The current work investigates solitary wave solutions for the fractional modified Degasperis-Procesi equation and the fractional gas dynamics equation with Caputo's derivative by using a modified extended direct algebraic method. This method transforms the targeted fractional partial differential equations (FPDEs) into more manageable nonlinear ordinary differential equations, which are then turned into systems of nonlinear algebraic equations with a series-based solution assumption. Using Maple 13, the solitary wave solutions are then obtained by solving the obtained systems. The method produces multiple innovative solitary wave solutions for both equations, which are graphically depicted as 3D and 2D graphs and provide important insights into their behaviors. These insights help us to comprehend wave behavior and the physical processes represented by these equations. Furthermore, the suggested technique exhibits dependability and efficacy in dealing with complicated FPDEs, which bodes well for future studies on the subject.

The present research applies an improved version of the modified Extended Direct Algebraic Method (mEDAM) called $r$+mEDAM to examine soliton phenomena in a notable mathematical model, namely the (2+1)-dimensional Nizhnik-Novikov-Veselov Model (NNVM), which possesses potential applications in exponentially localized structure interactions. The generalized hyperbolic and trigonometric functions are used to disclose a variety of soliton solutions, including kinks, anti-kink, bell-shaped and periodic soliton. Some 3D graphs are plotted for visual representations of these solutions which highlight their adaptability. The results provide a basis for practical usage and expansions to related mathematical models or physical systems. They also expand our understanding of the NNVM's dynamics, providing insights into its behavior and prospective applications.

This analysis used the backpropagation Levenberg-Marquardt technique coupled through neural networks (BPLMT-NN). The magnetohydrodynamic (MHD) viscous nanofluid flow due to the rotating disk (MHD-VNRD) through the slip effect was investigated. In the existence of a velocity slip condition, this communication investigated the boundary coating flow of viscous nanofluid under MHD conditions. The flow was produced by a disk that was revolving. A fluid effects electricity under the effect of a magnetic field that is transverse. The magnetic field that is generated is neglected when the magnetic Reynolds number is low. The properties of Brownian and thermophoresis motion were demonstrated using a nanofluid simulation. Hypotheses about the boundary coating and low magnetic Reynolds number were made while formulating the problem. To transform nonlinear partial differential equations into a scheme of ordinary differential equations, the similarity transformation was utilized. On the profiles of velocity, temperature, and concentration, a data set for the suggested (BPLMT-NN) was created for the impacts of several important parameters and was illustrated via the explicit Runge-Kutta technique. Using the (BPLMT-NN) testing, training, and validation approach, the estimated result of various situations was endorsed, and the suggested model was evaluated for fitness. After that, the proposed (BPLMT-NN) was validated using mean square error (MSE), regression analysis, and histogram investigations. The novelty of the proposed BPLMT-NN technique has various applications, such as disease diagnosis, robotic control systems, ecosystem evaluation, etc. We conducted analyses of some statistical data like gradient, performance, and epoch of the proposed fluid model. Based on closeness, as well as recommended and reference results, the suggested approach has made a distinction with precision level varying from ${10}^{-09}$ to ${10}^{-11}$.

In this study, we presented the conformable Laplace transform iterative method to find the approximate solution of the systems of nonlinear temporal-fractional differential equations in the sense of the conformable derivative. The advantage of the suggested approach was to compute the solution without discretization and restrictive assumptions. Three distinct examples were provided to show the applicability and efficacy of the proposed approach. To examine the exact and approximate solutions, we utilized the 2D and 3D graphics. Furthermore, the outcomes produced in this study were consistent with the exact solutions; hence, this strategy efficiently and effectively determined exact and approximate solutions to nonlinear temporal-fractional differential equations.

This study delved into the dynamics of wave solutions within the Estevez-Mansfield-Clarkson equation in fractional nonlinear space-time. Utilizing conformable fractional derivatives, the equation governing shallow water phenomena and fluid dynamics was transformed into a nonlinear ordinary differential equation. Applying the Riccati Bernoulli sub-ODE approach yielded a finite series representation. Notably, our findings revealed novel solitary wave solutions characterized by kink, anti-kink, periodic, and shock functions. Visualized through 3D and contour graphs, kink and periodic waves emerged as distinct observable manifestations. Intriguingly, the diversity of results surpassed previous results, contributing to a deeper understanding of the intricate dynamics inherent in the system.