Lie Group Analysis Research Articles

Explicit Solutions of a Class of (3+1)-Dimensional Nonlinear Model

In this article, we employ Lie group analysis to obtain symmetry reduction of a class of (3+1)-dimensional nonlinear model. This nonlinear model plays a critical role in the study of nonlinear sciences. By the exp$(-\varphi(z))$-expansion method, we construct explicit solutions for the proposed equation. Four types of explicit solutions are obtained, which are hyperbolic, exponential, trigonometric and rational function solutions.

Lie group analysis and dynamical behavior for classical Boussinesq–Burgers system

In this paper we consider classical Boussinesq–Burgers system. Lie group method is used to analyze the system in detail, and the system is reduced into ODEs under the Lie symmetries. Then exact traveling wave solutions of classical Boussinesq–Burgers system are obtained by simplest equation method. Further classical Boussinesq–Burgers system is shown to be self-adjoint and the new conservation laws of system are constructed for the first time. The dynamical behaviors of classical Boussinesq–Burgers system are analyzed.

ANALYSIS OF FLOW AND HEAT TRANSFER IN THE BOUNDARY LAYER OF FLUID WITH ESSENTIAL DEPENDENCE OF VISCOSITY FROM TEMPERATURE

The introduction of new environmentally safe, durable and fire-resistant thermal insulation based on basalt fiber requires the improvement of technological processes for the production of basalt filaments in order to increase their energy efficiency. The quality of basalt fiber significantly depends on the properties of the basalt melt in the process. The process is considered to be more perfect than the more homogeneous and isotropic final properties of the melt. The conditions of flow and heat transfer in the bath and feeder of the melting furnace have a significant impact on the final properties of the melt.
 The paper presents the results of studies of heat transfer in the boundary layer on a flat plate with the flow of a liquid, the viscosity of which depends significantly on temperature. The system of differential equations, which describes the steady-state flow regime, is solved using symmetry analysis (analysis of Lie groups).
 On the basis of the developed mathematical model, new results were obtained which characterize the regularities of the flow and heat exchange of a highly viscous fluid in the range of temperature variations from 900 to 1450 ° C. The conditions for the formation of a low-mobility layer of liquid near a solid surface are determined. The influence of the value of a low-mobility fluid layer on the temperature distribution in the boundary layer of basalt melt is established.
 The results obtained allow us to adjust the temperature regimes in the melting furnace and improve the process of manufacturing ultrathin basalt threads.

Power Series Solution of Time-Fractional Majda-Biello System Using Lie Group Analysis

In the present work, we consider a time fractional Majda-Biello system describing the equatorial baroclinic-barotropic waves which has great impact on weather forecast and seasonal climate variation. Using Lie group analysis, we derive the symmetric group of transformations for the governing system of equations. Under the invariant condition the system of fractional partial differential equations (FPDEs) is reduced to system of fractional ordinary differential equations (FODEs). Further, the power series solution of the reduced system of FODEs is constructed and it’s convergence is discussed.

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.

Magnetohydrodynamic fluid flow and heat transfer over a shrinking sheet under the influence of thermal slip

We study the heat transfer of a Magneto Hydrodynamic (MHD) boundary layer flow of a Newtonian fluid over a pervious shrinking sheet under the influence of thermal slip. The flow allows electric current to pass through. The governing PDEs are transformed into self-similar ODEs via Lie group analysis. We study the variations in the dimensionless quantities like velocity and temperature of the flow in terms of the different parameters involved in the problem. We discuss the thickness of the boundary layers under the influence of various parameters involved in the flow. Numerical simulations are carried out to explain and support the results obtained.

Symmetry and separability of the neutron diffusion equation

Separation of variables is one of the oldest techniques for solving certain classes of partial differential equations (PDEs). As is the case with many other solution techniques for differential equations, separation of variables may be codified within the broader framework of symmetry analysis. Though the separation of variables technique is frequently used in the nuclear engineering context with various equations describing neutron transport, its connection to the symmetries of those equations has not yet been thoroughly established. It is thus the purpose of this work to establish that connection using neutron diffusion as both an initial step toward analysis of more generally applicable equations, and as a connection to previous results in related problems. Using Lie group analysis, it is found that the traditional space-time separable solution of the neutron diffusion equation (featuring a single α-eigenvalue) corresponds to time translation and flux scaling symmetries. Additional solutions of this equation are also constructed using its broader symmetry set.

Lie group analysis of thermophoresis on a vertical surface in a porous medium

The magnetohydrodynamic and thermophoretic effects on a vertical surface in a porous medium are investigated. The fascinating aspects of thermo diffusion and diffusion-thermo impacts are given. Mathematical modelling via Lie group method was applied. Thereafter, the infinitesimal generators of governing equations are computed. Using suitable similarity variables the existing system which was non-linear is converted into expressions having no dimensions. The resulting expressions were solved numerically using shooting method and the characteristics of embedded parameters such as temperature, velocity, concentration profiles have been displayed graphically. We have compared our findings with those of previous ones to assure the affirmity of our analysis. The change in Sherwood number for progressive Dufour solutal values are also analyzed.

Lie group analysis, analytic solutions and conservation laws of the (3 + 1)-dimensional Zakharov-Kuznetsov-Burgers equation in a collisionless magnetized electron-positron-ion plasma

We work on the (3+1)-dimensional Zakharov-Kuznetsov-Burgers equation for the ion-acoustic waves in a collisionless magnetized electron-positron-ion plasma. We derive the Lie point symmetry generators and Lie symmetry groups. We construct certain solutions which are related to the known solutions. Using the symmetry generators, we obtain the reduction equations, through one of which we derive some power-series solutions and travelling-wave solutions including the shock solutions via the power-series and polynomial expansion methods. Shock waves are pictured out. Effects of the normalized ion gyrofrequency, $\Omega_{i}$ , the normalized kinematic viscosity, $\eta$ , the real parameter measuring the deviation from the Maxwellian equilibrium, $ \kappa$ , the ratio of the ion temperature to electron temperature, $\sigma_{1}$ , and the ratio of the electron temperature to positron temperature, $ \sigma_{2}$ , on the amplitude of the shock wave, |a|, are found: i) |a| is a hyperbolic function of $\eta$ ; ii) $\vert a\vert \rightarrow 0$ when $\Omega_{i}$ , $\kappa$ or $\eta \rightarrow 0$ ; iii) | a| goes to a constant when $ \Omega_i \rightarrow \pm \infty$ ; iv) |a| becomes a constant when $\kappa \rightarrow \pm \infty$ ; v) $\vert a\vert \rightarrow 0$ when $\sigma_{1} \rightarrow \pm \infty$ ; vi) |a| keeps unchanged when $\sigma_{2}$ varies and $\sigma_{1}= \frac{-3(2\kappa + \lambda^{2} - 2\kappa\lambda^{2})}{5(-1+2\kappa)}$ where $\lambda$ denotes the ion-acoustic wave propagation speed; vii) $\vert a\vert \rightarrow 0$ when $\sigma_{2} \rightarrow \pm \infty$ and $\sigma_{1} \neq\frac{-3(2\kappa +\lambda^{2}-2\kappa\lambda^{2})}{5(-1+2\kappa)}$ . We present the conditions of the nonlinear self-adjointness. Based on the nonlinear self-adjointness, we construct the conservation laws which are related to $\Omega_{i}$ , $\eta$ , $\kappa$ , $\sigma_{1}$ and $\sigma_{2}$ .

Predicting the Temperature Distribution in Longitudinal Fins of Various Profiles with Power Law Thermal Properties Using the Variational Iteration Method

In this article, the variational iteration method (VIM) is used to analyze heat transfer in longitudinal fins of various profiles with temperature dependent thermal conductivity and heat transfer coefficient. In order to show the efficiency of the VIM, the results obtained using the VIM are compared with the previously obtained results using the differential transform method (DTM) and Lie group analysis of a fin problem with the rectangular profile. After establishing confidence in VIM, heat transfer is analyzed to determine the temperature profiles in longitudinal fins of various profiles with power law thermal properties. The temperature distribution is compared between various profiles. The effects of some physical parameters such as the thermo-geometric fin parameter and thermal conductivity gradient on temperature distribution are illustrated.

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.

Lie symmetry analysis of a variable coefficient Calogero–Degasperis equation

We consider a general class of variable coefficient Calogero–Degasperis equations. The complete Lie group classification is performed with the aid of the appropriate equivalence group. Lie symmetries are used to derive a number of reductions by constructing the corresponding optimal lists of one-dimensional subalgebras of the Lie symmetry algebras. Furthermore, a number of non-Lie reductions are given. One of the reduced equations is the variable coefficient potential KdV equation which is studied from the point of view of Lie group analysis.

Symmetry properties, conservation laws, reduction and numerical approximations of time-fractional cylindrical-Burgers equation

In this paper the Lie group analysis of the time-fractional cylindrical Burgers equation (time-FCB), which is a fundamental PDE occurring in various areas of applied mathematics, such as fluid mechanics, non-linear acoustics, gas dynamics, traffic flow and etc. is given. For this purpose the Riemann–Liouville derivative is used to implement the Lie algorithm for finding the symmetry operators. A reduced form of the equation is given by using the similarity variables obtained from a symmetry and Erdelyi–Kober operator. In the next step conservation laws are derived via a generalization of Noether’s theorem. Finally the Chebyshev wavelets for time-fractional differential equations (FDEs) is applied for solving the considered equation.

MHD Slips Flow of a Micro-polar Fluid Due to Moving Plate in Porous Medium with Chemical Reaction and Thermal Radiation: A Lie Group Analysis

An attempt is made to investigation of MHD natural convection flow of a micropolar fluid due to a permeable moving plate in porous medium in the presence of chemical reaction and radiation. The Lie group analysis used to create similarity transformations and by using those transformations, the system of governing highly non-linear partial differential equations is converted into a set of non-linear coupled ordinary differential equations. The ordinary differential equations thus obtained have been solved by utilizing Keller box method. Numerical solutions for the skin-friction coefficient, Nusselt and Sherwood numbers as well as the velocity, micro-rotation, temperature and concentration are presented graphically for different parametric conditions and discussed in detail. The results show that the skin-friction coefficient reduces with greater values of micro-polar or material parameter and the heat transfer rate advances with reduction in micropolar and radiation parameters. The mass transfer rate accelerates with higher values of chemical reaction parameter.

Symmetry properties and explicit solutions of some nonlinear differential and fractional equations

One generalized Burgers hierarchy is derived by applying the Cole-Hopf transforation, whose dark-equation hierarchy is also generated by the dark-equation method, from which a generalized Burgers equation and a generalized Kupershmidt equation, respectively, are followed to obtain. Through Lie-group analysis method we produce similarity reductions, exact solutions of the generalized Burgers and the Kupershmidt equations. Specially, we investigate the similarity reductions of the fractional Kupershmidt equation and its exact solutions. In addition, we obtain the conservation laws of the Kupershmidt equation and its adjoint equation. Finally, we give rise to symmetries, primary branch solutions as well various recursion operators of degenerated equations from the Kupershmidt equation.

Exact Solutions of Generalizes the Harry Dym Equation Using Lie Group Analysis

Exact Solutions of Generalizes the Harry Dym Equation Using Lie Group Analysis

Lie group analysis of 2‐dimensional space‐fractional model for flow in porous media

Using the Lie group analysis approach, we study the symmetry properties of 2‐dimensional space‐fractional filtration equation with the Riesz potential. This equation is derived by a fractional generalization of Darcy law and permits to describe the pressure evolution during 1‐phase fluid flow through naturally fractured porous medium. We construct the prolongation of the point transformation group on the Riesz potential and obtain a generalization of the Leibniz rule for the Riesz potential, which are necessary for investigating symmetry properties using the invariance principle. The Lie group of linearly autonomous point transformations is constructed for the considered equation. In a limiting case of zero‐order potential, the obtained symmetry group coincides with the group of point transformations admitted by the classical integer‐order 2‐dimensional linear heat equation. Also, the asymptotic behavior of a group invariant solution is investigated.

Lie symmetry analysis, conservation laws and analytical solutions for a generalized time-fractional modified KdV equation

ABSTRACTIn this paper, a generalized time fractional modified KdV equation is investigated, which is used for representing physical models in various physical phenomena. By Lie group analysis method, the invariance properties and the vector fields of the equation are presented. Then the symmetry reductions are provided. Moreover, we construct the explicit solutions of the equation by using sub-equation method. Based on the power series theory, the approximate analytical solution for the equation are also constructed. Finally, the new conservation theorem is applied to constructed conservation laws for the equation.

Lie symmetry analysis and invariant solutions of $$\varvec{(3+1)}$$ ( 3 + 1 ) -dimensional Calogero–Bogoyavlenskii–Schiff equation

It is shown that the novel Lie group of transformations method is a competent and prominent tool in solving nonlinear partial differential equations(PDEs) in mathematical physics. Lie group analysis is used to carry out the similarity reduction and exact solutions of the (3 + 1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation. This research deals with the similarity solutions of CBS equation. We have obtained the infinitesimal generators, commutator table of Lie algebra, symmetry group and similarity reduction for the CBS equation. For the different Lie algebra, Lie symmetry method reduced (3 + 1)-dimensional CBS equation into new (2 + 1)-dimensional partial differential equations and again using Lie symmetry method these PDEs are reduced into various ordinary differential equations(ODEs).