Lie Group Method Research Articles

Lie reductions for the Euler-Bernoulli model of flexible beam robots

The Euler-Bernoulli model of flexible beam robots is a four-dimensional under-determined partial differential equation, related to define the unique link's deflection of this type of robots. In this paper, Lie point symmetries of the equation have been investigated by a Lie group method. Then, reductions of the Euler-Bernoulli equation to ordinary equations have been obtained by repeat the method . Two of the reduced equations are of the first order.

Symmetry Reductions of (2 + 1)-Dimensional CDGKS Equation and Its Reduced Lax Pairs

With the aid of symbolic computation, we obtain the symmetry transformations of the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation by Lou’s direct method which is based on Lax pairs. Moreover, we use the classical Lie group method to seek the symmetry groups of both the CDGKS equation and its Lax pair and then reduce them by the obtained symmetries. In particular, we consider the reductions of the Lax pair completely. As a result, three reduced (1 + 1)-dimensional equations with their new Lax pairs are presented and some group-invariant solutions of the equation are given.

Conservation Laws, Symmetry Reductions, and New Exact Solutions of the (2 + 1)-Dimensional Kadomtsev-Petviashvili Equation with Time-Dependent Coefficients

The (2 + 1)-dimensional Kadomtsev-Petviashvili equation with time-dependent coefficients is investigated. By means of the Lie group method, we first obtain several geometric symmetries for the equation in terms of coefficient functions and arbitrary functions oft. Based on the obtained symmetries, many nontrivial and time-dependent conservation laws for the equation are obtained with the help of Ibragimov’s new conservation theorem. Applying the characteristic equations of the obtained symmetries, the (2 + 1)-dimensional KP equation is reduced to (1 + 1)-dimensional nonlinear partial differential equations, including a special case of (2 + 1)-dimensional Boussinesq equation and different types of the KdV equation. At the same time, many new exact solutions are derived such as soliton and soliton-like solutions and algebraically explicit analytical solutions.

Lie Group Analysis for a Mixed Convective Flow and Heat Mass Transfer Over a Permeable Stretching Surface with Soret and Dufour Effects

ABSTRACTThe Lie group transformation method is applied for solving the problem of mixed convection flow with mass transfer over a permeable stretching surface with Soret and Dufour effects. The application of Lie group method reduces the number of independent variables by one and consequently the system of governing partial differential equations reduces to a system of ordinary differential equations with appropriate boundary conditions. Further, the reduced non-linear ordinary differential equations are solved numerically by using the shooting method. The effects of various parameters governing the flow and heat transfer are shown through graphs and discussed. Our aim is to detect new similarity variables which transform our system of partial differential equations to a system of ordinary differential equations. In this work a special attention is given to investigate the effect of the Soret and Dufour numbers on the velocity, temperature and concentration fields above the sheet.

Self-similar scalar cosmologies

We study several scalar cosmological models under the self-similar approach. We deduce, by stating and proving general theorems, which is the exact form that must follow the scalar field and the potential in the frameworks of a single scalar field and non-interacting (with a perfect fluid) models. The proofs are carried out by two methods: the matter collineation approach and the Lie group method. The results obtained are absolutely general and valid for all Bianchi models and the flat FRW one. In order to study how the “constant” G may vary we propose, in a phenomenological way, how to incorporate a variable G in the framework of scalar models by modifying the Klein-Gordon equation. This approach is more general than the usual one in the context of the FRW symmetries. We deduce the exact form to be followed by each quantity in these new models. Therefore, to study how the “constants” G and Λ may vary, we propose three scenarios where such constants are considered as time functions: modified general relativity with a perfect fluid, the scalar cosmological models (“quintessence”) in the non-interacting case and a scalar-tensor model with a dynamical Λ. As an example, we study the case of Bianchi VI0 geometry.

Lie group method for solving generalized Hirota–Satsuma coupled Korteweg–de Vries (KdV) equations

Method of Lie group has been applied to study Hirota–Satsuma coupled Korteweg–de Vries (KdV) system of partial differential equations. Lie group method is applied to determine symmetry reductions of the nonlinear partial differential equations. The resulting nonlinear ordinary differential equations are solved analytically and the obtained solutions are presented graphically.

Non-steady homogeneous deformations: Computational techniques using Lie theory, and application to ellipsoidal markers in naturally deformed rocks

The dynamic theory of deformable ellipsoidal inclusions in slow viscous flows was worked out by J.D. Eshelby in the 1950s, and further developed and applied by various authors. We describe three approaches to computing Eshelby's ellipsoid dynamics and other homogeneous deformations. The most sophisticated of our methods uses differential-geometric techniques on Lie groups. This Lie group method is faster and more precise than earlier methods, and perfectly preserves certain geometric properties of the ellipsoids, including volume. We apply our method to the analysis of naturally deformed clasts from the Gem Lake shear zone in the Sierra Nevada mountains of California, USA. This application demonstrates how, given three-dimensional strain data, we can solve simultaneously for best-fit bulk kinematics of the shear zone, as well as relative viscosities of clasts and matrix rocks.

Self-similar chameleon Jordan-Brans-Dicke cosmological models

In this paper we study the chameleon Jordan-Brans-Dicke (JBD) cosmological models under the hypothesis of self-similarity. Since there are several ways to define the matter Lagrangian for a perfect fluid: L m =−ρ and L m =γρ, we show that they bring us to obtain two completely different cosmological models. In the first approach, L m =−ρ, there is ordinary matter conservation, while in the second approach, L m =γρ, we get matter creation processes. We deduce for each approach the behaviour of each physical quantity, under the self-similar hypothesis, by employing the Lie group method. The results are quite general and valid for any homogeneous geometry (FRW, Bianchi types, etc.). As example, we calculate exact solutions for each approach by considering the case of a Bianchi II geometry. In this way we can determine the exact behaviour of each physical quantity and in particular of G eff and U (the potential that mimics the cosmological constant).We compare the solutions with the obtained ones in the framework of the usual JBD models.

Lie group analysis of nonlinear inviscid flows with a free surface under gravity

We applied the Lie group method in studying nonlinear inviscid flows with a free surface under gravity. This method reduces the number of independent variables by one. Therefore, for a system of partial differential equations with three independent variables we applied the method twice to yield a system represented by ordinary differential equations with appropriate corresponding conditions. We obtained analytical solutions for this system. Solutions for the free surface and the velocity components are obtained in closed form. The results are illustrated graphically for different parameters.

Preserving first integrals with symmetric Lie group methods

The discrete gradient approach is generalized to yield first integral preserving methods for differential equations in Lie groups.

New solutions for solving problem of particle trajectories in linear deep-water waves via Lie-group method

The nonlinear equations of the two-dimensional inviscid incompressible fluid in a constant gravitational field describing the wave propagation on the water surface are considered. The Lie-group method has been applied for determining symmetry reductions of the system of partial differential equations. Lie-group method starts out with a general infinitesimal group of transformations under which the given partial differential equations are invariant. The determining equations are a set of linear differential equations, the solution of which gives the transformation function or the infinitesimals of the dependent and independent variables. After the group has been determined, a solution to the given partial differential equations may be found from the invariant surface condition such that its solution leads to similarity variables that reduce the number of independent variables of the system. Effects of the wavelength λ and time t on the particle path have been studied and the results are plotted.

EXACT SOLUTIONS OF A FLAT FULL CAUSAL BULK VISCOUS FRW COSMOLOGICAL MODEL THROUGH FACTORIZATION

We study the classical flat full causal bulk viscous Friedmann–Robertson–Walker (FRW) cosmological model through the factorization method. The method shows that there exists a relationship between the viscosity parameter s and the parameter γ entering the equations of state of the model. Also, the factorization method allows to find some new exact parametric solutions for different values of the viscous parameter s. Special attention is given to the well-known case s = 1/2, for which the cosmological model admits scaling symmetries. Furthermore, some exact parametric solutions for s = 1/2 are obtained through the Lie group method.

Melting phenomenon in magneto hydro-dynamics steady flow and heat transfer over a moving surface in the presence of thermal radiation

The Lie group method is applied to present an analysis of the magneto hydro-dynamics (MHD) steady laminar flow and the heat transfer from a warm laminar liquid flow to a melting moving surface in the presence of thermal radiation. By using the Lie group method, we have presented the transformation groups for the problem apart from the scaling group. The application of this method reduces the partial differential equations (PDEs) with their boundary conditions governing the flow and heat transfer to a system of nonlinear ordinary differential equations (ODEs) with appropriate boundary conditions. The resulting nonlinear system of ODEs is solved numerically using the implicit finite difference method (FDM). The local skin-friction coefficients and the local Nusselt numbers for different physical parameters are presented in a table.

Analytical and numerical solutions of the equation for the beam propagation in a photovoltaic-photorefractive media

We use the classical Lie-group method to study the evolution equation describing a photovoltaic-photorefractive media with the effects of diffusion process and the external electric field. We reduce it to some similarity equations firstly, and then obtain some analytically exact solutions including the soliton solution, the exponential solution and the oscillatory solution. We also obtain the numeric solitons from these similarity equations. Moreover, We show theoretically that these solutions have two types of trajectories. One type is a straight line. The other is a parabolic curve, which indicates these solitons have self-deflection.

An introduction to Lie group integrators – basics, new developments and applications

We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalised to Lie groups.

Saddle-Node Heteroclinic Orbit and Exact Nontraveling Wave Solutions for (2+1)D KdV-Burgers Equation

We have undertaken the fact that the periodic solution of (2+1)D KdV-Burgers equation does not exist. The Saddle-node heteroclinic orbit has been obtained. Using the Lie group method, we get two-(1+1)-dimensional PDE, through symmetric reduction; and by the direct integral method, spread F-expansion method, and <svg style="vertical-align:-2.3205pt;width:47.662498px;" id="M1" height="20.6" version="1.1" viewBox="0 0 47.662498 20.6" width="47.662498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,17.65)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,5.944,17.65)"><path id="x1D43A" d="M713 296l-5 -25q-47 -7 -59 -20t-23 -72l-15 -79q-9 -48 -3 -74q-15 -3 -55 -13t-63 -15t-59.5 -10t-70.5 -5q-149 0 -243 80t-94 220q0 169 127.5 276.5t336.5 107.5q91 0 206 -36l-10 -165l-29 -1q1 85 -47.5 126t-146.5 41q-153 0 -243.5 -97t-90.5 -242&#xA;q0 -122 68.5 -198.5t188.5 -76.5q121 0 139 75l20 86q13 58 -1.5 70.5t-99.5 20.5l5 26h267z" /></g> <g transform="matrix(.012,-0,0,-.012,18.25,9.488)"><path id="x2032" d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" /></g> <g transform="matrix(.017,-0,0,-.017,22.412,17.65)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.017,-0,0,-.017,29.416,17.65)"><use xlink:href="#x1D43A"/></g><g transform="matrix(.017,-0,0,-.017,41.723,17.65)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg>-expansion method, we obtain exact nontraveling wave solutions, for the (2+1)D KdV Burgers equation, and find out some new strange phenomenons of sympathetic vibration to evolution of nontraveling wave.

Symmetry Reduced and New Exact Nontraveling Wave Solutions of (2+1)-Dimensional Potential Boiti-Leon-Manna-Pempinelli Equation

With the aid of Maple symbolic computation and Lie group method, (2+1)-dimensional PBLMP equation is reduced to some (1+1)-dimensional PDE with constant coefficients. Using the homoclinic test technique and auxiliary equation methods, we obtain new exact nontraveling solution with arbitrary functions for the PBLMP equation.

Second-Order Geometric Sliding Mode Attitude Observer with Application to Quadrotor on a Test Bench

A sliding mode observer design framework is proposed based on the Lie group method of numerical integration on manifolds, and a Second-Order Geometric Sliding Mode Attitude Observer (SOGSMAO) is designed for angular velocity estimation of quadrotor attitude. The algorithm constructs feedback in the angular velocity space and the space of equivalent Lie algebra of unit quaternion space, respectively. It avoids not only the complexity of constructing feedback in unit quaternion space but also the process of mandatory rescaling which is seen to deteriorate the accuracy of the angular velocity estimates during sliding. The performance of SOGSMAO is compared with traditional quaternion based sliding mode observer in which multiplicative quaternion correction is used and the results show that SOGSMAO gains better tracking performance. Then SOGSMAO is realized on a test bed and the effectiveness of the observer algorithm is verified by experimental studies.

Lie Group Analysis and Similarity Solutions for Mixed Convection Boundary Layers in the Stagnation-Point Flow toward a Stretching Vertical Sheet

An analysis for the mixed convection boundary layers in the stagnation-point flow toward a stretching vertical sheet is carried out via symmetry analysis. By employing Lie group method to the given system of nonlinear partial differential equations, we can obtain information about the invariants and symmetries of these equations. This information can be used to determine the similarity variables that will reduce the number of independent variables in the system. The transformed ordinary differential equations are solved numerically for some values of the parameters involved using fifth-order Improved Runge-Kutta Method (IRK5) coupled with shooting method. The features of the flow and heat transfer characteristics are analyzed and discussed in detail. Both cases of assisting and opposing flows are considered. This paper' results in comparison with known results are excellent.

Solving Nonlinear Differential Algebraic Equations by an Implicit Lie-Group Method

We derive an implicit Lie-group algorithm together with the Newton iterative scheme to solve nonlinear differential algebraic equations. Four numerical examples are given to evaluate the efficiency and accuracy of the new method when comparing the computational results with the closed-form solutions.