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Group Theory
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
Group theory is a mathematical generalization of the study of symmetry. Before delving into the numerous definitions and fundamental results of group theory, we begin this chapter with a discussion of symmetries, or more precisely, operations that preserve symmetries in geometrical objects and equations. For in-depth treatments of group theory and its applications see [1, 2, 3, 5, 7, 8, 12, 13, 14, 18, 19, 20, 21, 22, 23, 24, 27, 29, 32, 36, 40, 41] and the other references cited throughout this chapter. Our discussion of group theory follows these in-depth treatments.
Convective heat transport features of Darcy Casson fluid flow in a vertical channel: a Lie group approach
Published in Waves in Random and Complex Media, 2022
Zahoor Iqbal, Musharafa Saleem
Marius Sophus Lie developed the Lie group theory in the 1870s. The best technique for determining the transformations of similarity and corresponding representations of similarity of the governing equations is to analyze the Lie group. In branches of fluid mechanics, aerodynamics, plasma physics, meteorology, and chemical engineering, several researchers have used the Lie group or group theoretical methods to analyze different flow geometries. Symmetry groups, or simply symmetries, are invariant transformations that do not change the equation's structural structure. To locate all the infinitesimal of the differential equation (DE) [1,2] Lie group analysis is a constructive mathematical review. To get the solutions, there is no need for any previous hypotheses [3–5]. This research is to find a transformation that mapped itself to a DE. The Lie group's principle provides the solution of resemblance, which is the invariant solution of original and boundary value issues. One of the symmetry's most valuable properties is that it allows independent variables to be minimized. It can evaluate the DE in many directions until the differential equation's symmetry group is explored. Symmetry groups, for example, can be used (i) to extract new solutions from old ones (ii) to reduce the order of the equations given (iii) to decide whether a DE can be linearized and, if one exists, to create an explicit linearization and (iv) to drive kept quantities [6].
Qualitative analysis of magnetohydrodynamics Powell–Eyring fluid with variable electrical conductivity
Published in International Journal of Modelling and Simulation, 2023
Sradharam Swain, Suman Sarkar, Bikash Sahoo, Oluwole D. Makinde
Differential equations (DEs) have a long and illustrious history, beginning with calculus in 1665–1666. Several mathematicians, including Leibniz, Bernoulli, Ricatti, and Clairaut, contributed to the development of a wide variety of DEs. They showed remarkable improvement throughout those DEs. But many problems occurring in physics could not be solved using DEs. The origin of partial differential equations (PDEs) dates back to the eighteenth century. During 1880–1900, Norwegian mathematician Sophus Lie [21] developed an important technique known as Lie group analysis to solve PDEs. The Lie group theory introduced novel ideas like reducing, analysing, and classifying of PDEs. This method also works very well with non-linear PDEs. In this method ([22]), the PDEs retain their physical behaviour even after being reduced to the ODEs. These symmetries can be considered collections of related continuous mappings. The laws of conservation of mass, momentum, and energy are the fundamental principles that govern the hydrodynamic approach. In most cases, a scaling transformation with a single parameter is included in the equations that regulate the framework. The invariance constraint of the control system of the equations produced by the hydrodynamic technique was achieved using the Lie group of scaling transformations. Scaling analysis connects one spatial dimension to another spatial dimension. Meanwhile, Rehman et al. [23] studied the two dimensional Powell–Eyring fluid past a stretching sheet using Lie group analysis. Ferdows [24] used Lie group analysis to investigate free convective non-uniform MHD flow past a vertical stretching sheet. Swain et al. [25] studied special third grade fluid over the vertical stretching sheet via Lie scaling group of transformations. Further research on Lie group analysis on fluid flow problems can be seen in [26,27].
Einstein's vacuum field equation: Painlevé analysis and Lie symmetries
Published in Waves in Random and Complex Media, 2021
Lakhveer Kaur, Abdul-Majid Wazwaz
Lie group method of infinitesimal transformations is one of the most efficient classical method for finding symmetry reductions of PDEs. For a detailed study of Lie group theory, the interested reader is referred to the well-known books [14–16]. The technique has earlier been used effectively to obtain the exact solutions of various nonlinear PDEs, hence we avoid to discuss the method in detail. In this section, we have found the symmetry groups using Lie's classical method.