, and mitigating disaster risks. It also addresses the conservation laws associated with the modified KP equation. The generalised auxiliary equation (GAEM) scheme is used to compute the new solitary wave patterns of the modified KP equation, which explains the dynamics of nonlinear waves in Thomas–Fermi plasma. The idea of nonlinear self-adjointness is used to compute the conservation laws of the examined model. The graphical behaviour of some solutions is represented by adjusting the suitable value of the parameters involved.

The Closed-Form Solutions of an SIS Epidemic Reaction–Diffusion Model with Advection in a One-Dimensional Space Domain

This work investigates a class of susceptible–infected–susceptible (SIS) epidemic model with reaction–diffusion–advection (RDA) by utilizing the Lie group methods. The Lie symmetries are computed for the three widely used incidence functions: standard incidence, mass action incidence, and saturated incidence. The Lie algebra for the SIS-RDA epidemic model is four-dimensional for the standard incidence function, three-dimensional for mass action incidence, and two-dimensional for saturated incidence. The reductions and closed-form solutions for the SIS-RDA epidemic model for the standard incidence infection mechanism are established. The transmission dynamics of an infectious disease utilizing closed-form solutions is presented. To illustrate the paths of susceptible and infected populations, we consider the Cauchy problem. Moreover, a sensitivity analysis is conducted to provide insights into potential policy recommendations for disease control.

Partially Nonclassical Method and Conformal Invariance in the Context of the Lie Group Method

Partially Nonclassical Method and Conformal Invariance in the Context of the Lie Group Method

Analyzing invariants and employing successive reductions for the extended Kadomtsev Petviashvili equation in (3+1) dimensions.

In this research, we employ the potent technique of Lie group analysis to derive analytical solutions for the (3+1)-extended Kadomtsev-Petviashvili (3D-EKP) equation. The systematic application of this method enables the identification of Lie point symmetries associated with the equation, leading to the derivation of an optimal system of one-dimensional subalgebras relevant to the equation. This optimal system is utilized to obtain several invariant solutions. The Lie group method is subsequently applied to the reduced governing equations derived from the given equation. We complement our findings with Mathematica simulations illustrating some of the obtained solutions. Furthermore, a direct approach is used to investigate local conservation laws. Importantly, our study addresses a gap in the exploration of the 3D-EXP equation using group theoretic methods, making our findings novel in this context.

Invariant analysis of the two-cell tumor growth model in the brain

In this study, we explore the invariance properties of a tumor growth model involving two distinct cell populations. These populations are characterized by different diffusion coefficients but share a common killing rate. This particular model serves as a representation of tumor growth within the brain. By employing the Lie group method, we unveil a two-dimensional symmetry algebra for cases where both diffusion coefficients are allowed to vary arbitrarily. Interestingly, this method reveals a nine-dimensional symmetry algebra when the diffusion coefficients are held constant. In both scenarios involving varying and constant diffusion coefficients, we conduct similarity reductions to deduce group invariant solutions, thus elucidating the model’s behavior. Notably, our findings demonstrate that the tumor’s growth remains exponential irrespective of the presence or absence of a killing rate. This remarkable phenomenon holds for various configurations of diffusion coefficients. To validate our observations, we employ Mathematica simulations, which corroborate the model’s exponential growth behavior and emphasize the role of killing rates, diffusion coefficients, and growth rate parameters in driving this exponential trend. Also, the conserved flows and conserved quantities of the model are demonstrated.

Novel high-order explicit energy-preserving schemes for NLS-type equations based on the Lie-group method

In this paper, by taking the NLS-type equations as examples, we propose a novel high-order explicit energy-preserving Lie-group method for Hamiltonian PDEs. The main idea is to combine recently developed generalized SAV (G-SAV) approach (Cheng et al., 2021) and explicit Lie group method, in which the fourth-order or higher-order Lie group method only composes two exponentials in each stage, and this is the key point that the proposed methods can preserve energy of Hamiltonian PDEs. This is our first attempt to construct high-order explicit energy-preserving methods without using projection technique (Hairer et al., 2006 [14]), and some stability conclusions of our proposed methods for the NLS-type equations are also presented. Finally, we present 2D and 3D numerical simulations to demonstrate the stability and accuracy.

Dynamics of invariant solutions of the DNA model using Lie symmetry approach

The utilization of the Lie group method serves to encapsulate a diverse array of wave structures. This method, established as a robust and reliable mathematical technique, is instrumental in deriving precise solutions for nonlinear partial differential equations (NPDEs) across a spectrum of domains. Its applications span various scientific disciplines, including mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and several others. This research focuses specifically on the crucial molecule DNA and its interaction with an external microwave field. The Lie group method is employed to establish a five-dimensional symmetry algebra as the foundational element. Subsequently, similarity reductions are led by a system of one-dimensional subalgebras. Several invariant solutions as well as a spectrum of wave solutions is obtained by solving the resulting reduced ordinary differential equations (ODEs). These solutions govern the longitudinal displacement in DNA, shedding light on the characteristics of DNA as a significant real-world challenge. The interactions of DNA with an external microwave field manifest in various forms, including rational, exponential, trigonometric, hyperbolic, polynomial, and other functions. Mathematica simulations of these solutions confirm that longitudinal displacements in DNA can be expressed as periodic waves, optical dark solitons, singular solutions, exponential forms, and rational forms. This study is novel as it marks the first application of the Lie group method to explore the interaction of DNA molecules.

Small-Scale Cosmology Independent of the Standard Model

‘Small-scale cosmology’ is a theory designed to incorporate the linear redshift versus distance relation, which is inferred from observations, into the theoretical framework independent of the global Robertson–Walker–Friedman (RWF)-type models. The motivation behind this is that the RWF cosmological models, based on the assumptions of homogeneity and a constant matter density, as well as the concept of expanding space inherent to them are not applicable on the scales of observations from which the linear Hubble law is inferred. Therefore, explaining the Hubble law as the small redshift limit of the RWF model or as an effect of expanding space is inconsistent. Thus, the Hubble linear relation between the redshift of an extragalactic object and its distance should be considered an independent law of nature valid in the range of the distances where the RWF cosmology is not valid. In general, the theory, based on that concept, can be developed in different ways. In the present paper, ‘small-scale cosmology’ is formulated as a theory operating in the (redshift–object coordinates) space, which allows developing a conceptual and computational basis of the theory along the lines of that of special relativity. In such a theory, the condition of invariance of the Hubble law with respect to a change in the observer acceleration plays a central role. In pursuing this approach, the effectiveness of group theoretical methods is exploited. Applying the Lie group method yields transformations of the variables (the redshift and space coordinates of a cosmological object) between the reference frames of the accelerated observers. In this paper, the transformations are applied to studying the effects of the solar system observer acceleration on the observed shape, distribution and rotation curves of galaxy clusters.

Influence of magnetic field-dependent viscosity on Casson-based nanofluid boundary layers: A comprehensive analysis using Lie group and spectral quasi-linearization method

This study examines the effects of magnetic-field-dependent (MFD) viscosity on the boundary layer flow of a non-Newtonian sodium alginate-based Fe3O4 nanofluid over an impermeable stretching surface. The non-Newtonian Casson and homogeneous nanofluid models are utilized to derive the governing flow and heat transfer equations. Applying Lie group transformations to dimensional partial differential equations yields nondimensional ordinary differential equations, which are then numerically solved using the spectral quasi-linearization technique. The analysis primarily focuses on the impacts of the MFD viscosity parameter, nanoparticle volume fraction of Fe3O4, and magnetic parameters on the flow and heat transfer characteristics. The local skin friction and heat transfer rate behaviors influenced by viscosity changes due to the magnetic field are discussed. It is found that MFD viscosity significantly impacts flow and thermal energies, enhancing skin friction coefficients and reducing Nusselt numbers in the boundary layer region.

Optimal System, Symmetry Reductions and Exact Solutions of the (2 + 1)-Dimensional Seventh-Order Caudrey–Dodd–Gibbon–KP Equation

In this paper, the (2+1)-dimensional seventh-order Caudrey–Dodd–Gibbon–KP equation is investigated through the Lie group method. The Lie algebra of infinitesimal symmetries, commutative and adjoint tables, and one-dimensional optimal systems is presented. Then, the seventh-order Caudrey–Dodd–Gibbon–KP equation is reduced to nine types of (1+1)-dimensional equations with the help of symmetry subalgebras. Finally, the unified algebra method is used to obtain the soliton solutions, trigonometric function solutions, and Jacobi elliptic function solutions of the seventh-order Caudrey–Dodd–Gibbon–KP equation.

Evaluation and implementation of Lie group integration methods for rigid multibody systems

AbstractAs commonly known, standard time integration of the kinematic equations of rigid bodies modeled with three rotation parameters is infeasible due to singular points. Common workarounds are reparameterization strategies or Euler parameters. Both approaches typically vary in accuracy depending on the choice of rotation parameters. To efficiently compute different kinds of multibody systems, one aims at simulation results and performance that are independent of the type of rotation parameters. As a clear advantage, Lie group integration methods are rotation parameter independent. However, few studies have addressed whether Lie group integration methods are more accurate and efficient compared to conventional formulations based on Euler parameters or Euler angles. In this paper, we close this gap using the $\mathbb{R}^{3}\times SO(3)$ R 3 × S O ( 3 ) Lie group formulation and several typical rigid multibody systems. It is shown that explicit Lie group integration methods outperform the conventional formulations in terms of accuracy. However, it turns out that the conventional Euler parameter-based formulation is the most accurate one in the case of implicit integration, while the Lie group integration method is computationally the more efficient one. It also turns out that Lie group integration methods can be implemented at almost no extra cost in an existing multibody simulation code if the Lie group method used to describe the configuration of a body is chosen accordingly.

Invariance properties of the microstrain wave equation arising in microstructured solids

The Lie group method stands as a significant, powerful, and straightforward mathematical tool for discovering precise invariant solutions and traveling wave solutions in prevalent nonlinear evolution equations across engineering, applied mathematics, and physics. The interplay of dispersive effects arising from material microstructure, combined with nonlinearities, results in the formation of solitary waves. In this article, we apply the Lie group method to derive invariant and comprehensive traveling wave solutions for the strain wave equation in microstructured solids. This yields a diverse array of exact solutions, including traveling wave solutions, solitons, shock waves, periodic, singular, and rational solutions, all of which bear potential significance in various engineering applications.

The lie group analysis method for heat transfer in steady boundary layer flow field of nanofluid

In order to reveal the mechanism of the abnormal movement (Brownian motion enhances thermal scattering) of nanoparticles on the fluid enhanced heat transfer, the two-phase model was used to study the abnormal convection and diffusion of viscous nanofluids in the flat boundary layer of porous medium. Firstly, for the two-dimensional steady boundary layer stagnation point flow of incompressible Newtonian-nanofluids, the nonlinear governing equations of the flow field and temperature field of nanofluids are established from the Oberbeck-Boussinesq approximate equations. Secondly, the modern Lie group analysis method is introduced, we give the Lie symmetry determining equation of the flow field partial differential equations and the characteristics of the solutions. Further, using the relationship between the Lie symmetries and the conserved quantities, the conservation vector form of the flow field and the group invariant solution are derived in detail, and the reduced order model of the nanofluid flat boundary layer is obtained. Finally, the correctness of the analytical results obtained by the Lie group method was verified for different values of the flow parameter Prandtl. Research has shown that the Lie group method can be used to analytically solve the velocity and temperature distribution functions of abnormal motion of nanoparticles. The fluid temperature increases with the increase of the volume fraction parameter of nanoparticles, but decreases with the increase of the Prandtl value of the base fluid, and decreases with the increase of the plate stretching speed. The Lie group analysis method in this paper provides reference value for numerical simulation solutions of various heat and mass transfer in nanofluids.

Lie group analysis, solitons, self-adjointness and conservation laws of the nonlinear elastic structural element equation

This study is based on the Lie group method for the nonlinear elastic structural element equation (ESE Equation). We obtain a three-dimensional Lie algebra. By utilizing this Lie algebra a four-dimensional optimal system is constructed. The governing ESE Equation is converted to nonlinear ordinary differential equations (ODEs) via symmetry reduction. We use a modified auxiliary equation (MAE) procedure to deal with nonlinear ODEs. These ODEs reveal the dynamics of the periodic and soliton solutions. We obtain soliton solutions through rational, trigonometric, and hyperbolic functions. Wolfram Mathematica simulations vividly illustrate the wave characteristics of the derived solutions, affirming their properties as singular periodic solutions, a singular solution, an optical dark soliton solution, and a singular soliton solution. We also obtain the local conservation laws by a new conservation theorem introduced by Ibragimov.

Symmetry analysis for the (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves

This article delves into the wave dynamics of the (3+1)-dimensional nonlinear model, which serves as a representation of shallow water waves. This model finds relevance in addressing various natural phenomena such as tides, storms, atmospheric flows, and tsunamis, all linked to shallow water waves. These waves, often called long water waves, exhibit a considerable wavelength relative to their depth. The Lie group method ensures a wide range of wave structures. This method is a recognized and dependable mathematical technique for obtaining precise solutions to nonlinear partial differential equations across various domains. Its applications span fields such as mathematical physics, nonlinear dynamics, oceanography, engineering sciences, and numerous other disciplines. Furthermore, we elucidate the physical implications of certain solutions by generating 2D and 3D graphs using the corresponding parameter values.

A reduced-order lie group-based race car model for systematic trajectory optimization on 3D tracks

AbstractThis paper derives the dynamic equations of a reduced-order race-car model using Lie-group methods. While these methods are familiar to computational dynamicists and roboticists, their adoption in the vehicle dynamics community is limited. We address this gap by demonstrating how this framework integrates smoothly with the Articulated-Body Algorithm (ABA) and provides a fresh and systematic formulation of vehicle dynamics. For the first time, we model the car body as the end effector of a serial robot with a floating base connected to the track via virtual revolute and prismatic joints. Our formulation also accounts for the effects of 3D track geometry, providing a natural embedding of the car into the 3D track. We rigorously reconcile the ABA steps with key aspects of vehicle dynamics, including road-tire interactions, aerodynamic forces, and load transfers. The resulting model, simple yet accurate, is a powerful tool to efficiently solve Minimum-Lap-Time Planning problems. To demonstrate the effectiveness of our approach, we show numerical results obtained on the Nürburgring circuit. Our optimization problem is formulated via a direct collocation method and solved using the CasADi optimization suite. To validate the results, we test our reduced-order model against a full-fledged multi-body model recently developed by the same authors. The comparison confirms the validity of our reduced-order model, proving both the accuracy of the solution and the computational efficiency achieved.

The Upshot of Nonlinear Thermal Emission on a Conducting Jeffrey Nanofluid Flow over a Stretching Sheet: A Lie Group Approach

The upshot of nonlinear thermal radiation of steady state MHD heat transfer of Jeffrey nanofluid together with prescribed boundary conditions of interest was studied. Essential fluid properties, dimensionless switch parameters with the assistance of the Lie group method were used to transform the convenient partial differential equations that describe the present model into a system of ordinary differential equations. The generalized flow of the present model incorporates Jeffrey parameters and nonlinear thermal radiation. The Mathematical model is first renovated into ordinary differential equations by Lie symmetry group alteration. The renovated equations were solved numerically using a bvp4c MATLAB solver. The Jeffrey parameter serves as a stabilizer on the velocity of the fluid while thermal radiation parameter and heat source-sink parameter improves the flow temperature. Lewis number, chemical reaction parameter diminished the mass transfer flow. Equally skin friction, Sherwood number and Nusselt number were expanded.

Analysis of MHD Fluid Flow and Heat Transfer Inside an Inclined Deformable Filter Chamber: Lie Group Method

This paper investigates MHD fluid flow and distribution of heat inside a filter chamber during a process of filtering particles from the fluid. A flow model of MHD viscous incompressible fluid inside a filter is studied to seek semi-analytical solutions which are analysed to find flow and heat dynamics that lead to optimal outflow (maximum filtrates) during filtration. Lie group method is used to reduce a system of four partial differential equations describing fluid flow and temperature distribution inside the filter chamber to a system of two ordinary differential equations. The reduced system is then solved by perturbation process to obtain semi-analytical solutions for flow velocity and temperature variation inside the chamber. To understand the flow dynamics and heat distribution of the underlying case study better, the influence of different parameters during filtration resulting from the filter design, flow dynamics and heat effects are graphically presented and analysed in order to identify a combination of flow parameters that yields the best filtration process. The findings show that to maximise filtrates production, more fluid injection is required during filtration. Moreover, from the findings, it is evident that the temperature increase inside the chamber arising from the Joule heating effect is ideal since it increases internal work done and hence increases filtrates production.

Explicit high accuracy energy-preserving Lie-group sine pseudo-spectral methods for the coupled nonlinear Schrödinger equation

The newly developed generalized scalar auxiliary variable (GSAV) approaches are very popular methods that can be applied to construct high-accuracy energy-preserving schemes for conservative systems. However, these proposed high-order schemes are fully-implicit and have the disadvantages of huge calculating quantities and long computing time for solving nonlinear systems. Therefore, we develop in this paper explicit energy-preserving schemes to deal with the coupled nonlinear Schrödinger equation by combining the Lie-group method and GSAV approaches. The given schemes are very efficient, have high accuracy, and can inherit the modified energy of the system. Ample numerical results are presented to confirm the developed schemes' efficiency, accuracy, and conservation.