Lie Group Analysis Method Research Articles

A Note On Lie Symmetry Analysis, Optimal System,New Solitary Wave Solutions And Conservation Laws Of The Pavlov Equation

In this essay, based on the Lie group analysis method, the investigation has been carried out on the symmetry properties of the (2 + 1)-dimensional Pavlov equation. Its maximal symmetry group in Lie's sense and the corresponding one-dimensional optimal system of subgroups are presented. Furthermore, by using Ibragimov's method which is a generalization of Noether's theorem, the conservation laws for the intended equation are determined. Finally, utilizing the traveling wave solutions method, some exact solutions of this nonlinear partial differential equation are constructed.

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.

Qualitative analysis of magnetohydrodynamics Powell–Eyring fluid with variable electrical conductivity

ABSTRACT The non-Newtonian fluid model helps to visualize the fluid movements in modern industrial materials for the enhancement of work productivity. Therefore, this numerical investigation examines the flow behaviour of Powell–Eyring fluid over the stretching sheet with variable electric conductivity and thermal radiation. We have used the Lie group analysis method to reduce the governing momentum and energy equations into ordinary differential equations (ODEs). The resultant system is then numerically solved using the shifted Chebyshev collocation method. The accuracy of the numerical method has been verified by comparing the current work to existing literature, and it is found to be in excellent agreement. The effects of rheological parameters like stretching sheet parameter, magnetic field parameter, material parameters, suction/injection parameter, radiation parameter and Prandtl number on fluid velocity and temperature profiles are examined in detail through tables and graphs. The main motivation behind this study is to examine the effects of Lorentz force on fluid velocity and temperature in the presence of thermal radiation, which has several industrial applications. It is observed that the velocity profile decreases when the magnetic field parameter increases. Furthermore, it is noticed that increasing thermal radiation parameter increases the temperature profile.

Time-fractional generalized fifth-order KdV equation: Lie symmetry analysis and conservation laws

The purpose of this study is to apply the Lie group analysis method to the time-fractional order generalized fifth-order KdV (TFF-KdV) equation. We examine applying symmetry analysis to the TFF-KdV equation with the Riemann–Liouville (R–L) derivative, employing the G′/G-expansion approach to yield trigonometric, hyperbolic, and rational function solutions with arbitrary constants. The discovered solutions are unique and have never been studied previously. For solving non-linear fractional partial differential equations, we find that the G′/G-expansion approach is highly effective. Finally, conservation laws for the equation are well-built with a full derivation based on the Noether theorem.

Lie group analysis method for wall climbing robot systems

Lie group analysis method for wall climbing robot systems

A self-similar solution for unsteady adiabatic and isothermal flows behind the shock wave in a non-ideal gas using Lie group analysis method with azimuthal or axial magnetic field in rotating medium

The similarity solutions using Lie group analysis for shock wave propagation in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field in the case of isothermal and adiabatic flows are obtained. All possible cases of similarity solutions are discussed using the Lie group analysis for the isothermal and adiabatic flows. The arbitrary constants involved in the generators of local Lie groups bring various possible cases of solutions with exponential law and power law shock paths. Similarity solution for isothermal and adiabatic flows with power law shock path is discussed in detail. The density of ambient medium is taken to be constant. The axial and azimuthal fluid velocities and magnetic field in the ambient medium are assumed to be varying according to power law. The effect on shock wave strength and that on the flow variables due to variation of the Alfven Mach number, adiabatic index of the gas, non-idealness parameter, rotational parameter and initial magnetic field variation exponent are investigated. It is found that these parameters have decaying effects on shock wave. The obtained results in the case of isothermal and adiabatic flows are also compared with each other.

Lie Symmetry Analysis and Explicit Solutions for the Time-Fractional Regularized Long-Wave Equation

This paper systematically investigates the Lie group analysis method of the time-fractional regularized long-wave (RLW) equation with Riemann–Liouville fractional derivative. The vector fields and similarity reductions of the time-fractional (RLW) equation are obtained. It is shown that the governing equation can be transformed into a fractional ordinary differential equation with a new independent variable, where the fractional derivatives are in Erdèlyi–Kober sense. Furthermore, the explicit analytic solutions of the time-fractional (RLW) equation are obtained using the power series expansion method. Finally, some graphical features were presented to give a visual interpretation of the solutions.

Symmetries and exact solutions of the population balance equation with growth and breakage processes, using group analysis

Population balance equation which may be employed to handle a wide variety of particle processes has certainly received unprecedented attention, but the absence of explicit exact solutions necessitates the use of numerical approaches. The one‐dimensional population balance equation with time independent but size dependent growth and breakage rate is investigated. The corresponding system of equation is solved analytically by use of the Lie group analysis method. Symmetries, reduced equations, invariant solutions, explicit exact, and unphysical solutions are presented in this paper.

Group formalism of Lie transformations, exact solutions and conservation laws of nonlinear time-fractional Kramers equation

In this paper, we will concentrate on a systematic investigation of finding Lie point symmetries of the nonlinear [Formula: see text]-dimensional time-fractional Kramers equation via Riemann–Liouville and Caputo derivatives. By using the Lie group analysis method, the invariance properties and the symmetry reductions of the time-fractional Kramers equation are provided. It is shown that by using one of the symmetries of the underlying equation, it can be transformed into a nonlinear [Formula: see text]-dimensional fractional differential equation with a new dependent variable and the derivative in Erdélyi–Kober sense. Furthermore, we construct some exact solutions for the time-fractional Kramers equation using the invariant subspace method. In addition, adapting Ibragimov’s method, using Noether identity, Noether operators and formal Lagrangian, we construct conservation laws of this equation.

Lie group analysis and invariant solutions of (3+1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation

We study the (3[Formula: see text]+[Formula: see text]1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation by using the Lie group analysis method. The reduction equations are given out by selecting different constants and new explicit solutions are also presented by the Riccati equation expansion and the power series expansion.

Lie group analysis method for the transverse shift of electromagnetic wave on interface of media

The spin Hall effect of electromagnetic wave has important application prospects in the fields of Optical-electromechanical conversion, condensed matter and high energy physics. Based on the Euler–Lagrange equations of electromagnetic field by four-dimensional covariation, a deflection model of electromagnetic wave reflecting and refracting on interface of media is established in this paper. By using Lie group analysis method, the allowable infinitesimal symmetries, conserved quantities of partial differential equations and motion equation of energy center are derived. More importantly, their detailed analytical expressions and properties are given. The content of this paper can reveal the mechanism and influencing factors for transverse shift of the electromagnetic wave energy center, and provide a powerful theoretical basis for the numerical simulation of nonlinear photoelectric wave equation and spin regulation.

LIE SYMMETRY ANALYSIS TO FISHER'S EQUATION WITH TIME FRACTIONAL ORDER

The aim of this letter is to apply the Lie group analysis method to the Fisher's equation with time fractional order. We considered the symmetry analysis, explicit solutions to the time fractional Fisher's(TFF) equations with Riemann-Liouville (R-L) derivative. The time fractional Fisher's is reduced to respective nonlinear ordinary differential equation(ODE) of fractional order. We solve the reduced fractional ODE using an explicit power series method.

Lie symmetries, exact solutions and conservation laws of the Oskolkov–Benjamin–Bona–Mahony–Burgers equation

In this paper, the Oskolkov–Benjamin–Bona–Mahony–Burgers (OBBMB) equation has been investigated by Lie symmetry analysis. Lie group analysis method is implemented to derive the vector fields and symmetry reductions. The OBBMB equation has been reduced into nonlinear ordinary differential equation (ODE) by exploiting symmetry reduction method. Using the similarity reduction equation, the exact solutions are obtained by extended [Formula: see text]-method. Finally, the new conservation theorem proposed by Ibragimov has been utilized to construct the conservation laws of the aforesaid equation.

Symmetries, conservation laws and exact solutions of the time-fractional diffusivity equation via Riemann–Liouville and Caputo derivatives

ABSTRACT In this paper, we deal with the diffusivity equation, by using the Lie group analysis method. The infinitesimal generators of this equation are investigated. The concept of non-linear self-adjointness is employed to construct the conservation laws for fractional evolution equations using its Lie point symmetries. The approach is demonstrated on diffusivity equation with the Riemann–Liouville and Caputo time-fractional derivatives. It is shown that this equation is non-linearly self-adjoint and therefore desired conservation laws can be obtained using appropriate formal Lagrangian. Lie group method provides an efficient tool to solve time-fractional diffusivity equation. For this, the similarity reductions are performed with the similarity variables obtained from symmetry operators. As a result, the reduced fractional ordinary differential equations (FODEs) are deduced, and some group invariant solutions in the explicit form are obtained as well. We apply the invariant subspace method for constructing more particular solutions of the desired equation. It allows one to reduce an fractional partial differential equation to a system of non-linear FODEs.

Symmetry reductions, exact solutions, stability analysis and conservation laws of time-fractional Sharma–Tasso–Olever equation

In this paper, Lie group analysis method is applied to consider a vector field and symmetry reductions for time-fractional Sharma–Tasso–Olver equation, exact hyperbolic wave solutions, power series solutions and its convergence are investigated. Stability and boundedness analysis of trivial solution to the reduced ordinary differential equation is shown by constructing appropriate Lyapunov function. Conservation laws of the equation are well constructed with a detailed derivation making use of Noether’s operator.

LIE SYMMETRY ANALYSIS TO THE WEAKLY COUPLED KAUP–KUPERSHMIDT EQUATION WITH TIME FRACTIONAL ORDER

The aim of this paper is to apply the Lie group analysis method to the weakly coupled Kaup–Kupershmidt (KK) equation with time fractional order. We considered the symmetry analysis, explicit solutions to the weakly coupled time fractional KK (TF-KK) equation with Riemann–Liouville (RL) derivative. The weakly coupled TF-KK equation is reduced to a nonlinear ordinary differential equation (ODE) of fractional order. We solve the reduced fractional ODE using the sub-equation method.

Symmetry analysis of some nonlinear generalised systems of space–time fractional partial differential equations with time-dependent variable coefficients

In this paper, the Lie group analysis method is applied to carry out the Lie point symmetries of some space–time fractional systems including coupled Burgers equations, Ito’s system, coupled Korteweg–de-Vries (KdV) equations, Hirota–Satsuma coupled KdV equations and coupled nonlinear Hirota equations with time-dependent variable coefficients with the Riemann–Liouville derivative. Symmetry reductions are constructed using Lie symmetries of the systems. To the best of our knowledge, nobody has so far derived the invariants of space–time nonlinear fractional partial differential equations with time-dependent coefficients.

Symmetry analysis and some new exact solutions of Born–Infeld equation

This paper propounds the Lie group analysis method for finding exact solutions of Born–Infeld (BI) equation arising in nonlinear electrodynamics. We obtain generators of infinitesimal transformations, commutator table of Lie algebra, the complete geometric vector field, group symmetries and reduction equations. For the set of geometric vector field, we find an optimal system of the vector fields. Each element in this system helps to reduce the main equation into an ordinary differential equation, which provides analytical solution to the BI equation. We perform numerical simulation to obtain an appropriate visual appearance and dynamic behavior of the traced solutions. The nature of the solutions is investigated both analytically and physically through their evolutionary profile by considering appropriate choices of arbitrary constants.

Lie point symmetries for biological magneto-Jeffrey fluid flow in expanding or contracting permeable walls: a blood vessel model

ABSTRACTThis article addresses biological flow in expanding or contracting blood vessels. The alternate contraction and expansion of vessels is known to act as a physiological pump to generate flow. When muscles compress blood vessel walls, the valve at the end of the source is closed, and the downstream end opens; for this reason, blood is pumped in the downstream direction. To study this situation, a model has been developed here that consists of the unsteady two-dimensional flow of an incompressible magneto-Jeffrey fluid in a porous semi-infinite channel with expanding or contracting walls; the channel is closed from one end by a compliant membrane. The Lie group analysis method was used to transform a system of nonlinear partial differential equations into nonlinear ordinary differential equations that were solved using the perturbation method. The effects of Jeffrey parameters (, which is the ratio of relaxation time to retardation time, and Deborah parameter ) and other physical parameters are plotted and discussed. We find that the axial velocity of a Newtonian fluid is greater than that of a non-Newtonian fluid if the walls are in contraction and vice versa if the walls are in expansion. If the walls are expanded, the blood is distributed in a large area, and thus, the normal pressure decreases, while if the walls are contracted, blood is distributed in less space, and thus, the pressure increases. Normal pressure in expanding blood vessels is less than that in contracting blood vessels.

Invariant analysis and conservation laws for the time fractional [formula omitted]-dimensional Zakharov–Kuznetsov modified equal width equation using Lie group analysis

In this paper, the invariance properties of the time fractional (2+1)-dimensional Zakharov–Kuznetsov modified equal width (ZK-MEW) equation have been investigated using the Lie group analysis method. Lie point symmetries of the time fractional (2+1)-dimensional ZK-MEW equation have been derived by using the Lie group analysis method of fractional differential equations. Using the Lie symmetry analysis, the vector fields and the symmetry reduction of this equation are obtained. It is shown that the time fractional (2+1)-dimensional ZK-MEW equation can be transformed to an equation with Erdélyi–Kober fractional derivative. Finally using new conservation theorem with formal Lagrangian, the new conserved vectors are well constructed with a detailed derivation, which constitutes the conservation analysis for the time fractional (2+1)-dimensional ZK-MEW equation.