Lately, a new class of nanofluids, namely hybrid nanofluids, has been introduced that performs much better compared with the nanofluids when a healthier heat transfer rate is the objective of the study. Heading in the same direction, the present investigation accentuates the unsteady hybrid nanofluid flow involving CuO, Al₂O₃/C₂H₆O₂ achieved by an oscillating disk immersed in the porous media. In a study of the homogeneous and heterogeneous reactions, the surface catalyzed reaction was also considered to minimize the reaction time. The shape factors of the nanoparticles were also taken into account, as these play a vital role in assessing the thermal conductivity and heat transfer rate of the system. The assumed model is presented mathematically in the form of partial differential equations. The system is transformed by invoking special similarity transformations. The Keller Box scheme was used to obtain numerical and graphical results. It is inferred that the blade-shaped nanoparticles have the best thermal conductivity that boosts the heat transfer efficiency. The oscillation and surface-catalyzed chemical reactions have opposite impacts on the concentration profile. This analysis also includes a comparison of the proposed model with a published result in a limiting case to check the authenticity of the presented model.

This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point $ 0\leq c < d $ and fixed points at the end of the interval $ [0, d] $. For $ i = 0, 1 $, the boundary conditions are as follows: $ a_{i}, b_{i} > 0 $, $ a_{0} p(c) = -b_{0} p(d), \ a_{1} p^{'}(c) = -b_{1} p^{'}(d) $. Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixed-point theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved.

This manuscript aims to study the existence and uniqueness of solutions to a new system of differential equations. This system is a mixture of fractional operators and stochastic variables. The study has been completed under nonlocal functional boundary conditions. In the study, we used the fixed-point method to examine the existence of a solution to the proposed system, mainly focusing on the theorems of Leray, Schauder, and Perov in generalized metric spaces. Finally, an example has been provided to support and underscore our results.

This article analyzes a complex coupled system of multipoint nonlinear boundary value problems involving Caputo‐type fractional discrete differential equations with multiple fractional q −integrals. We establish the uniqueness and existence of solutions using a rigorous approach grounded in fixed‐point theory, specifically Banach’s fixed‐point theorem and the Leray–Schauder theorem. The theoretical findings are supported by several concrete examples, demonstrating the practical applicability of the developed analytical techniques and advancing solution methodologies for fractional discrete boundary value problems.

This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order $3<\hat{u} \leq 4$3 < u ˆ ≤ 4 and includes a term $\Theta (\beta,\mathscr{Z}(\beta ))$Θ ( β , Z ( β ) ) alongside boundary conditions. Through the application of a fixed point theorem in appropriate function spaces, we consider nonlocal conditions along with necessary assumptions under which solutions to the given FDE exist. Furthermore, we offer an example to illustrate the results.