N. S. Alharthi

This work examines the fractional Sawada–Kotera and Korteweg–de Vries (KdV)–Burgers equations, which are essential models of nonlinear wave phenomena in many scientific domains. The homotopy perturbation transform method (HPTM) and the Yang transform decomposition method (YTDM) are two sophisticated techniques employed to derive analytical solutions. The proposed methods are novel and remarkable hybrid integral transform schemes that effectively incorporate the Adomian decomposition method, homotopy perturbation method, and Yang transform method. They efficiently yield rapidly convergent series-type solutions through an iterative process that requires fewer computations. The Caputo operator, used to express the fractional derivatives in the equations, provides a robust framework for analyzing the behavior of non-integer-order systems. To validate the accuracy and reliability of the obtained solutions, numerical simulations and graphical representations are presented. Furthermore, the results are compared with exact solutions using various tables and graphs, illustrating the effectiveness and ease of implementation of the proposed approaches for various fractional partial differential equations. The influence of the non-integer parameter on the solutions behavior is specifically examined, highlighting its function in regulating wave propagation and diffusion. In addition, a comparison with the natural transform iterative method and optimal auxiliary function method demonstrates that the proposed methods are more accurate than these alternative approaches. The results highlight the potential of YTDM and HPTM as reliable tools for solving nonlinear fractional differential equations and affirm their relevance in wave mechanics, fluid dynamics, and other fields where fractional-order models are applied.

The fluid flow across two parallel permeable spinning sheets is studied in the current analysis. The hybrid nanofluid (HNF) is composed of aluminum alloys (Ti₆Al₄V-AA7075), ethylene glycol (50%) and water (50%). The unification of Ti₆Al₄V and AA7075 alloys delivers an effective functionality of robustness and durability against corrosion. The probable uses take in aviation elements, the automobile sector, athletic apparel and surgical instruments. Therefore, the aim of this study is to mathematically formulate the HNF flow consisting of Ti₆Al₄V-AA7075 alloys in the form of nonlinear coupled PDEs. The flow equations are reconditioned into the dimensionless form by employing a similar approach. The transform equations are further numerically solved through the PCM (parametric continuation method). For the validity of the results, the outcomes are compared to the published studies. The PCM results are presented in the figures. It is noticed that the effect of Reynold number, surface rotation and porosity parameters drops the flow rate. The inclusion of Ti₆Al₄V-NPs in base fluid remarkably enhances the mass and energy conduction rate of HNF. The thermal field upsurges with the effect of heat radiation and exponential heat source/sink, whereas drops with variation Ti₆Al₄V and AA7075 NPs.

In many science and engineering fields, integer-order differential equations are unable to provide a satisfactory explanation for a wide range of phenomena when compared to fractional-order differential equations. The fractional modified Korteweg-de Vries (mKdV) equations are investigated in this work by employing effective analytical methods within the Caputo operator. The findings for the given problems are computed using the Elzaki transformation, the homotopy perturbation method, and the Adomian decomposition method. With these techniques, the problems were first made simpler utilizing the Elzaki transform, and the problems were then comprehensively solved by employing the decomposition and perturbation approaches. A few numerical cases with their approximate analytical solutions are considered to demonstrate the conclusions drawn from the findings. To verify these approaches, we examined two cases and compared them with the real outcomes. By using these methods, the solution to the suggested problem is represented by recurrence relations. The selected issues have series solutions that can be found and have features that more quickly approach the exact results. It is found that there is a strong correlation between the derived results and the real results of every problem when the number of terms increases in the series solution of the problem. The use of efficient techniques that provide higher levels of accuracy with less computation makes the current work innovative. To further benefit the scientific community, the proposed methods can also be used in the future to solve other fractional nonlinear problems.

The Zakharov–Kuznetsov equation, along with its modified forms [the modified Zakharov–Kuznetsov and further modified Zakharov–Kuznetsov equations], is important modeling equations for interpreting different magnetoacoustic waves in plasma. In this study, we established stable broad-ranging soliton solutions, including well-known lump- and stripe-shaped soliton solutions, obtained using the Hirota bilinear (HB) method. The figures in this text show the effect of physical parameters in the plasma system—dispersion and turbulent dispersion coefficients—on wave height. Wave heights remain determinable for different values of the physical parameters and the free parameters associated with them. A three-dimensional (3D) plot of the ascertained solutions clarified the stability points of the equations we studied based on the bifurcation theory. The results for each modeling equation make a significant contribution to the analysis of dust-ion-acoustic waves in plasma in Saturn's magnetosphere. We assert that the HB method is reliable, appropriate, easy to use, and powerful for extracting closed-form soliton solutions.

The goal of the current study is to analyze several nonlinear two-dimensional time-fractional Rosenau–Hyman equations. The two-dimensional fractional Rosenau–Hyman equation has extensive use in engineering and applied sciences. The fractional view analysis of two-dimensional time-fractional Rosenau–Hyman equations is discussed using the homotopy perturbation approach, Adomian decomposition method, and Yang transformation. Some examples involving two-dimensional time-fractional Rosenau–Hyman equations are provided to better understand the suggested approaches. The solutions appear as infinite series. We offer a comparison between the accurate solutions and those that are generated employing the proposed approaches to demonstrate the effectiveness and applicability of the proposed techniques. The results are graphically illustrated using two-dimensional and three-dimensional graphs. It has been noted that the obtained results and the targeted problems real solutions are quite similar. Calculated solutions at various fractional levels describe some of the problems useful dynamics. A comparison between the numerical solutions of the models under study and the exact solutions in cases when a solution is known serves as a clear demonstration of the viability and dependability of the suggested approaches. Other fractional problems that arise in other fields of science and engineering can be solved using a modified version of the current techniques.