Symmetry Method Research Articles

Solution Methods for Nonlinear Ordinary Differential Equations Using Lie Symmetry Groups

For formulating mathematical models, engineering problems and physical problems, Nonlinear ordinary differential equations(NODEs) are used widely. Nevertheless, explicit solutions can be obtained for very few NODEs, due to lack of techniques to obtain explicit solutions. Therefore methods to obtain approximate solution for NODEs are used widely. Although symmetry groups of ordinary differential equations (ODEs) can be used to obtain exact solutions however, these techniques are not widely used. The purpose of this paper is to present applications of Lie symmetry groups to obtain exact solutions of NODEs . In this paper we connect different methods,theorems and definitions of Lie symmetry groups from different references and we solve first order and second order NODEs. In this analysis three different methods are used to obtain exact solutions of NODEs. Using applications of these symmetry methods, drawbacks and advantages of these different symmetry methods are discussed and some examples have been illustrated graphically. Focus is first placed on discussing about the notion of symmetry groups of the ODEs. Focus is then changed to apply them to find general solutions for NODEs under three different methods. First we find suitable change of variables that convert given first order NODE into variable separable form using these symmetry groups. As another method to find general solutions for first order NODEs, we find particular type of solution curves called invariant solution curves under Lie symmetry groups and we use these invariant solution curves to obtain the general solutions. We find general solutions for the second order NODEs by reducing their order to first order using Lie symmetry groups.

Lie symmetry and exact homotopic solutions of a non-linear double-diffusion problem

The Lie symmetry method is applied, and exact homotopic solutions of a non-linear double-diffusion problem are obtained. Additionally, we derived Lie point symmetries and corresponding transformations for equations representing heat and mass transfer in a thin liquid film over an unsteady stretching surface, using MAPLE. We used these symmetries to construct new (Lie) similarity transformations that are different from those that are commonly used for flow and mass transfer problems. These new (Lie) similarity transformations map the partial differential equations of a mathematical model under consideration to ordinary differential equations along with boundary conditions. Lie similarity transformations are shown to lead to new solutions for the considered flow problem. These solutions are obtained using the homotopy analysis method to analytically solve the ordinary differential equations that resulted from the reduction of considered flow equations through Lie similarity transformations. With the aid of these solutions, effects of various parameters on the flow and heat transfer are discussed and presented graphically in this study.

Lie algebra and conservation laws for the time-fractional heat equation

The Lie symmetry method is applied to derive the point symmetries for the N-dimensional fractional heat equation. We find that the numbers of symmetries and Lie brackets are reduced significantly as compared to the integer order for all dimensions. In fact for integer order linear heat equation the number of solution symmetries is equal to the product of the order and space dimension, whereas for the fractional case, it is half of the product on the order and space dimension. The Lie algebras for the integer and fractional order equations are mentioned using the subsequent computations of Lie brackets and by inspection. Interestingly, it is observed that for the one-dimensional fractional heat equation, the Lie algebra obtained by inspection of symmetries is similar to the result obtained by computation of Lie brackets, which is . The Lie algebra using the symmetries of the two-dimensional heat equation is observed to be , whereas using the Lie brackets the algebra is deduced to be . Hence, it can be concluded that the Lie algebra obtained from the nonzero Lie brackets can be conflated to the algebra which is obtained by inspection. Further, the subsequent Lie algebras are mentioned for the three and four-dimensional integer and fractional equations and the conservation laws are explicitly stated.

A Lie group treatment on a generalized evolution Fisher type equation with variable coefficients

In this study, a general second-order evolution equation of the Fisher-type, with time-dependent variable coefficients, is considered. This equation contains many well-known equations, and obtained results may be applicable in investigating other evolution equations. Lie symmetries and corresponding invariant solutions of the considered problem are studied by a non-traditional Lie symmetry method (LSM). Prolongation of the model is an essential part of the Lie symmetry method, which in the current work, we analyze by the heir-equations. Finally, different types of solutions depending on the variable coefficients, such as the exponential solutions, trigonometric solutions, Bessel solutions, and Mathieu solutions, are extracted.

Shot Boundary Detection Based on Global Features and the Target Features

Video processing plays an important role in the intelligent monitoring and management system of agricultural information. Video shot boundary detection is the basic symmetry step underlying video processing techniques. According to the current shot boundary detection algorithm, the feature changes between gradual transition frames are difficult to detect, and the misdetection situation is caused by ignoring the attention of the target feature during the feature extraction. A novel symmetry multi-step comparative scheme of shot boundary detection algorithm based on global features and target features is proposed. First, the RGB color histogram features of the video frame are extracted. Second, foreground object detection for the video frames is performed using the Gaussian Mixture Model (GMM), and the scale-invariant features transformation (SIFT) of the foreground targets is extracted. Finally, global features and target features fusion through weights, calculating the difference between adjacent frames across multiple steps, generate a pattern distance map. The pattern distance map of the gradual transition and the cut detection is different; we can judge the gradual transition and the cut detection according to the pattern distance map. Experiments show that the proposed symmetry method improves by about 2% in recall and accuracy compared to other algorithms.

Travelling wave solutions, symmetry reductions and conserved vectors of a generalized hyper-elastic rod wave equation

This work presents a generalized hyper-elastic rod wave (gHRW) equation from the Lie symmetry method’s standpoint. The equation illustrates dispersive waves generating in hyper-elastic rods. Using multiplier approach we find conserved vectors of the underlying equation. We subsequently obtain first integrals of the conserved vectors under the time–space group invariant u(t,x)=H(x−νt). Finally, by analysing various attainable instances of the arbitrary coefficient function g(u), we perform symmetry reductions of gHRW equation to lower order ordinary differential equations and in some instances obtain analytic solutions for special values of arbitrary constants.

An Extended (2 + 1)-dimensional Coupled Burgers System in Fluid Mechanics: Symmetry Reductions; Kudryashov Method; Conservation Laws

The Burgers-type equations are noticed in plasma astrophysics, ocean dynamics, atmospheric science, computational fluid mechanics, cosmology, condensed matter physics, statistical physics, nonlinear acoustics, vehicular traffic, electronic transport, etc. This prompts us to examine an extended (2 + 1)-dimensional coupled Burgers system in fluid mechanics. We determine novel exact solutions by the Lie symmetry method in conjunction with Kurdyshov method. Finally, conservation laws of the abovementioned system are generated. The findings can well mimic complex waves and their dealing dynamics in fluids.

An example from turbulence how not to use the invariant function method of Lie-group symmetries

An example from turbulence how not to use the invariant function method of Lie-group symmetries

Development of a Non-Integral Form of Coordination Number Equation Based on Pair Distribution Function and Gaussian Function

The coordination number (CN) is an important structure property of liquid metals. A simple yet extremely precise method for calculating CN is proposed, the classical CN methods are evaluated systematically, and the mathematical forms of the symmetry method are corrected. Using the Gaussian function construct, the first coordination shell of the pair distribution function (PDF), the right-hand side of the first peak of the pair distribution function is extrapolated, and the CN expression is simplified with a Gaussian function to obtain its non-integral form. The first coordination shell is used to explain the Tao coordination number model (Tao CN) and obtain a Modified Tao CN. The Gaussian function is combined with the Tao CN, obtaining the function expression for the peak with peak position. These are important for the structural research of liquid metals. The CN of 27 liquid metals is calculated by these methods. The average relative deviation of the Gaussian function extrapolation method is ±6.46%, of the Modified Tao CN is ± 18.51%; those of the four classical methods range from ±15% to ±42%. The Modified Tao CN and extrapolation methods to calculate CN are more accurate for calculating CN than the classical method; they are more suitable for use in quantitative applications of CN. The equations derived in this work can be applied to the problem of integration of distribution functions to obtain simple mathematical models.

Comment on "Turbulence Statistics of Arbitrary Moments of Wall-Bounded Shear Flows: A Symmetry Approach".

A recent Letter by Oberlack et al. [Phys. Rev. Lett. 128, 024502 (2022)] claims to have derived new symmetry-induced solutions of the non-modelled statistical Navier-Stokes equations of turbulent channel flow. A high accuracy match to DNS data for all streamwise moments up to order 6 is presented, both in the region of the channel-center and in the inertial sublayer close to the wall. Here we will show that the findings and conclusions in that study are highly misleading, as they give the impression that a significant breakthrough in turbulence research has been achieved. But, unfortunately, this is not the case. Besides trivial and misleading aspects, we will demonstrate that even basic turbulence-relevant correlations as the Reynolds-stress cannot be fitted to data using the proposed symmetry-induced scaling laws. The Lie-group symmetry method as used by Oberlack et al. cannot bypass the closure problem of turbulence. It is just another assumption-based method that requires modelling and is not, as claimed, a first-principle method that leads directly to solutions. Next to PRL, two more papers by Oberlack et al. are called out for correction or a retraction.

Lie group analysis and novel solutions for the generalized variable-coefficients Sawada-Kotera equation

In this paper, the generalized fifth-order variable-coefficients Sawada-Kotera equation arising in coastal seas, fjords, lakes, and the atmospheric boundary layer is studied by using the symmetry method. As a result, four-vector fields are obtained and a commutative Lie group of transformations. Then, by using suitable combinations of the Lie vector fields three distinct similarity reductions in the form of nonlinear ordinary differential equations are yielded. By solving the reduced equations using the known techniques and the Jacobi expansion method many novel periodic and solitary wave solutions are considered. From a physical point of view, the dynamic behavior of two distinct wave structures, periodic and kink soliton, was investigated for different choices of the variable coefficients and it was clear that the wave propagation shape is affected by the change of the variable function.

Fibonacci Wavelet Method for the Numerical Solution of Nonlinear Reaction-Diffusion Equations of Fisher-Type

This article aims to propose an efficient Fibonacci wavelet-based collocation method for solving the nonlinear reaction-diffusion equation of Fisher-type. The underlying numerical scheme starts by formulating operational matrices of integration corresponding to the Fibonacci wavelets. Besides, we study the error analysis and convergence theorem of the proposed technique. Subsequently, a set of algebraic equations are formed corresponding to the given problem, which could be handled via any conventional method, for instance, the Newton iteration technique. To demonstrate the efficiency of the proposed wavelet-based numerical method, we compare the obtained absolute,L∞,L2, and root mean square (RMS) error norms with the existing Lie symmetry method and cubic trigonometricB-spline (CTB) differential quadrature method in tabular form. From the numerical outcomes, it is ascertained that the proposed numerical technique is computationally more effective and yields precise outcomes in comparison to the existing ones.

EXPLICIT OPTICAL DROMIONS WITH KERR LAW HAVING FRACTIONAL TEMPORAL EVOLUTION

In this work, we derived the (2+1)-dimensional Schrödinger equation from the (2+1)-dimensional Klein–Gordon equation. We also obtained the fractional order form of this equation at the same time so as to discover the connection between them. For the (2+1)-dimensional Klein–Gordon equation, symmetries and conservation laws are pres ented. For different gauge constraint, from the perspective of conservation laws, the corresponding symmetries are obtained. After that, based on the fractional complex transform, soliton solutions of the time fractional (2+1)-dimensional Schrödinger equation are displayed. Some figures are showed behaviors of soliton solutions. It is important to discover the relationships between these equations and to obtain their explicit solutions. These solutions will perhaps provide a theoretical basis for the explanation of complex nonlinear phenomena. From the results of this paper, it is clear that the Lie symmetry method is a particularly important tool for dealing with differential equations.

Lie symmetry analysis, conservation laws and diverse solutions of a new extended (2+1)-dimensional Ito equation

<abstract><p>In this paper, a new class of extended (2+1)-dimensional Ito equations is investigated for its group invariant solutions. The Lie symmetry method is employed to transform the nonlinear Ito equation into an ordinary differential equation. The general solution of the solvable linear differential equation with different parameters is obtained, and the plot of the solvable linear differential equation is given. A power series solution for the equation is then derived. Furthermore, a conservation law for the equation is constructed by utilizing a new Ibragimov conservation theorem.</p></abstract>

Editorial of Papers Published in 2020–2021 in the Mathematics and Symmetry/Asymmetry Section

This editorial is a short review of papers accepted in the Mathematics and Symmetry/Asymmetry section in 2020–2021 about the symmetry methods [...]

Symmetry and Approximation Methods

The overwhelming majority of mathematical problems, describing realistic systems and processes, contain two parts: first, the problem needs to be characterized by an effective mathematical model and, second, the appropriate solutions are to be found.

A Mechanical Equipment Fault Diagnosis Model Based on TSK Fuzzy Broad Learning System

In an intelligent manufacturing context, the smooth operations of mechanical equipment in the production process of enterprises and timely fault diagnosis during operations have become increasingly important. However, the effect of traditional fault diagnosis depends on the feature extraction quality and experts’ empirical knowledge, which is inefficient and costly, and cannot match the needs of mechanical equipment fault diagnosis in intelligent manufacturing. The TSK fuzzy system has a strong approximation capability and the ability to interpret expert knowledge. The broad learning system (BLS) has strong feature extraction and fast computation capabilities. In this paper, we present a new model—the TSK fuzzy broad learning system (TSK-BLS). The model integrates the advantages of the BLS and the fuzzy system at the same time, which can be calculated quickly and accurately by pseudo-inverse and symmetry methods. On the other hand, the model is an embedded model-building mechanism, which extends the application scope of BLS theory. The model was tested on a bearing fault data set, provided by Case Western Reserve University, and the model’s fault diagnosis accuracy was as high as 0.9967. The results were compared with those of the convolutional neural network (CNN) and the BLS models, whose fault diagnosis accuracies are 0.6833 and 0.9133, respectively. Comparison showed that the proposed fault diagnosis model—TSK-BLS, achieved significant improvements.

Symmetries and symmetry reductions of the combined KP3 and KP4 equation

To find symmetries, symmetry groups and group invariant solutions are fundamental and significant in nonlinear physics. In this paper, the finite point symmetry group of the combined KP3 and KP4 (CKP34) equation is found by means of a direct method. The related point symmetries can be obtained simply by taking the infinitesimal form of the finite point symmetry group. The point symmetries of the CKP34 equation constitute an infinite dimensional Kac-Moody–Virasoro algebra. The point symmetry invariant solutions of the CKP34 equation are obtained via the standard classical Lie point symmetry method.

Special Issue “Advanced Symmetry Methods for Dynamics, Control, Optimization and Applications”

Nonlinear systems described by differential equations are of great theoretical significance and do frequently arise in practice [...]

Multi-dimensional optimal system and conservation laws for Chaplygin gas Cargo-LeRoux model

The well-known Cargo-LeRoux model for the isentropic Chaplygin gas is investigated using classical Lie symmetry method. Optimal systems up to six dimensions are constructed using the adjoint transformation and the invariants of the admitted Lie algebras. We develop exact solutions to the Cargo-LeRoux model by using the one-dimensional optimal system and discussed the physical behavior of such solutions graphically. Conservation laws to the governing equations are also obtained by using the nonlinear self-adjointness of the model. The performance of the developed solutions is explored through the evolutionary behavior of a discontinuity wave.