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This communication studies the importance of convective heat transfer is intensified remarkably in various disciplines of modern engineering sciences and technological development such as heat exchangers, refrigeration, air conditioning, food processing, damage to crops, and many more. The main focus of this study is to develop mathematical modeling of the 2D stagnation point flow of configured by an extended heated stretchable sheet subject to nonlinear thermal radiation with the revised nanofluid model. Moreover, the influence of Newtonian heating, MHD flow, and Brownian movement features are invoked for analysis. The essential nonlinear PDEs of this assessment are modeled with the aid of boundary layer theory and then renovated into nonlinear ODEs by invoking appropriate similarity solutions with help of MATHEMATICA 11.0 programming language. The physical insight of relevant flow parameters is highlighted through graphical illustration. Finally, this investigation greatly impacts engineering and industrial applications of the materials, mainly in geophysical and geothermal systems, storage devices, space science, and several other disciplines.

This paper examines two methods for solving the nonlinear fractional Phi-four problem with variable coefficients. One of the distinct states of the Klein–Gordon model yields the Phi-four equation. It is also used to simulate the kink and anti-kink solitary wave connections that have recently emerged in biological systems and nuclear particle physics. The approaches that are being suggested consist of the Yang transform, the homotopy perturbation approach, the decomposition approach, and the fractional derivative as stated by Caputo. The advantages of the proposed techniques are their capability of combining two dominant approaches for attaining precise and approximate solutions of nonlinear equations. It is important to keep in mind that the suggested methods can perform better in general as they need less computational effort than the alternative methods, while keeping a high level of numerical precision. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. Additionally, several simulations are used to show the physical behaviors of the found solutions with regard to fractional order. The article’s results possess complimentary properties that relate to the symmetry of partial differential equations.

The fractional structures of variables using Riemann–Liouville notion have been analyzed by various authors. The novel idea of this article is to introduce the new notion of weighted behavior on random variables using integral inequalities. In view of these, we obtain some new generalized fractional inequalities by using this new fractional integration of continuous random variables.

This article investigates the seventh-order Lax’s Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV equation was constructed to represent a wide range of physical processes involving the evolution and interaction of nonlinear waves. In the Caputo sense, the fractional derivative is considered. We employed the Yang transform, the Adomian decomposition method and the homotopy perturbation method to obtain the solution to the time-fractional Lax’s Korteweg–de Vries problem. We examined and compared a particular example with the actual result to verify the approaches. By utilizing these methods, we can construct recurrence relations that represent the solution to the problem that is being proposed, and we are then able to present graphical representations that enable us to visually examine all of the results in the proposed case for different fractional order values. Furthermore, the results of the current approach exhibit a good correlation with the precise solution to the problem being studied. Furthermore, the present study offers an example of error analysis. The numerical outcomes obtained by applying the provided approaches demonstrate that the techniques are easy to use and have superior computational performance.