Our investigation delves into a specific category of nonlinear pseudo-parabolic partial differential equations (PDEs) that emerges from physical models. This set of equations includes the one-dimensional (1D) Oskolkov equation, the Benjamin-Bona-Mahony-Peregrine-Burgers (BBMPB) equation, the generalized hyperelastic rod wave (HERW) equation, and the Oskolkov Benjamin Bona Mahony Burgers (OBBMB) equation. We employ the new extended direct algebraic (NEDA) method to tackle these equations. The NEDA method serves as a powerful tool for our analysis, enabling us to obtain solutions grounded in various mathematical functions, such as hyperbolic, trigonometric, rational, exponential, and polynomial functions. As we delve into the physical implications of these solutions, we uncover complex structures with well-known characteristics. These include entities like dark, bright, singular, combined dark-bright solitons, dark-singular-combined solitons, solitary wave solutions, and others.

In this paper, we are putting forward the periodic solution of non-linear oscillators by means of variational iterative method (VIM) using Laplace transform. Here, we present a comparative study of the new technique based on Laplace transform and the previous techniques of maximum minimum approach (MMA) and amplitude frequency formulation (AFF) for the analytical results. For the non-linear oscillators, MMA, AFF and VIM by Laplace transform give the same analytical results. Comparison of analytical results of VIM by Laplace transform with numerical results by fourth-order Runge–Kutta (RK) method conforms the soundness of the method for solving non-linear oscillators as well as for the time and boundary conditions of the non-linear oscillators.