Lie Group Method Research Articles

Evolution of singular surface and interaction with a strong shock in reacting polytropic gases using Lie group theory

We apply the Lie group method, which leaves the partial differential equations (PDEs) for the plane and radially symmetric flow of reacting polytropic gases invariant, to obtain an exact solution. Using the singular surface theory and compatibility conditions, we derive an evolution equation that governs the amplitude of the singular surface. The interaction between a singular surface and a strong shock is considered. The amplitudes of reflected and (or) transmitted wave and the jump in shock acceleration after the interaction are determined.

Solutions of Direct and Inverse Even-Order Sturm-Liouville Problems Using Magnus Expansion

In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has ability to solve direct regular (and some singular) SLPs of even orders (tested for up to eight), with a mix of (including non-separable and finite singular endpoints) boundary conditions, accurately and efficiently. The present technique is successfully applied to overcome the difficulties in finding suitable sets of eigenvalues so that the inverse SLP problem can be effectively solved. The inverse SLP algorithm proposed by Barcilon (1974) is utilized in combination with the Magnus method so that a direct SLP of any (even) order and an inverse SLP of order two can be solved effectively.

The calculation of stopping power and range for radium, thorium and uranium using new electronic potential energy function

In this paper, the electronic stopping power and range calculations of radium-226, thorium-229 and uranium-236 isotopes, which are natural alpha emitters, were performed for the alpha particles in the 200 keV-10 MeV energy range. The effective charge approach based on the Bethe-Bloch theory for the stopping power, the continuous slowing down approach and the Simpson 1/3 method were used for calculation of the range. To this end, new electronic potential energy functions were primarily obtained by using the Lie group method for the target elements. For electronic charge density, relativistic atomic natural orbitals based on Gaussian type orbitals were selected. The results were compared with the existing data sets in the literature. The potential energy curves obtained were irregular at certain distances, but were quite close to that of the multi-atomic target systems. The stripping distances of the target atoms and the corresponding potential energy values were determined from the potential energy functions. It was observed that the stopping power and range values were generally compatible with the SRIM and ICRU 49 data at specific error rates. By adding some correction factors to the calculations, it is predicted that the discrepancies and the error rates may be reduced. The results acquired with this innovative work can be used in the interaction of charged particles with matter, applications related to alpha decay and shielding.

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.

Group invariant solution for a fluid-driven fracture with a non-Darcy flow in porous medium

The evolution of a two-dimensional, pre-existing, fluid-driven permeable fracture is investigated. Fluid leak-off through the permeable fracture interface into the rock formation is modelled by a non-Darcy model — the Forchheimer model. The injected fluid is a viscous incompressible Newtonian fluid and the flow in the fracture is considered laminar. When lubrication theory and the Perkins–Kern–Nordgren approximation are employed, a system of nonlinear differential equations for the half-width of the fracture and the leak-off depth is obtained. The Lie group method is used to reduce the system of partial differential equations to a boundary value problem for a second order ordinary differential equation and to obtain group invariant and numerical solutions, based on the Forchheimer number.

Series solutions and bifurcation of traveling waves in the Benney–Kawahara–Lin equation

In this paper, the Lie group method is performed on the BKL equation. By reducing the BKL equation to four classes of ODEs, we obtain new power series solutions of it. Furthermore, as an important reduced equation, the traveling wave equation is investigated in detail. Treating it as a singular perturbation system in $${\mathbb {R}}^4$$ , we apply the geometric singular perturbation theory and dynamical system methods to study its phase space geometry. By using some techniques, such as tracking the unstable manifold of the saddle, discussing transversality of the stable and unstable manifolds and investigating some complicated nonlocal bifurcation, we obtain wave speed conditions to guarantee the existence of various bounded traveling waves of the BKL equation.

Comment on the paper “Lie group method for the modified model of MHD flow and heat transfer of a non-Newtonian fluid with prescribed heat flux over a moving porous plate, Xinhui Si, Lili Yuan, Liancun Zheng, Yanan Shen, Limei Cao, Journal of Molecular Liquids 220 (2016) 768–777”

The present comment concerns some doubtful results included in the above paper.

Symmetries and conservation laws of the Yao\u2013Zeng two-component short-pulse equation

This paper uses the Lie group method to analyze the symmetries of the Yao–Zeng two-component short-pulse equation which describes the propagation of polarized ultrashort light pulses in cubically nonlinear anisotropic optical fibers. Similarity reductions and exact solutions are obtained by constructing an optimal system of one-dimensional subalgebras. Moreover, the explicit solutions are constructed by the power series method and the convergence of power series solutions is proved. In addition, nonlinear self-adjointness and conservation laws of this system are discussed.

Analytical research of (3+1)-dimensional Rossby waves with dissipation effect in cylindrical coordinate based on Lie symmetry approach

Rossby waves, one of significant waves in the solitary wave, have important theoretical meaning in the atmosphere and ocean. However, the previous studies on Rossby waves commonly were carried out in the zonal area and could not be applied directly to the spherical earth. In order to overcome the problem, the research on (3+1)-dimensional Rossby waves in the paper is placed into the spherical area, and some new analytical solutions of (3+1)-dimensional Rossby waves are given through the classic Lie group method. Finally, the dissipation effect is analyzed in the sense of the above mentioned new analytical solutions. The new solutions on (3+1)-dimensional Rossby waves have important value for understanding the propagation of Rossby waves in the rotating earth with the influence of dissipation.

Lie Symmetry Analysis and Some Exact Solutions of (2+1)-dimensional KdV-Burgers Equation

In this work, the Lie group method is used to perform the similarity reduction and to obtain the exact solutions of the $$(2+1)$$ -dimensional KdV-Burgers equation. Similarity transformation method reduces $$(2+1)$$ -dimensional KdV-Burgers equation into $$(1+1)$$ -dimensional PDEs, later it reduces these PDEs into various ordinary differential equations and helps to find exact solutions of $$(2+1)$$ -dimensional KdV-Burgers equation. With the help of reduced equations, we have obtained the exact explicit solutions. Moreover, later by power series method, the exact analytic solutions of the KdV-Burgers equation are obtained.

Direct similarity reductions and new exact solutions of the short pulse equation

In this paper, we present some similarity reductions of the short pulse equation(SPE) based on the direct similarity reduction method proposed by Clarkson and Kruskal. These similarity reductions have a more general form than those obtained by using the Lie group method. Especially, we obtain one new similarity reduction which can not be obtained by Lie group method. Furthermore, we derive one new exact analytic solutions by the method of undetermined coefficients.

Exact optical solitons in metamaterials with anti-cubic law of nonlinearity by Lie group method

In this paper, using Lie point symmetry analysis, we study the dynamics of the solitons for the nonlinear Schrodinger’s equation with anti-cubic nonlinearity. Some new doubly periodic solutions are obtained that degenerate to dark and bright soliton solutions. In our of best knowledge, the obtained solutions are new. Those obtained results have important applications in the understanding the nonlinear propagation theory of solitons in metamaterials.

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.

Conditions for translation and scaling invariance of the neutron diffusion equation

Lie group methods are applied to the time-dependent, monoenergetic neutron diffusion equation in materials with spatial and time dependence. To accomplish this objective, the underlying 2nd order partial differential equation (PDE) is recast as an exterior differential system so as to leverage the isovector symmetry analysis approach. Some of the advantages of this method as compared to traditional symmetry analysis approaches are revealed through its use in the context of a 2nd order PDE. In this context, various material properties appearing in the mathematical model (e.g., a diffusion coefficient and macroscopic cross section data) are left as arbitrary functions of space and time. The symmetry analysis that follows is restricted to a search for translation and scaling symmetries; consequently the Lie derivative yields specific material conditions that must be satisfied in order to maintain the presence of these important similarity transformations. The principal outcome of this work is thus the determination of analytic material property functions that enable the presence of various translation and scaling symmetries within the time-dependent, monoenergetic neutron diffusion equation. The results of this exercise encapsulate and generalize many existing results already appearing in the literature. While the results contained in this work are primarily useful as phenomenological guides pertaining to the symmetry behavior of the neutron diffusion equation under certain assumptions, they may eventually be useful in the construction of exact solutions to the underlying mathematical model. The results of this work are also useful as a starting point or framework for future symmetry analysis studies pertaining to the neutron transport equation and its many surrogates.

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.

Exact solutions for some time‐fractional evolution equations using Lie group theory

In this paper, some classes of nonlinear partial fractional differential equations arising in some important physical phenomena are considered. Lie group method is applied to investigate the symmetry group of transformations under which the governing time‐fractional partial differential equation remains invariant. The symmetry generators are used for constructing similarity variables, which leads to a reduced ordinary differential equation of Erdélyi‐Kober fractional derivatives. Furthermore, a particular exact solution for each governing equation(s) is constructed. Moreover, the physical significance of the solution is investigated graphically based on numerical simulations in order to highlight the importance of the study.

Lie symmetry analysis, conservation laws and numerical approximations of time-fractional Fokker–Planck equations for special stochastic process in foreign exchange markets

In this paper the transition joint probability density function of the solution of the Ornstein–Uhlenbeck process is presented by a deterministic parabolic time-fractional PDE (FPDE), named time-fractional Fokker–Planck equation. The article generally is divided to two parts: theoretical and approximated analysis. In the theoretical sections Lie group method and invariant subspace method are applied for finding exact solutions and conservation laws of the considered equation. In the next part, by Chebyshev wavelets’s method the numerical solutions are driven. Then the usefulness of this approximated method is comparing with the exact solutions by some plotted graphs.

Lie symmetry analysis and conservation law of variable-coefficient Davey–Stewartson equation

A variable-coefficient Davey–Stewartson (vcDS) equation is investigated in this paper. Infinitesimal generators and symmetry groups are presented by the Lie group method, and the optimal system is presented with adjoint representation. Based on the optimal system, similarity reductions to partial differential equations (PDEs) are obtained, then some PDEs are reduced to ordinary differential equations (ODEs) by two-dimensional subalgebras, and the similarity solutions are provided, including periodic solutions and elliptic function solutions. With Lagrangian, it is shown that vcDS is nonlinearly self-adjoint. Furthermore, based on nonlinear self-adjointness, conservation laws for vcDS equation are derived.

Symmetry analysis of a (2+1)-D system

The classical Lie group method and the (2+1)-D generalized symmetry method in vector analysis are adopted to find infinitesimal symmetries for a (2+1)-D generalized Painleve Burgers system, and its various reduced systems are obtained.

Study for Darcy-Brinkman model connected Oldroyd–B Constitutive Model related with Solutions of Velocity of Viscoelastic Fluid Flow in a circular Pipe filled with Porous Media

The basic purpose of this research paper is to investigate the solution of velocity of viscoelastic fluid flow with Porous media in a circular pipe to use the Lei- group technique to the system of PDE’s comprising the continuity, momentum and constitutive; equations, under appropriate initial and boundary conditions. The associated problem is organized by using the Darcy-Brinkman model connected with Oldroyd–B constitutive model. The investigation is arranged through analytical solutions of the PDE’s system connected with initial and boundary value problem. The analytical solution is established under the symmetries for the system through Lie group method.