The basic idea of the ‘partially nonclassical method’, developed in the present paper, is to apply the invariance requirement of the Lie group method using not all differential consequences of the invariant surface condition but only part...
In this study, the nonlinear term in the two-dimensional Bratu equation has been replaced by its Taylor’s expansion. Hence, the resulting nonlinear partial differential equation has been studied using the Lie group method. The symmetry reductio...
A comprehensive study of the negative-order Kadomtsev–Petviashvili (nKP) partial differential equation by Lie group method has been presented. Initially the infinitesimal generators and symmetry reduction, which were obtained by applying the Lie grou...
In this study, the Lie group method for constructing exact and numerical solutions of the generalized time-dependent variable coefficients Burgers’, Burgers’–KdV, and KdV equations with initial and boundary conditions is presented. Lie group theory i...
The present paper recalls a formulation of non-conservative system dynamics through the Lagrange–d’Alembert principle expressed through a generalized Euler–Poincaré form of the system equation on a Lie group. The paper illust...
An efficient technique is proposed to solve the one-dimensional Burgers’ equation based on multiquadric radial basis function (MQ-RBF) for space approximation and a Lie-Group scheme for time integration. The comparisons of the numerical results...
In the numerical integration of the second-order nonlinear boundary value problem (BVP), the right boundary condition plays the role as a target equation, which is solved either by the half-interval method (HIM) or a new derivative-free Newton method...
Standard correlation analysis is one of the frequently used methods in financial markets. However, this matrix can give erroneous results in the conditions of chaos, fractional systems, entropy, and complexity for the variables. In this study, we emp...
Hessenberg differential algebraic equations (Hessenberg-DAEs) with a high index play a critical role in the modeling of mechanical systems and multibody dynamics. Motivated by the widely used Lie-group differential algebraic equation (LGDAE) method,...
Multi-object tracking (MOT) has attracted increasing attention and achieved remarkable progress. However, accurately tracking objects with homogeneous appearance, heterogeneous motion, and heavy occlusion remains a challenge because of two problems:...
A non-iterative method for the difference of means is presented to calculate the log-Euclidean distance between a symmetric positive-definite matrix and the mean matrix on the Lie group of symmetric positive-definite matrices. Although affine-invaria...
Specifying time-dependent correlation matrices is a problem that occurs in several important areas of finance and risk management. The goal of this work is to tackle this problem by applying techniques of geometric integration in financial mathematic...
A Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some p...
Under axial and azimuthal magnetic inductions, the similarity solutions for a cylindrical shock wave in a weakly conducting ideal gas are determined using the Lie group invariance method. The axial and azimuthal magnetic inductions and density are pr...
The study of non-linear partial differential equations is a complex task requiring sophisticated methods and techniques. In this context, we propose a neural network approach based on Lie series in Lie groups of differential equations (symmetry) for...
In this paper, the
We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the method of nonco...
We review some of the main achievements of the orbit method, when applied to Poisson–Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider
This paper deals with the use of Lie group methods to solve optimization problems in blind signal processing (BSP), including Independent Component Analysis (ICA) and Independent Subspace Analysis (ISA). The paper presents the theoretical fundamental...
This paper proposes a solution to the sideways heat conduction problem (SHCP) based on the time and space integration direction. Conventional inverse problems depend highly on the available data, particularly when the observed data are contaminated w...
This study investigates the potential Kadomtsev–Petviashvili equation incorporating a power-type nonlinearity (PKPp), a model that features prominently in various nonlinear phenomena encountered in physics and applied mathematics. A complete No...
This paper studies a mixed PDE containing the second time derivative and a quadratic nonlinearity of the Monge–Ampère type in two spatial variables, which is encountered in geophysical fluid dynamics. The Lie group symmetry analysis of t...
The present paper deals with neural algorithms to learn the singular value decomposition (SVD) of data matrices. The neural algorithms utilized in the present research endeavor were developed by Helmke and Moore (HM) and appear under the form of two...
In this paper, we study a generalization of the well-known Kelvin-Voigt viscoelasticity equation describing the mechanical behaviour of viscoelasticity. We perform a Lie symmetry analysis. Hence, we obtain the Lie point symmetries of the equation, al...
Systems of Lane-Emden equations arise in the modelling of several physical phenomena, such as pattern formation, population evolution and chemical reactions. In this paper we construct Noether and partial Noether operators corresponding to a Lagrangi...
This paper aims to test the structure of interest rates during the period from 1 September 1981 to 28 December 2020 by using Lie algebras and groups. The selected period experienced substantial events impacting interest rates, such as the economic cr...
In this paper, by applying the Lie group method and the direct symmetry method, Lie algebras of the Benjiamin Ono equation are obtained, and we find that results of the two methods are same. Based on the Lie algebra, Lie symmetry groups, relationship...
Geometry modeling methods can conserve the geometry characters of a system, which helps the dynamic equations more concisely and is good for long simulations. Reduced attitude, Lie group and Lie algebra are three different expressions of geometry. Mo...
19 April 2021
(1) Background: The present study analyzed the need for cognitive closure and narrative creativity in adolescents. The aim was to demonstrate a strong relationship between narrative creativity and the need for cognitive closure. We analyzed a group o...
We study the known coherent states of a quantum harmonic oscillator from the standpoint of the originally developed noncommutative integration method for linear partial differential equations. The application of the method is based on the symmetry pr...
This paper proposes a dead-reckoning (DR) method for vehicles using Lie theory. This approach treats the pose (position and attitude) and velocity of the vehicle as elements of the Lie group
Aimed at the alignment problem of strapdown inertial navigation system (SINS) on the swing base, a novel coarse alignment method using special orthogonal group optimal estimation is proposed. There are two main contributions in this paper. First, bas...
In recent years, the simplified computation of position and velocity changes in nonlinear systems using Lie groups and Lie algebra has been widely used in the study of robot localization systems. The unscented Kalman filter (UKF) can effectively deal...
There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group
In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space...
A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of posi...
It is very important to create mathematical models for real world problems and to propose new solution methods. Today, symmetry groups and algebras are very popular in mathematical physics as well as in many fields from engineering to economics to so...
Neural ordinary differential equations (neural ODEs) are a well-established tool for optimizing the parameters of dynamical systems, with applications in image classification, optimal control, and physics learning. Although dynamical systems of inter...
Starting from the very first principles we derive explicit parameterizations of the ortho-symplectic matrices in the real four-dimensional Euclidean space. These matrices depend on a set of four real parameters which splits naturally as a union of th...
The initial attitude error is challenging to satisfy the requirements of the linear model due to the complex nature of the ocean environment. This presents a challenge in the transfer alignment of the ship. In order to enhance the precision and veloc...
In this paper, we address the problem of estimating the position of a mobile such as a drone from noisy position measurements using the framework of Lie groups. To model the motion of a rigid body, the relevant Lie group happens to be the Special Euc...
The Lie symmetry group transformation method was used to investigate the partial differential equations that model the motion of a natural convective unsteady flow past to a non-isothermal vertical flat surface. The one-parameter Lie group transforma...
In this contribution, we are using Lie group theoretical methods
This paper studies a non-linear viscoelastic wave equation, with non-linear damping and source terms, from the point of view of the Lie groups theory. Firstly, we apply Lie’s symmetries method to the partial differential equation to classify the Lie...
This paper presents a sensor fusion method for navigation of unmanned underwater vehicles. The method combines Lie theory into Kalman filter to estimate and compensate for the misalignment between the sensors: inertial navigation system and Doppler V...
This paper investigates the role of solvable and nilpotent Lie algebras in the domains of cryptography and steganography, emphasizing their potential in enhancing security protocols and covert communication methods. In the context of cryptography, we...
The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. Ho...
In this paper, we mainly put the Lie symmetry analysis method on the Gibbons-Tsarev equation (GTe) to obtain some new results, including some Lie symmetries, one-parameter transformation groups, explicit invariant solutions in the form of power serie...
By applying the Lie symmetry method, group-invariant solutions are constructed for axially loaded Euler beams. The corresponding mathematical models of the beams are formulated. After introducing the infinitesimal transformations, the determining equ...
This article presents a comprehensive study of the (2+1)-dimensional Zakharov–Kuznetsov (ZK)
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