The impact of inter-particle spacing and the radius of gold nanoparticles on nanofluid flow have substantial significance across applications. Optimizing these parameters in biomedical engineering enhances the drug delivery systems, thus controlling the release of medicines and accurately targeting the targeted area. We explored nanofluid flow on a bi-directional elongated plate. The surface of the sheet was characterized with variable porosity with inclined magnetic field effects, which is the main novelty of the work. We focused on how nanoparticle radius and spacing affect the overall flow dynamics. Additionally, we incorporated the Cattaneo-Christov heat and mass flux model effects to discuss the mass and thermal diffusions using some flow conditions. The major equations were translated in dimensionless form and solved with artificial neural networks (ANNs). As outcomes, we uncovered that primary velocity has weakened with extension in stretching ratio and magnetic factors and has been amplified with progression in variable porous factor with absolute error (AE) in the range 10^-3 to 10^-7. Thermal panels have enlarged with escalation in thermophoresis, magnetic, radiation, and Brownian motion factors with absolute errors AEs in the range ${10}^{-3}$ to ${10}^{-7}$. Concentration panels have escalated with augmentation in the thermophoresis factor and activation energy factor and weakened with the expansion in Schmidt number, chemical reactivity factor, and Brownian motion factor. We conclude that the model's optimal performance has observed at epochs 111,225,194,270,179,220,339, and 221 for different scenarios. For all the scenarios, the gradient values are associated at $9.97\times {10}^{-8}$, $9.91\times {10}^{-9}$, $9.92\times {10}^{-8}$, $9.91\times {10}^{-8}$, $9.95\times {10}^{-8}$, $9.91\times {10}^{-8}$, $9.92\times {10}^{-8}$, and $9.91\times {10}^{-8}$.

We examined ternary hybrid Carreau nanofluid flow on a porous bi-directional elongating sheet. The nanoparticles of TiO₂, CoFe₂O₄, and MgO were mixed with water to get a ternary hybrid nanofluid. The flow was influenced by slip conditions of velocities along the x- and y-axes. The impacts of thermal- and space-dependent heat sources, thermal radiation, viscous dissipation, and Joule heating were used in this study. Moreover, magnetic effects were used along the z-axis, which was normal to the flow direction. The major equations were solved using the homotopy analysis method (HAM) in dimensionless form. As an outcome of this study, we discovered that with progression in velocity slip factors along x- and y-axes, magnetic factor, porosity factor, and local Weissenberg number, there was a reduction in primary and secondary velocities. With an upsurge in the stretching ratio factor, there was a reduction in primary flow and augmentation in secondary flow. Thermal distribution was augmented with the surge in thermal Biot number, thermal-dependent heat source factor, magnetic factor, space-dependent heat source parameter, radiation factor, and Eckert numbers along primary and secondary directions. The skin friction coefficients have augmented with growth in magnetic factor, porosity factor, and velocity slip factors along the x- and y-axes. The Nusselt number escalated with a surge in radiation factor, space-dependent heat source factor, thermal-dependent heat source factor, and Eckert numbers along x- and y-axes. Our results were validated through comparative analysis by matching our results with established data. A fine agreement was noticed among all the results. Our findings benefit aerospace, biomedical, and electronics industries by improving thermal management in porous media. Magnetic and slip conditions aid in advanced manufacturing, while enhanced Nusselt numbers support efficient heat exchanger design.

We focused on the analytical solution of strong nonlinearity and complicated time-fractional evolution equations, including the Sawada-Kotera equation, Ito equation, and Kaup-Kupershmidt equation, using an effective and accurate method known as the Aboodh residual power series method (ARPSM) in the framework of the Caputo operator. Therefore, the Caputo operator and the ARPSM are practical for figuring out a linear or nonlinear system with a fractional derivative. This technique was effectively proposed to obtain a set of analytical solutions for various types of fractional differential equations. The derived solutions enabled us to understand the mechanisms behind the propagation and generation of numerous nonlinear phenomena observed in diverse scientific domains, including plasma physics, fluid physics, and optical fibers. The fractional property also revealed some ambiguity that may be observed in many natural phenomena, and this is one of the most important distinguishing factors between fractional differential equations and non-fractional ones. We also helped clarify fractional calculus in nonlinear dynamics, motivating researchers to work in mathematical physics.

This paper presents a novel approach for exploring the dynamics of fractional Kersten-Krasil'shchik coupled KdV-mKdV systems by using the unification of the Adomian decomposition method and ZZ transformation. The suggested method combines the Aboodh transform and the Adomian decomposition method, both of which are trustworthy and efficient mathematical tools for solving fractional differential equations (FDEs). This method's theoretical analysis is addressed for nonlinear FDE systems. To find exact solutions to the equations, the method is applied to fractional Kersten-Krasil'shchik linked KdV-mKdV systems. The results show that the suggested method is efficient and practical for solving fractional Kersten-Krasil'shchik linked KdV-mKdV systems and that it may be applied to other nonlinear FDEs. The suggested method has the potential to provide new insights into the behavior of nonlinear waves in fluid and plasma environments, as well as the development of new mathematical tools for modeling and studying complicated wave phenomena.

This study presented a comprehensive analysis of nonlinear fractional systems governed by the advection-dispersion equations (ADE), utilizing the Mohand transform iterative method (MTIM) and the Mohand residual power series method (MRPSM). By incorporating the Caputo fractional derivative, we enhanced the modeling capability for fractional-order differential equations, accounting for nonlocal effects and memory in the systems dynamics. We demonstrated that both MTIM and MRPSM were effective for solving fractional ADEs, providing accurate numerical solutions that were validated against exact results. The steady-state solutions, complemented by graphical representations, highlighted the behavior of the system for varying fractional orders and showcased the flexibility and robustness of the methods. These findings contributed significantly to the field of computational physics, offering powerful tools for tackling complex fractional-order systems and advancing research in related fields.

This paper introduces a novel numerical approach for tackling the nonlinear fractional Phi-four equation by employing the Homotopy perturbation method (HPM) and the Adomian decomposition method (ADM), augmented by the Shehu transform. These established techniques are adept at addressing nonlinear differential equations. The equation's complexity is reduced by applying the Shehu Transform, rendering it amenable to solutions via HPM and ADM. The efficacy of this approach is underscored by conclusive results, attesting to its proficiency in solving the equation. With extensive ramifications spanning physics and engineering domains like fluid dynamics, heat transfer, and mechanics, the proposed method emerges as a precise and efficient tool for resolving nonlinear fractional differential equations pervasive in scientific and engineering contexts. Its potential extends to analogous equations, warranting further investigation to unravel its complete capabilities.

This paper focuses on novel approaches to finding solitary wave (SW) solutions for the modified Degasperis-Procesi and fractionally modified Camassa-Holm equations. The study presents two innovative methodologies: the Yang transformation decomposition technique and the homotopy perturbation transformation method. These methods use the Caputo sense fractional order derivative, the Yang transformation, the adomian decomposition technique, and the homotopy perturbation method. The inquiry effectively solves the fractional Camassa-Holm and Degasperis-Procesi equations, which also provides a detailed numerical and graphical comparison of the solutions found. The results, which include accurate solutions, derived solutions, and absolute error displayed in tabular style, demonstrate the effectiveness of the suggested procedures. These procedures are iterative, which results in several answers. The estimated absolute error attests to the correctness and simplicity of these solutions. Especially in plasma physics, these approaches may be expanded to handle various linear and nonlinear physical issues, including the evolution equations controlling nonlinear waves.

This paper presents a comparative study of two popular analytical methods, namely the Homotopy Perturbation Transform Method (HPTM) and the Adomian Decomposition Transform Method (ADTM), to solve two important fractional partial differential equations, namely the fractional heat transfer and porous media equations. The HPTM uses a perturbation approach to construct an approximate solution, while the ADTM decomposes the solution into a series of functions using the Adomian polynomials. The results obtained by the HPTM and ADTM are compared with the exact solutions, and the performance of both methods is evaluated in terms of accuracy and convergence rate. The numerical results show that both methods are efficient in solving the fractional heat transfer and porous media equations, and the HPTM exhibits slightly better accuracy and convergence rate than the ADTM. Overall, the study provides a valuable insight into the application of the HPTM and ADTM in solving fractional differential equations and highlights their potential for solving complex mathematical models in physics and engineering.

The current investigation deals with the bioconvective peristaltic flow of a Sutterby nanofluid in a channel. Here, symmetric channel walls are considered to be elastic. Thermal transport included effects such as thermal radiation, Joule heating, and dissipation. The characteristics of a first-order chemical reaction are integrated into mass transport. We utilized a large wavelength approximation with a small Reynolds number to simplify the system. After that, we used numerical techniques for the solution of a complex system of equations. Finally, the effects of several parameters are examined graphically. This research could have a big influence on optimizing heat and mass transfer in nanofluid-based systems, with potential implications for solar energy systems, thermal management devices, biosensors, fuel cell technology, pharmaceutical processing, and targeted drug delivery mechanisms.

In this paper, we use the Riccati–Bernoulli sub-ODE method in conjunction with the Bäcklund transformation to find out the exact solutions of the nonlinear time–space fractional Bogoyavlenskii equation. The obtained solutions encompass multiple kink solitary wave solutions that are quite unique and important in addition to solutions presented in hyperbolic, trigonometric, and rational function forms. This equation describes central factors influencing its behavior including fluid dynamics in shallow water waves and plasma, which demonstrates our conclusions have broad applications for such systems. We also study the effect of the fractional order parameter ( $\alpha$) on solutions and plot their behavior using MATLAB in two dimensions. This work also contributes to the knowledge of the physical structures of the fractional Bogoyavlenskyi equation apart from showcasing the potential of the Riccati–Bernoulli sub-ODE method when applied to nonlinear fractional differential equations.