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Parkinson’s disease (PD) is a prevalent neurodegenerative disorder, primarily associated with dopaminergic neuron depletion in the Substantia Nigra. Current treatment focuses on compensating for dopamine (DA) deficiency, but the blood–brain barrier (BBB) poses challenges for effective drug delivery. Using differentiated SH-SY5Y cells, we investigated the co-administration of DA and the antioxidant Grape Seed Extract (GSE) to study the cytobiocompability, the cytoprotection against the neurotoxin Rotenone, and their antioxidant effects. For this purpose, two solid lipid nanoparticle (SLN) formulations, DA-co-GSE-SLNs and GSE-ads-DA-SLNs, were synthesized. Such SLNs showed mean particle sizes in the range of 187–297 nm, zeta potential values in the range of −4.1–−9.7 mV, and DA association efficiencies ranging from 35 to 82%, according to the formulation examined. The results showed that DA/GSE-SLNs did not alter cell viability and had a cytoprotective effect against Rotenone-induced toxicity and oxidative stress. In addition, this study also focused on the evaluation of Alpha-synuclein (aS) levels; SLNs showed the potential to modulate the Rotenone-mediated increase in aS levels. In conclusion, our study investigated the potential of SLNs as a delivery system for addressing PD, also representing a promising approach for enhanced delivery of pharmaceutical and antioxidant molecules across the BBB.

An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable is two. As the Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset, the proposed method will be more appropriate for solving such symmetric models. In order to illustrate the efficiency of the new method, we applied it on some test examples. The obtained results confirm the theoretical behavior regarding the order of convergence. Furthermore, the numerical results are in good agreement with the exact solution and are more accurate than other existing results in the literature.

In this research, the analytical methods of the differential transform method (DTM), homotopy asymptotic method (HAM), optimal homotopy asymptotic method (OHAM), Adomian decomposition method (ADM), variation iteration method (VIM) and reproducing kernel Hilbert space method (RKHSM), and the numerical method of the finite difference method (FDM) for (analytical-numerical) simulation of 2D viscous flow along expanding/contracting channels with permeable borders are carried out. The solutions for analytical method are obtained in series form (and the series are convergent), while for the numerical method the solution is obtained taking into account approximation techniques of second-order accuracy. The OHAM and HAM provide an appropriate method for controlling the convergence of the discretization series and adjusting convergence domains, despite having a problem for large sizes of obtained results in series form; for instance, the size of the series solution for the DTM is very small for the same order of accuracy. It is hard to judge which method is the best and all of them have their advantages and disadvantages. For instance, applying the DTM to BVPs is difficult; however, solving BVPs with the HAM, OHAM and VIM is simple and straightforward. The extracted solutions, in comparison with the computational solutions (shooting procedure combined with a Runge–Kutta fourth-order scheme, finite difference method), demonstrate remarkable accuracy. Finally, CPU time, average error and residual error for different cases are presented in tables and figures.

In this paper, a symmetric eight-step predictor method (explicit) of 10th order is presented for the numerical integration of IVPs of second-order ordinary differential equations. This scheme has variable coefficients and can be used as a predictor stage for other implicit schemes. First, we showed the singular P-stability property of the new method, both algebraically and by plotting the stability region. Then, having applied it to well-known problems like Mathieu equation, we showed the advantage of the proposed method in terms of efficiency and consistency over other methods with the same order.

The fuel cell is an electrochemical energy converter that directly converts the chemical energy of the fuel into electrical current and heat. The fuel cell has been able to identify itself as a source of clean energy over the past few decades. In order to achieve the durability and stability of fuel cells, many parameters should be considered and evaluated Therefore, in this study, a single-channel high-temperature polymer exchange membrane fuel cell (HT-PEMFC) has been numerically simulated in three-dimensional, isothermal and single-phase approach. The distribution of the hydrogen and oxygen concentrations, as well as water in the anode and cathode, are shown; then the effect of different parameters of the operating pressure, the gas diffusion layer porosity, the electrical conductivity of the gas diffusion layer, the ionic conductivity of the membrane and the membrane thickness are investigated and evaluated on the fuel cell performance. The results showed that the pressure drop in the cathode channel was higher than the anode channel, so that the pressure drop in the cathode channel was higher than 9 bars but, in the anode channel was equal to 2 bars. By examining the species concentration, it was observed that their concentration at the entrance was higher and at the output was reduced due to participation in the reaction and consumption. Also, with increasing the operating pressure, the electrical conductivity of the gas diffusion layer and ionic conduction of the membrane, the performance of the fuel cell is improved.

A numerical study on natural convection in a cubical cavity with partial top and bottom openings is performed in this paper. One of the vertical walls of the cavity has higher temperature than that of the opposite one; the remaining walls are insulated perfectly. Three-dimensional simulations of governing equations have been performed using a finite volume technique. The results are presented for different parameters such as opening length and Rayleigh number. It is observed that heat transfer rate and fluid flow can be controlled via opening ratio size and Rayleigh number.

This article describes the impact of slip conditions on nanofluid flow through a stretching sheet. Nanofluids are very helpful to enhance the convective heat transfer in a boundary layer flow. Prandtl number also play a major role in controlling the thermal and momentum boundary layers. For this purpose, we have considered a model for effective Prandtl number which is borrowed by means of experimental analysis on a nano boundary layer, steady, two-dimensional incompressible flow through a stretching sheet. We have considered γAl2O3-H2O and Al2O3-C2H6O2 nanoparticles for the governing flow problem. An entropy generation analysis is also presented with the help of the second law of thermodynamics. A numerical technique known as Successive Taylor Series Linearization Method (STSLM) is used to solve the obtained governing nonlinear boundary layer equations. The numerical and graphical results are discussed for two cases i.e., (i) effective Prandtl number and (ii) without effective Prandtl number. From graphical results, it is observed that the velocity profile and temperature profile increases in the absence of effective Prandtl number while both expressions become larger in the presence of Prandtl number. Further, numerical comparison has been presented with previously published results to validate the current methodology and results.

In this article, entropy generation of an Eyring–Powell nanofluid through a permeable stretching surface has been investigated. The impact of magnetohydrodynamics (MHD) and nonlinear thermal radiation are also taken into account. The governing flow problem is modeled with the help of similarity transformation variables. The resulting nonlinear ordinary differential equations are solved numerically with the combination of the Successive linearization method and Chebyshev spectral collocation method. The impact of all the emerging parameters such as Hartmann number, Prandtl number, radiation parameter, Lewis number, thermophoresis parameter, Brownian motion parameter, Reynolds number, fluid parameter, and Brinkmann number are discussed with the help of graphs and tables. It is observed that the influence of the magnetic field opposes the flow. Moreover, entropy generation profile behaves as an increasing function of all the physical parameters.

In this article, entropy generation with radiation on non-Newtonian Carreau nanofluid towards a shrinking sheet is investigated numerically. The effects of magnetohydrodynamics (MHD) are also taken into account. Firstly, the governing flow problem is simplified into ordinary differential equations from partial differential equations with the help of similarity variables. The solution of the resulting nonlinear differential equations is solved numerically with the help of the successive linearization method and Chebyshev spectral collocation method. The influence of all the emerging parameters is discussed with the help of graphs and tables. It is observed that the influence of magnetic field and fluid parameters oppose the flow. It is also analyzed that thermal radiation effects and the Prandtl number show opposite behavior on temperature profile. Furthermore, it is also observed that entropy profile increases for all the physical parameters.

The main concern of the present article is to study steady magnetohydrodynamics (MHD) flow, heat transfer and entropy generation past a permeable rotating disk using a semi numerical/analytical method named Homotopy Analysis Method (HAM). The results of the present study are compared with numerical quadrature solutions employing a shooting technique with excellent correlation in special cases. The entropy generation equation is derived as a function of velocity, temperature and concentration gradients. Effects of flow physical parameters including magnetic interaction parameter, suction parameter, Prandtl number, Schmidt number, Soret and Dufour number on the fluid velocity, temperature and concentration distributions as well as entropy generation number are analysed and discussed in detail. Results show that increasing the Soret number or decreasing the Dufour number tends to decrease the temperature distribution while the concentration distribution is enhanced. The averaged entropy generation number increases with increasing magnetic interaction parameter, suction parameter, Prandtl number, and Schmidt number.

In this article, entropy generation on MHD Casson nanofluid over a porous Stretching/Shrinking surface has been investigated. The influences of nonlinear thermal radiation and chemical reaction have also taken into account. The governing Casson nanofluid flow problem consists of momentum equation, energy equation and nanoparticle concentration. Similarity transformation variables have been used to transform the governing coupled partial differential equations into ordinary differential equations. The resulting highly nonlinear coupled ordinary differential equations have been solved numerically with the help of Successive linearization method (SLM) and Chebyshev spectral collocation method. The impacts of various pertinent parameters of interest are discussed for velocity profile, temperature profile, concentration profile and entropy profile. The expression for local Nusselt number and local Sherwood number are also analyzed and discussed with the help of tables. Furthermore, comparison with the existing is also made as a special case of our study.

Entropy generation during peristaltic flow of nanofluids in a non-uniform two dimensional channel with compliant walls has been studied. The mathematical modelling of the governing flow problem is obtained under the approximation of long wavelength and zero Reynolds number (creeping flow regime). The resulting non-linear partial differential equations are solved with the help of a perturbation method. The analytic and numerical results of different parameters are demonstrated mathematically and graphically. The present analysis provides a theoretical model to estimate the characteristics of several Newtonian and non-Newtonian fluid flows, such as peristaltic transport of blood.