This article investigates the impacts of varying thermal conductivity in the semi-infinite medium due to laser heating sources with a timed pulse. The governing formulations are based on a Lord-Shulman (LS) thermoelastic model with one thermal relaxation time under changing thermal conductivity. To study the nonlinear problems and get the governing formulations, the time-domain finite element approach is used. Laplace transforms and the eigenvalues method are used to solve the problem in the linear case implementation. The impacts of varying thermal relaxation time, time duration of a laser pulse and thermal conductivity are investigated. The numerical outcome obtained through the finite element approach are compared with analytical solutions for constant thermal conductivity to validate the precision of the numerical solution. The numerical results of temperature, stress, and displacement variation are presented graphically.

This paper focuses on studying the oscillatory properties of a distinctive class of second-order advanced differential equations with distributed deviating arguments in a noncanonical case. Utilizing the Riccati method and the comparison method with first-order equations, in addition to other analytical methods, we have established criteria to test whether the solutions of the studied equation exhibit oscillatory behavior. To verify the validity of the results we obtained and determine their applicability, we present some examples to confirm the strength and accuracy of our proposed criteria.

In this article, we present a unified mathematical solver that is widely used in applied nonlinear differential equations based on the well-known tanh–coth method. We employ this solver to provide new solutions for the unstable and modified unstable nonlinear Schrödinger models. We also introduce He’s frequency technique to examine the dynamical behavior properties. The presented solutions include various mathematical structures, such as solitons, dissipative super-waves, bright and dark envelope waves, and explosive envelope waves. The behavior of the presented solutions for suitable free parameter values is depicted by two-dimensional and three-dimensional graphs. The proposed approach offers various benefits, such as eliminating time-consuming and difficult calculations. The results presented here can be utilized to characterize a wide range of physical phenomena in numerous disciplines of practical study, such as pulse propagation in optical fibers and quantum field. Finally, this study increases the current understanding of mathematical methods for solving nonlinear differential equations, which may be applied to other nonlinear wave equations in other scientific areas.