SYMMETRY ALGEBRAS OF THE CANONICAL LIE GROUP GEODESIC EQUATIONS IN DIMENSION FIVE

Nowadays, there is much interest in constructing exact analytical solutions of differential equations using Lie symmetry methods. Lie devised the method in the 1880s. These methods were substantially developed utilizing modern mathematical language in the 1960s and 1970s by several different groups of authors such as L.V. Ovsiannikov, G. Bluman, and P. J. Olver, and have since been implemented as a software package for symbolic computation on commonly used platforms such as Mathematica and MAPLE. In this work, we first develop an algorithmic scheme using the MAPLE platform to perform a Lie symmetry algebra identification and validate it on nonlinear systems of five second-order ordinary differential equations, namely the canonical connection systems of geodesic equations associated with the five-dimensional Lie algebras. In each case, the symmetry generators are determined, and the corresponding nonvanishing Lie brackets are computed. Moreover, the algebraic structure of a Lie algebra of symmetries is identified. Second, we take a theoretical approach and formulate the conditions for a Lie symmetry for the case of the canonical Lie group connection when the Lie algebra is solvable with a one-codimensional abelian nilradical. Such conditions have a complicated system of PDEs, as it has several equations and coefficients to be determined. However, we push the integration of such PDEs as much as possible and investigate to what extent they clarify the concrete results obtained by MAPLE. A comparison with qualitative analysis is demonstrated

Classification of Symmetry Lie Algebras of the Canonical Geodesic Equations of Five-Dimensional Solvable Lie Algebras

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A_{5,7}^{abc} to A_{18}^a. For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized

Symmetries of the Canonical Geodesic Equations of Five-Dimensional Nilpotent Lie Algebras

In this paper, symmetries of the canonical geodesic equations of indecomposable nilpotent Lie groups of dimension five are constructed. For each case, the associated system of geodesics is provided. In addition, a basis for the associated Lie algebra of symmetries as well as the corresponding non-zero Lie brackets are listed and classified. This is a joint work with Ryad Ghanam and Gerard Thompson

A Study of the Soliton Solutions with an Intrinsic Fractional Discrete Nonlinear Electrical Transmission Line

This study aims to identify soliton structures as an inherent fractional discrete nonlinear electrical transmission lattice. Here, the analysis is founded on the idea that the electrical properties of a capacitor typically contain a non-integer-order time derivative in a realistic system. We construct a non-integer order nonlinear partial differential equation of such voltage dynamics using Kirchhoff’s principles for the model under study. It was discovered that the behavior for newly generated soliton solutions is impacted by both the non-integer-order time derivative and connected parameters. Regardless of structure, the fractional-order alters the propagation velocity of such a voltage wave, thus bringing up a localized framework under low coupling coefficient values. The generalized auxiliary equation method drove us to these solitary structures while employing the modified Riemann–Liouville derivatives and the non-integer order complex transform. As well as addressing sensitivity testing, we also investigate how our model’s altered dynamical framework shows quasi-periodic properties. Some randomly selected solutions are shown graphically for physical interpretation, and conclusions are held at the end

Exploring Wave Interactions and Conserved Quantities of KdV–Caudrey–Dodd–Gibbon Equation Using Lie Theory

This study introduces the KdV–Caudrey–Dodd–Gibbon (KdV-CDGE) equation to describe long water waves, acoustic waves, plasma waves, and nonlinear optics. Employing a generalized new auxiliary equation scheme, we derive exact analytical wave solutions, revealing rational, exponential, trigonometric, and hyperbolic trigonometric structures. The model also produces periodic, dark, bright, singular, and other soliton wave profiles. We compute classical and translational symmetries to develop abelian algebra, and visualize our results using selected parameters

Soliton solutions, Lie symmetry analysis and conservation laws of ionic waves traveling through microtubules in live cells

This study explores a fourth-order nonlinear symmetric solution to ionic waves in living microtubules. Lie group analysis and the new extended direct algebraic approach are used to build solitary wave solutions. We reduce the partial differential equation based on symmetry into ordinary differential equations. Solved equations yield novel single wave patterns. For physical explanation, specific solitary wave structures are visually shown. The solutions include the anti-kink, kink shape, the absolute value of the spike shape, singular kink wave shape, and dark-singular soliton solution. Utilizing the multiplier technique, we establish the equation’s local conservation laws. Some solution sketches depict the model’s vision

Classification of Symmetry Lie Algebras of the Canonical Geodesic Equations of Five-Dimensional Solvable Lie Algebras

In this investigation, we present symmetry algebras of the canonical geodesic equations of the indecomposable solvable Lie groups of dimension five, confined to algebras A 5 , 7 a b c to A 18 a . For each algebra, the related system of geodesics is provided. Moreover, a basis for the associated Lie algebra of the symmetry vector fields, as well as the corresponding nonzero brackets, are constructed and categorized

Analytical Analyses for a Fractional Low-Pass Electrical Transmission Line Model with Dynamic Transition

This research explores the solitary wave solutions, including dynamic transitions for a fractional low-pass electrical transmission (LPET) line model. The fractional-order (FO) LPET line mathematical system has yet to be published, and neither has it been addressed via the extended direct algebraic technique. A computer program is utilized to validate all of the incoming solutions. To illustrate the dynamical pattern of a few obtained solutions indicating trigonometric, merged hyperbolic, but also rational soliton solutions, dark soliton solutions, the representatives of the semi-bright soliton solutions, dark singular, singular solitons of Type 1 and 2, and their 2D and 3D trajectories are presented by choosing appropriate values of the solutions’ unrestricted parameters. The effects of fractionality and unrestricted parameters on the dynamical performance of achieved soliton solutions are depicted visually and thoroughly explored. We furthermore discuss the sensitivity assessment. We, however, still examine how our model’s perturbed dynamical framework exhibits quasi periodic-chaotic characteristics. Our investigated solutions are compared with those listed in published literature. This research demonstrates the approach’s profitability and effectiveness in extracting a range of wave solutions to nonlinear evolution problems in mathematics, technology, and science

Approximation of the Fixed Point for Unified Three-Step Iterative Algorithm with Convergence Analysis in Busemann Spaces

In this manuscript, a new three-step iterative scheme to approximate fixed points in the setting of Busemann spaces is introduced. The proposed algorithms unify and extend most of the existing iterative schemes. Thereafter, by making consequent use of this method, strong and Δ-convergence results of mappings that satisfy the condition (Eμ) in the framework of uniformly convex Busemann space are obtained. Our results generalize several existing results in the same direction