Symmetries are crucial to the investigation of nonlinear physical processes, particularly the evaluation of a differential problem in the real world. This study focuses on the investigation of the Kadomtsev–Petviashvili $\left(\mathrm{KP}\right)$ model within a (3+1)-dimensional domain, governing the behavior of wave propagation in a medium characterized by both nonlinearity and dispersion. The inquiry employs two distinct analytical techniques to derive multiple soliton solutions and multiple solitary wave solutions. These methods include the modified Sardar sub-equation technique and the Darboux transformation $\left(\mathrm{DT}\right)$. The modified Sardar sub-equation technique is used to obtain multiple soliton solutions, while the $\mathrm{DT}$ is introduced to develop two bright and two dark soliton solutions. These solutions are presented alongside their corresponding constraint conditions and illustrated through 3-D, 2-D, and contour plots to physically portray the derived solutions. The results demonstrate that the employed analytical techniques are useful and have not yet been explored in the context of the analyzed models. The proposed methodologies are valuable and can be applied to additional nonlinear evolutionary models employed to describe nonlinear physical models within the domain of nonlinear science.

In nature, symmetry is all around us. The symmetry framework represents integer partial differential equations and their fractional order in the sense of Caputo derivatives. This article suggests a semi-analytical approach based on Aboodh transform ($\mathbb{A}$T) and the homotopy perturbation scheme (HPS) for achieving the approximate solution of time-fractional porous media and heat transfer equations. The $\mathbb{A}$T converts the fractional problems into simple ones and obtains the recurrence relation without any discretization or assumption. This nonlinear recurrence relation can be decomposed via the use of the HPS to obtain the iterations in terms of series solutions. The initial conditions play an important role in determining the successive iterations and yields towards the exact solution. We provide some numerical applications to analyze the accuracy of this proposed scheme and show that the performance of our scheme has strong agreement with the exact results.

In this paper, we suggest a modification for the residual power series method that is used to solve fractional-order Helmholtz equations, which is called the Shehu-transform residual power series method ($\mathbb{S}$T-RPSM). This scheme uses a combination of the Shehu transform ($\mathbb{S}$T) and the residual power series method (RPSM). The fractional derivatives are taken with respect to Caputo order. The novelty of this approach is that it does not restrict the fractional order and reduces the need for heavy computational work. The results were obtained using an iterative series that led to an exact solution. The 3D graphical plots for different values of fractional orders are shown to compare $\mathbb{S}$T-RPSM results with exact solutions.

The goal of this paper is to consider a differential equation system written as an interesting equivalent form that has not been used before. Using Perov’s fixed point theorem in generalized metric spaces, the existence and uniqueness of the solution are obtained for the proposed system. The approximation of the solution is given, and as a novelty, the approximation of its derivative is also obtained using the same iteration steps.

This paper investigates second-order functional dynamic equations with mixed nonlinearities on an arbitrary unbounded above-time scale, T. We will use a unified time scale approach and the well-known Riccati technique to derive oscillation criteria of the Nehari-type for second-order dynamic equations. The findings demonstrate a significant improvement in the literature on dynamic equations. The symmetry is beneficial and influential in defining the right style of study for the qualitative behavior of solutions to dynamic equations. We include an example to demonstrate the significance of our results.