In this article, we study solvable Lie algebras A(m) -Q(n) , whose left nilpotent algebras are m-dimensional abelian Lie algebras A(m) and whose right nilpotent algebras are n-dimensional quasifiliform Lie algebras Q(n) . We determine the Lie brackets of A(m) -Q(n) , and give the complete classification of A(2) -Q(n) . In addition, we give the complete classification of solvable Lie algebras Q(n) -A(1) .
The nonlinear Von Kármán (VK) system plays a significant role in mechanics, nonlinear dynamics and physical applications. In the present paper, we develop the combination of Lie group analysis and analytic method for dealing with exact solutions to the nonlinear VK system, all of the point symmetries of the system are obtained, the symmetry transformations and exact solutions of the VK system are investigated. Especially, some new types of solutions are presented, including the traveling wave solutions and analytic solutions. This paper can be regarded as a sequel of Ref. [1], but their research methods and results are completely different.
This study investigates the mixed convection flow of a squeezing couple-stress nanofluid with Cattaneo-Christov heat and mass flux formulations, addressing the limitations of classical Fourier and Fick laws in capturing non-equilibrium thermal and mass transport phenomena. The mathematical framework incorporates the effects of Joule heating, chemical reaction, nonlinear thermal radiation, a magnetic field, and heat generation/absorption. The governing partial differential equations are transformed into a system of nonlinear ordinary differential equations using similarity transformations, which are subsequently solved numerically via an efficient implicit finite difference scheme. A convergence analysis is performed to ensure grid-independent and stable numerical results. Comprehensive analyses are presented to elucidate the influence of pertinent dimensionless parameters on velocity, temperature, and concentration distributions. Furthermore, variations in the skin friction coefficient, Nusselt number, and Sherwood number are systematically examined. The results demonstrate significant potential for applications in advanced thermal management systems, microfluidic devices, energy conversion technologies, and biomedical engineering, where precise control of heat and mass transfer in non-Fourier fluids is crucial.
We investigate Lie symmetries and their associated inverse problems in the framework of Hamiltonian field theories formulated on time scales. Time scale calculus provides a unified setting for continuous and discrete dynamics, offering a natural description of nonlinear systems that exhibit hybrid temporal behavior. Within this context, we establish Hamilton’s canonical equations on time scales and derive the corresponding Lie symmetry transformations together with their infinitesimal generators. Based on invariance principles, we obtain conserved quantities and formulate necessary and sufficient conditions under which weak and strong Lie symmetries exist. We then address the inverse symmetry problem, namely how a prescribed invariant determines the underlying symmetry structure of nonlinear time-scale Hamiltonian systems. A nontrivial example is presented to illustrate the effectiveness of the developed theory. The results generalize classical Lie symmetry methods to the time-scale setting and contribute a new perspective to the analysis of nonlinear dynamics in regimes where discrete and continuous effects coexist.
The Nano/microelectromechanical systems (N/MEMS), particularly graphene-based devices, exhibit highly nonlinear and oscillatory dynamic behavior that is complicated to solve accurately utilizing the conventional numerical and theoretical approaches. This study aims to design an advanced and efficient computational framework to analyze the dynamic response of graphene N/MEMS systems. In this paper, a hybrid morlet wavelet neural network (MWNN) algorithm is proposed to approximate the solution of the governing nonlinear model. The Morlet wavelet-based activation function is employed due to its strong capability to capture localized oscillatory patterns inherent in N/MEMS dynamics. To further enhance accuracy and convergence, a hybrid optimization strategy is integrated, where advanced optimization algorithms are used to initialize and update neural network parameters efficiently. This hybridization significantly improves convergence speed, stability, and prediction accuracy compared to a single neural network approach. The proposed framework is applied to estimate the dynamic displacement response of the graphene N/MEMS system under varying nonlinear stiffness and electrostatic actuation parameters. The obtained results are validated against reference numerical solutions, demonstrating high accuracy and robustness. Performance evaluation is carried out using Theil’s inequality coefficient (TIC), mean square error (MSE), and mean absolute deviation (MAD), confirming the effectiveness of the proposed approach.
