Lie Group Approach Research Articles

Residual symmetry and interactions between solitons and periodic waves of the Drinfeld-Sokolov-Satsuma-Hirota system

Abstract In this paper, the Drinfeld-Sokolov-Satsuma-Hirota (DSSH) system is studied by using residual symmetry and consistent Riccati expansion (CRE) method, respectively. The residual symmetry of the DSSH system is localized to a Lie point symmetry in a properly prolonged system, based on which we get a new Bäcklund transformation for this system. New symmetry reduction solutions of the DSSH system are obtained by applying the classical Lie group approach on the prolonged system. Moreover, the DSSH system is proved to be CRE integrable and new interesting interaction solutions between solitons and periodic waves are generated and analyzed.

Gold-blood nanofluid flow in cone-disk system for Tiwari and Das model in the presence of thermal radiation using lie group approach

Gold-blood nanofluid flow in cone-disk system for Tiwari and Das model in the presence of thermal radiation using lie group approach

Numerical analysis of inclined magnetic force impact on cross liquid flowing with widening shallow and heat generating

ABSTRACT The creation, utilisation, conversion, and exchange of thermal energy are all covered under the thermal engineering subject of heat transfer. Heat transfer plays a major role in engineering and industrial sectors like food packaging, food additive manufacturing, electronic cooling, microturbine, etc. Scientists across the world have tried to explore the impact of heat transfer on fluid past an expandable surface due to its exciting applications in industries like polymer production, paper production, crystal glass manufacturing, etc. This article has unique exploration, and it emphasises interpreting the Cross fluid flow along stretching surface with convective heat transport analysis by using the Lie group methodology. We investigated the implication of inclined upper-convected Maxwell fluid magnetohydrodynamic (MHD) movement through a penetrable strained plate. To solve nonlinear differential equations and clearly determine their absolute invariants, the Lie group approach is introduced. For the numerical solution of non-linear ordinary differential equations with boundary conditions, the fourth-order Runge–Kutta procedure with shelling performance is utilised. Plots showing the impacts of motivated and non-motivated MHD are researched and examined for the alterations of many parameters, and statistical graphs are given to explain the physical values. It is evident that opposing behaviour occurs in connection to temperature captivation and generation parameters together with motivated and non-motivated MHD. Under an amplified, heat-generating parameter and Deborah number, the rate of heat transmission increases. The Lie group investigation of Cross liquid in the presence/absence of inclined and non-inclined has not been investigated yet in the available literature.

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.

Highly accurate differentiation of the exponential map and its tangent operator

Exponential coordinates are widely used in simulation codes for flexible multibody systems based on a Lie group approach. The accurate and efficient evaluation of the exponential map, the tangent operator and its higher order derivatives is crucial. This paper presents a systematic derivation process based on the matrix series expansion of the exponential map and of the tangent operator. This approach is general as it can be applied to any matrix Lie group. For the Lie groups SO(3) and SE(3), the closed form expression of these operators can also be established and is summarized in the paper. It is shown that the closed form of the operators is affected by round-off errors for small rotation amplitudes, whereas the series form is affected by truncation errors at high rotation amplitudes. The computational efficiency of the two approaches is also discussed. A switching strategy between the closed form and the series form is then proposed to obtain an adjustable compromise between accuracy and computational cost.

Attitude control for a full-scale flexible electric solar wind sail spacecraft on heliocentric and displaced non-Keplerian orbits

The electric solar wind sail (E-sail) spacecraft is a fuel-free propulsion system with promising potential in deep space explorations and solar activity monitoring missions. It is challenging to conduct attitude control for a full-scale flexible E-sail spacecraft due to its complex dynamical characteristics, including the spatial multi-scale, high-speed spinning, and large deformations. In this work, a double-loop attitude control strategy for the flexible E-sail spacecraft with high accuracy and efficiency is proposed. The E-sail spacecraft is firstly modeled by the referenced nodal coordinate formulation, where the rotations are described by the Lie group approach. A state-feedback controller developed from the Lyapunov stability theory is prescribed as the inner loop. The gain is obtained by solving a semidefinite programming problem with linear matrix inequality constraints, which is globally optimal and numerically reliable. A multi-variable proportional-integral-derivative controller is then introduced as the outer loop for flexibly enhancing the control performance. Several missions of the E-sail spacecraft, including attitude maneuvers on heliocentric and displaced non-Keplerian orbits, are studied to show that the desired attitude adjustments are accurately and efficiently achieved without exciting high-frequency oscillations. The double-loop control strategy has great prospects in designing and implementing orbital transformation and long-term stable Sun-orbiting missions.

On a Conducting Jeffrey Nanofluid Flow over a Stretching Sheet with Porous Material and Diffusion-Thermo: A Lie Group Approach

In this effort, a steady state conducting fluid of Jeffrey Nano type over a stretching sheet due to porous material together with prescribed boundary condition of interest was studied.

Computational study of bioconvection rheological nanofluid flow containing gyrotactic microorganisms: a model for bioengineering nanofluid fuel cells

ABSTRACT The recent advancements in using micro-organisms effectively for biofuels are the key reason for conducting this study. Therefore, the current theoretical work addresses the Oswald – DeWaele power-law models of bioconvection nanofluid flow containing both nanoparticles and gyrotactic microorganisms. This study has examined two major elements of gyrotactic microorganisms: swimming behavior and the development of bioconvection structures. In addition, the Lie-group approach of transformations is used to establish similarity representations of the governing equations rather than the existing transformations. To achieve the model’s aims, the Legendre-based collocation method (LBCM) was adopted. The findings are analyzed using graphs to examine the patterns of the relevant parameters in the problem. Compared with existing results, the present analysis is effective, fast-converging, and accurate. The bioconvection parameters have a significant impact on the density of motile microorganisms, nanoparticle volume fraction, flow momentum, and temperature distribution. Furthermore, the current work finds application in nanofluid fuel.

Optimal subalgebra, conservations laws and symmetry reductions with analytical solutions using Lie symmetry analysis and geometric approach for the (1+1)-dimensional Manakov model

In this article, the (1+1)-dimensional Manakov model has been examined for finding its exact closed form solitonic solutions with the help of symmetry generators. These symmetry generators are explored using the Lie symmetry analysis, commonly known as the classical Lie group approach and the geometric approach. In a geometric approach, the extended Harrison and Estabrook’s differential forms have been used for obtaining the infinitesimal generators of the Manakov model. As there are infinite possibilities for the linear combination of infinitesimal generators, so by using Olver’s standard approach a one-dimensional optimal system of subalgebra has been established. Additionally, the ‘new conservation theorem’ put forth by Ibragimov has been utilized in order to devise the conservation laws for the (1+1)-dimensional Manakov model. Finally, the exact closed form solutions are obtained with the help of Lie symmetries corresponding to the defined model.

GROUP ANALYSIS OF A NONLINEAR HEAT-LIKE EQUATION

We study a nonlinear heat like equation from a lie symmetry stand point. Heat equation have been employed to study ow of current, information and propagation of heat. The Lie group approach is used on the system to obtain symmetry reductions and the reduced systems studied for exact solutions. Solitary waves have been constructed by use of a linear span of time and space translation symmetries. We also compute conservation laws using multiplier approach and by a conservation theorem due to Ibragimov.

Convective heat transport features of Darcy Casson fluid flow in a vertical channel: a Lie group approach

This work is devoted for the flow analysis and to explore the characteristics of thermal energy transport in non-Newtonian Casson fluid flow in a channel. The flow between the two walls is controlled by constantly slipping between them (fixed on a cold plate or a suction wall). The flow of Casson fluid is explored under the effects of MHD and mixed convection. The impacts of Joule heating and viscous dissipation are incorporated to investigate the thermal transport in the flow. Beside this, the heat transport behavior is properly examined by taking heat source/sink effects into account. The partial differential equations (PDEs) scheme is resolved with the Lie group method. After applying the Lie group theory, we have a system of ordinary differential equations (ODEs) numerically processed. The effects of physical constraints are analyzed with ND-Solve technique in MATHEMATICA and illustrated graphically with reasonable physical arguments. It is detected that because of escalation in the Eckert number the thermal transport in the flow significantly raised. While the flow velocity retarded in case of higher Hartmann number.

Study of flow of Buongiorno nanofluid in a conical gap between a cone and a disk

The cone–disk apparatus consists of a cone that touches the disk at its apex and is used in medical evices, viscosimeters, conical diffusers, etc. Theoretically, a three-dimensional flow of a nanofluid in a conical gap of a cone–disk apparatus is studied for four different physical configurations. Buongiorno nanofluid model, consisting of thermophoresis and Brownian diffusion mechanisms, is used to describe the convective heat transport of the nanofluid. The continuity equation, the Navier–Stokes momentum equation, the heat equation, and the conservation of nanoparticle volume fraction equation constitute the governing system for the flow of nanofluids. The Lie group approach is used to obtain self-similar equations. Solutions are computed for an appropriate rotational Reynolds number and four different gap angles to examine flow, mass, and heat transport features. The skin friction coefficients and torque are computed and analyzed. Multivariate nonlinear regression analysis is also performed. A co-rotating disk and cone configuration has been shown to produce less torque due to the increased centrifugal force. Of the four cone–disk apparatus configurations, the maximum heat/mass transport occurs for a rotating disk with a static cone for all selected gap angles, and the least drag in the radial direction is attained for a rotating cone with a static disk. In addition, there is a minimal drag along the tangential direction for the counter-rotating disk and cone configuration. Brownian diffusion and thermophoresis of the nanoparticles lead to a higher fluid temperature and, thus, lower Nusselt numbers are obtained.

Lie group approach for constructing all reciprocal transformations: The two‐dimensional stationary gas dynamics equations

Recently, an infinitesimal approach for finding reciprocal transformations has been proposed. The method uses the group analysis approach and consists of similar steps as for finding an equivalence group of transformations. The new method provides a systematic tool for finding classes of reciprocal transformations (group of reciprocal transformations). Similar to the classical group analysis, this approach can be also applied for finding all reciprocal transformations (not only composing a group) of the equations under study. The present paper provides this algorithm. As an illustration, the method is applied to the two‐dimensional stationary gas dynamics equations. Equivalence group, group of reciprocal transformations, and completeness of all discrete reciprocal transformations are presented in the paper. The results are stated in form of a theorem.

Algebraic solutions for pricing American put options under the constant elasticity of variance (CEV) model: Application of the Lie group approach

In this work, the algebraic properties of the American put option under the constant elasticity of variance (CEV) model of financial markets are studied with the implementation of the symmetry approach based on Lie’s group theory. The CEV model is described mathematically in terms of the parabolic time–space partial differential equation (PDE) along with the suitable choice of a terminal condition. The classical Lie symmetry analysis is carried out for the governing PDE. Based on the infinitesimal Lie symmetry generators, new classes of group invariant solutions are derived from the mathematical model and the general as well as the specific cases of the CEV model are thoroughly examined. Finally, the results are plotted against the various emergent parameters and their effect are analyzed and discussed.

Lie Group Approach to Dynamic-Model-Aided Navigation of Multirotor Unmanned Aerial Vehicles

In this study, the Lie group approach was used for state estimation of the dynamic-model-aided navigation (DMAN) of a small multirotor unmanned aerial vehicle. The unit quaternions constituting a Lie group called the three-sphere space (<i>S</i>&#x00B3;) were used to represent the attitude in the dynamic equations for the process and measurement models. The linearization of these models is presented in terms of Lie algebra corresponding to <i>S</i>&#x00B3;. The use of Lie algebra to describe the attitude increment conforms to the linearity assumption, on which the measurement update of the extended Kalman filter (EKF) is based. In this study, it was experimentally validated that the Lie group approach combined with DMAN performs better than the EKF that uses the conventional linearization of the process and measurement models under the assumption that the nonlinearity effect is negligible for a small attitude increment. It was demonstrated that the navigation states estimated using the proposed model are better than or comparable to those obtained using the current methods, and the proposed method significantly improves the internal properties.

Langrangian formulation and solitary wave solutions of a generalized Zakharov–Kuznetsov equation with dual power-law nonlinearity in physical sciences and engineering

This paper presents analytical studies carried out explicitly on a higher-dimensional generalized Zakharov–Kuznetsov equation with dual power-law nonlinearity arising in engineering and nonlinear science. We obtain analytic solutions for the underlying equation via Lie group approach as well as direct integration method. Moreover, we engage the extended Jacobi elliptic cosine and sine amplitude functions expansion technique to seek more exact travelling wave solutions of the equation for some particular cases. Consequently, we secure, singular and nonsingular (periodic) soliton solutions, cnoidal, snoidal as well as dnoidal wave solutions. Besides, we depict the dynamics of the solutions using suitable graphs. The application of obtained results in various fields of sciences and engineering are presented. In conclusion, we construct conserved currents of the aforementioned equation via Noether’s theorem (with Helmholtz criteria) and standard multiplier technique through the homotopy formula.

Estimation of robot states with poisson process based on EKF approximate of Kushner filter: a completely coordinate free Lie group approach

In this paper, a Lie coordinate-free torque based Euler–Lagrange equations of motion are developed for a 3-D link (3-DOF) robot. Intentional torque and jerky torque (non-intentional torque) are considered as the inputs to the dynamic profile of the robot. The jerky torque is modelled as a superposition of compound Poisson processes, which is a unique feature. The state vector of the robot, i.e., angular position and angular velocity vector, is thus a Markov process whose transition probability generator can be expressed in terms of the rate of the compound Poisson process that defines the jerky torque. Proof of frame invariance is provided to support the coordinate-free robot dynamics profile. Noise-free measurement is investigated as an ideal case. Angular position measurement is considered with white Gaussian noise. Further, an implementable finite-dimensional EKF approximate to Kushner–Kallianpur filter is obtained to estimate the robot state vector. Finally, the simulations are implemented on commercially available Omni bundle robot.

An analysis of Maxwell fluid through a shrinking sheet with thermal slip effect: a Lie group approach

Two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid passing through the shrinking sheet is considered. With the impact of thermal slip, thermal radiation and heat source-sink conditions, the UCM fluid model is integrated. The method of the Lie scaling group is used to transform the strongly nonlinear governing partial differential equations (PDEs) into the ordinary differential equations (ODEs). The transformed ODEs are numerically solved using NDSolve command of MATHEMATICA and graphically presented with their results. The Deborah number’s influence on the velocity profile $$f^{\prime } (\eta )$$ is studied for different values and different behavior observed. The Hartmann number M and the mass transfer parameter S have decreased the boundary layer thickness. The Prandtl number has increased the temperature profile $$\theta (\eta )$$ . In contrast, the thermal boundary layer thickness was decreased by the heat source-sink parameter Q , the radiation parameter R and the thermal slip parameter L. Table 1 shows the verification of the results.

Invariant analysis, conservation laws, and some exact solutions for (2+1)-dimension fractional long-wave dispersive system

The present study is devoted to investigate the (2+1)-dimensional time-fractional long-wave dispersive (fLWD) model, which describes the dynamical behaviors of shallow water waves propagate along two horizontal directions with fast memories. We first establish the symmetries and conservation laws for this model by improving the fractional Lie group approach and fractional Noether’s formulae. Then, using the general fractional Erdelyi–Kober operator, we reduce the original equations into a time-fractional integro-differential system. Finally, the invariant subspace method is applied to this model to obtain more explicit solutions and the dynamical behaviors are analyzed through numerical simulations. This work extends the basic method to solve the higher dimensional system with mixed order of integer and fractional derivatives.

Evolution of $$C^{1}$$-wave and its collision with the blast wave in one-dimensional non-ideal gas dynamics

The aim of the present paper is to analyze the collision of discontinuity wave with blast wave in non-ideal gas flow through Lie group approach. The particular solution of the system of quasilinear hyperbolic PDEs, governing one-dimensional unsteady cylindrically symmetric motion in non-ideal gas, is obtained. The equation governing the evolutionary process of weak discontinuity in non-ideal gas is derived which is utilized to discuss the growth and decay process of the weak discontinuity. Furthermore, the collision of discontinuity wave with the blast wave is examined inside the state described by the exact solution of the blast wave. Also, the existence and uniqueness properties of reflection and transmission coefficients of the waves along with the discontinuity in the shock acceleration are discussed.