This research focuses on solving the nonlinear second-order jet engine vibration equation utilizing a hybrid analytical technique named the least square homotopy perturbation method (LSHPM). The numerical and graphical comparison of the solutions obtained using the homotopy perturbation method (HPM), LSHPM, and the MATLAB built-in solver bvp5c is presented across four distinct cases. Additionally, a comparative analysis between the solutions derived from LSHPM and those reported in previous literature is also presented. The tabular and graphical representation of the solutions, along with the numerical validation through residual error analysis, are given. Furthermore, the convergence analysis of the LSHPM for its stability and solution reliability is provided. The graphical and numerical representation of the residual error analysis reveals that LSHPM exhibits superiority over HPM in terms of rapid convergence and accuracy. The strong agreement between the results obtained from HPM and bvp5c with those of LSHPM demonstrates that LSHPM offers a more efficient, reliable, and fast convergent solution of the initial and boundary value problem.

This work studies the exact solutions of the integrable (3+1)-dimensional combined potential Kadomtsev-Petviashvili (pKP) equation with the B-type Kadomtsev-Petviashvili (BKP) equation, which is used to characterize several nonlinear oscillations occurring in hydrodynamics, plasma physics, and nonlinear optics. A bilinear representation of the pKP-BKP model is used to study the properties of different wave solutions. A variety of ansatzes are utilized to derive lump cross-kink waves, lump cross-periodic waves, rogue waves, as well as two, three, and multi-wave solutions pertinent to the model. In addition, a traveling wave transformation is applied to transform the problem into an ordinary differential equation. The new auxiliary equation methodology yields solutions including rational, exponential, hyperbolic, and trigonometric functions. Graphical visualizations using 2D plots, contour plots, and 3D plots show the dynamics of the obtained solutions. These solutions are of great importance in nonlinear fiber optics and telecommunications, which contribute to our understanding of the fundamental physical models.