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This study explores the effects of using space-fractional derivatives and adding multiplicative noise, modeled by a Wiener process, on the solutions of the space-fractional stochastic regularized long wave equation. New fractional stochastic solutions are constructed, and the consistency of the obtained solutions is examined using the transition between phase plane orbits. Their bifurcation and dependence on initial conditions are investigated. Some of these solutions are shown graphically, illustrating both the individual and combined influences of fractional order and noise on selected solutions. These effects appear as alterations in the amplitude and width of the solutions, and as variations in their smoothness.

This paper presents a hybrid based nanofluid flow past over a stretching/shrinking surface. The magnetic field effect is applied normal to the surface. Additionally, the melting heat transfer effect is also taken into considerations. For the performance of heat transport phenomenon, the heat generation/absorption effect is added. The suitable similarity transformations are used to transform the partial differential equations into dimensionless form of ordinary differential equations. For obtaining solutions of the problem, a bvp4c technique is used to handle the transformed differential equations along with boundary conditions. Dual nature study is performed which is the first and second solutions of the problem. In the first solution, increasing the melting temperature improves the rate of heat transfer. Higher values of the shrinking parameter cause an increase in the velocity profile in the first solution and decreases in the second solution. Further, the temperature of the liquid improves as the thermal radiation and heat generation parameters increase, respectively. The temporal stability analysis represents that only one of the two solutions is stable as time evolves. The numerical results are acquired in the form of tabulated data and graphical structures.

In this paper, we investigate the time M-fractional modified nonlinear Schrödinger equation that describes the propagation of rogue waves in deep water. Periodic, solitary, and kink (or anti-kink) wave solutions are discussed using the bifurcation theory for planar integrable systems. Some new wave solutions are constructed using the first integral for the traveling wave system. The degeneracy of the obtained solutions is investigated by using the transition between orbits. We visually explore some of the solutions using graphical representations for different values of the fractional order.

In this study, an advanced computational artificial neural network (ANN) procedure is designed using the novel characteristics of the Levenberg–Marquardt backpropagation (LBMBP), i.e., ANN-LBMBP, for solving the waste plastic management in the ocean system that plays an important role in the economy of any country. The nonlinear mathematical form of the waste plastic management in the ocean system is categorized into three groups: waste plastic material W(χ), marine debris M(χ), and reprocess or recycle R(χ). The learning based on the stochastic ANN-LBMBP procedures for solving mathematical waste plastic management in the ocean is used to authenticate the sample statics, testing, certification, and training. Three different statistics for the model are considered as training 70%, while for both validation and testing are 15%. To observe the performances of the mathematical model, a reference dataset using the Adams method is designed. To reduce the mean square error (MSE) values, the numerical performances through the ANN-LBMBP procedures are obtained. The accuracy of the designed ANN-LBMBP procedures is observed using the absolute error. The capability, precision, steadfastness, and aptitude of the ANN-LBMBP procedures are accomplished based on the multiple topographies of the correlation and MSE.