In this study, numerical results of a fractional-order multi-dimensional model of the Navier–Stokes equations will be achieved via adoption of two analytical methods, i.e., the Adomian decomposition transform method and the q-Homotopy analysis transform method. The Caputo–Fabrizio operator will be used to define the fractional derivative. The proposed methods will be implemented to provide the series form results of the given models. The series form results of proposed techniques will be validated with the exact results available in the literature. The proposed techniques will be investigated to be efficient, straightforward, and reliable for application to many other scientific and engineering problems.

This study aims to show the role of the stagnation point flow in solar optimization in the presence of a Riga plate. This requirement is conceivable in the case of solar energy management with a suitable solar collector covering and visual thermal optimization. Solar energy radiation and thermal convection of glycol (C₃H₈O₂)-based aluminum oxide (Al₂O₃) and copper (Cu) nanoparticles were used for a solar collector, and were studied in terms of the stagnation point flow theoretically. Stagnation refers to the state of a solar thermal system in which the flux varies in the collection loop to control the extra heating. The CVFEM code was used to analyze the flow in the case of represented stagnation using the FEA-Tools multiple physics software that manages partial derivative equations (PDEs). The streamlined patterns and energy contours for different cases were studied in detail. The transformation equations were treated with the numerical method (RK-4 technique) and showed strong agreement of the physical results corresponding to the initial conditions and boundaries. The results showed that hybrid nanofluids have the advanced capability to enhance the thermal performance of the base solvent and provide uniform distribution to the solar panel. The solar optimization and uniform thermal expansion results are displayed graphically.

To satisfy the electricity needs of a village in Tangi, northwest Pakistan, the present research can design and evaluate the environmental and economical aspects of an optimal hybrid photovoltaic-biogas-hydropower-battery energy sustainable system (PV-BG-HP-BESS). This framework integrates various renewable energy sources, delivering a modern, efficient approach to sustainable energy solutions. The HOMER Pro software is utilized to optimize the most economical and effective hybrid energy system. The results showed that the proposed hybrid system comprising 91.4 kWp PV modules, 19.6 kW hydropower, a 50 kW biogas generator (BG), 36 batteries, and a 60.6 kW converter was the most economical choice. This system, which used the cyclic charging (CC) method, had a cost of energy (COE) of 0.0728 $/kWh and a total net present cost (NPC) of $152,242. The suggested hybrid energy system for rural areas of Pakistan includes photovoltaic (PV), biogas (BG), hydro, and battery components to provide a dependable and sustainable power supply. This system minimizes the need for expensive fossil fuels while simultaneously minimizing environmental impact by lowering pollutants and greenhouse gas emissions. The system's annual electricity production is 294,782 kWh, with PV leads at 59.4%, BG at 6.02%, and hydro at 34.6%, ensuring uninterrupted power generation even in remote areas. The unmet load, extra electricity, and capacity shortage illustrate the reliability of the system and make it possible to address rural electrification challenges while supporting sustainable development and economic growth. Moreover, the outcomes of the proposed hybrid system dominate the previous studies in multiple objectives, including cost and sensitivity analysis, when compared.

The gradient of surface temperature is known as Marangoni convection and plays an important role in silicon melt, spray, atomic reactors, and thin fluid films. Marangoni convection has been considered in the liquid film spray of carbon nanotube (CNT) nanofluid over the unsteady extending surface of a cylinder. The two kinds of CNTs, single-wall carbon nanotubes (SWCNTs) and multiple-wall carbon nanotubes (MWCNTs), formulated as water-based nanofluids have been used for thermal spray analysis. The thickness of the nanofluid film was kept variable for a stable spray rate and pressure distribution. The transformed equations of the flow problem have been solved using the optimal homotopy analysis method (OHAM). The obtained results have been validated through the sum of the total residual errors numerically and graphically for both types of nanofluids. The impact of the physical parameters versus velocity, pressure, and temperature pitches under the influence of the Marangoni convection have been obtained and discussed. The obtained results are validated using the comparison of OHAM and the (ND-solve) method.

In this study, a relatively new method to solve partial differential equations (PDEs) called the fractional reduced differential transform method (FRDTM) is used. The implementation of the method is based on an iterative scheme in series form. We test the proposed method to solve nonlinear fractional Burgers equations in one, two coupled, and three dimensions. To show the efficiency and accuracy of this method, we compare the results with the exact solutions, as well as some established methods. Approximate solutions for different values of fractional derivatives together with exact solutions and absolute errors are represented graphically in two and three dimensions. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional partial differential equations over existing methods.

The primary goal of this study is to create and characterise solitary wave solutions for the conformable Fractional Coupled Boussinesq-Whitham-Broer-Kaup Equations (FCBWBKEs), a model that governs shallow water waves. Through wave transformations and the chain rule, the authors used the modified Extended Direct Algebraic Method (mEDAM) for transforming FCBWBKEs into a more manageable Nonlinear Ordinary Differential Equation (NODE). This accomplishment is particularly noteworthy because it surpasses the drawbacks linked to both the Caputo and Riemann–Liouville definitions in complying to the chain rule. The study uses visual representations such as 3D, 2D, and contour graphs to demonstrate the dynamic nature of solitary wave solutions. Furthermore, the investigation of diverse wave phenomena such as kinks, shock waves, periodic waves, and bell-shaped kink waves highlights the range of knowledge obtained in the study of shallow water wave behavior. Overall, this study introduces novel methodologies that produce valuable and consistent results for the problem under consideration.