In this work, we study a time-fractional ion sound and Langmuir waves system (FISLWS) with Atangana–Baleanu derivative (ABD). We use a fractional ABD operator to transform our system into an ODE. We investigate multiwaves, periodic cross-kink, rational, and interaction solutions by the combination of rational, trigonometric, and various bilinear functions. Furthermore, 3D, 2D, and relevant contour plots are presented for the natural evolution of the gained solutions under the selection of proper parameters.

The aim of this work is to build novel analytical wave solutions of the nonlinear space-time fractional (2+1)-dimensional breaking soliton equations, with regards to the modified Riemann–Liouville derivative, by employing mathematical schemes, namely, the improved simple equation and modified F-expansion methods. We used the fractional complex transformation of the concern fractional differential equation to convert it for the solvable integer order differential equation. After the successful implementation of the presented methods, a comprehensive class of novel and broad-ranging exact and solitary travelling wave solutions were discovered, in terms of trigonometric, rational and hyperbolic functions. Hence, the present methods are reliable and efficient for solving nonlinear fractional problems in mathematics physics.

This research explores new soliton solutions to the Atangana–Baleanu derivative (ABD) fractional system of equations for ion sound and Langmuir waves (FISLW). We utilize the fractional ABD operator to convert our system into an ordinary differential equations. In recent years, machine learning (ML) evolves significantly in the context of data analysis and computing different solutions, which typically enables systems to operate wisely. Now, we are going to use numerous ML tools including matplotlib. pyplot as plt, scipy.integrate, mpl toolkits.mplot3d, and Axes3D to generate various types of optical solutions by using complete discriminant of the polynomial method. We will also analyze solutions for the hyperbolic function, trigonometric function, Jacobian elliptic function (JEF), and other solitary wave solutions. Solitons have extensive uses in pure and applied mathematics, including nonlinear partial differential equations: the Boussinesq equation, the nonlinear Schrödinger equation, and the sine-Gordon equation, Lie groups, Lie algebras, and differential and algebraic geometry. In addition, we study the chaotic behaviour, i.e., 2D, 3D, time series, Poincarè maps, and sensitivity analysis of our governing model. Sensitivity analysis explores how changes in a system’s variables affects its behaviour.

Partial differential system featuring momentum and energy balance is often used to describe simulations of flow initiation and thermal shifting in boundary layers. The buoyancy force in terms of temperature is factored in the momentum balance equation. Buoyancy force causes the flow quantity to fluctuate along the streamwise direction 𝑋, therefore problem can be best analyzed through nonsimilar modeling. In this analysis nonsimilar model is evolved for radiative mixed convection of a magnetized power-law nanoliquid flow on top of a vertical plate installed in a stationary fluid. The upward linear stretching initiated the flow in vertical direction. Assuming nano-fluids are composite of copper (Cu) and alumina (Al₂O₃) nanoparticles, the viscous dissipation in this case is negligible. The nonsimilar system is dealt analytically by local nonsimilarity (LNS) via numerical algorithm bvp4c. Surface temperature and flow field are shown visually in relation to factors like mixed convection, magnetic field strength, nanoparticle volume fraction, radiation parameters, and Prandtl number. The repercussions of magnetic and mixed convection parameters on rate of energy transfer and friction coefficient are represented in tabular forms. The results obtained are compared to the published literature. It is found that the existence of nanoparticles significantly improves the temperature profile of considered nanoliquid. It is also observed that when the estimates of the magnetic parameter increase, the velocity profile decreases. Enhancement in nanoparticles concentration and mixed convection parameter improves the velocity profile. According to the author's findings, no published research has yet addressed the description of non-similar analysis for the considered problem. This study might be beneficial to researchers concentrating on industrial nanofluid applications.