In this work, the catalytic activity of three different sizes of gold nano particles (AuNPs) (12, 30, and 45 nm) synthesized by the citrate reduction process studied in the conventional redox reaction of K₃Fe (CN₆)⁻³ to K₄Fe (CN₆)⁻⁴ using NaBH₄(reductant) at four different temperatures (5 °C, 10 °C, 15 °C, and 20 °C) and measured by UV–visible spectrophotometry. Comparative kinetic analysis of different sizes of AuNPs including rate constant, activation energy, Entropy values and Frequency of collisions are reported for the first time. Transmission electron microscopy analysis is employed to investigate morphology and particle size. Spherical nanoparticles of size 12, 30, and 45 nm were observed. The UV–visible spectra were recorded at regular intervals, and it was seen that the peak of K₃Fe (CN₆)⁻³ decreased gradually with time, at the same time surface plasmon resonance of AuNPs remained constant. As reaction catalysts, AuNPs maintain their optical density which shows their stability during the course of reaction. The kinetic parameters i.e., rate constant, and activation energy (k, t_1/2, E_a) were determined for three distinct sizes of gold nanoparticles (AuNPs) using the reductant at various concentrations. The value of k increases by increasing reductant concentration. This rise was significant for the small AuNPs. Increasing gold nanoparticle size (12, 30, 45 nm) decreased rate constant. As the size of AuNPs decreased the E_a reduced as well, i.e. 17.325 k cal mol⁻¹ for 12 nm, 19 k cal mol⁻¹ for 30 nm and 21 k cal mol⁻¹ for 45 nm AuNPs. For 50 mM of NaBH₄, k for 45 nm AuNPs is 0.10728 s⁻¹, but for 12 nm AuNPs, the value of k is 0.145 s⁻¹, indicating that the 12 nm AuNPs have the greatest rate constant values. The rate of reaction rises with an increase in reductant concentration and temperature, but this increase is significant in the case of small-sized nanoparticles, i.e., 12 nm, which have a high surface area and low E_a. Besides being a model redox reaction, the reduction of K₃Fe (CN₆)⁻³ to K₄Fe (CN₆)⁻⁴ has industrial use in making fertilizers and paint industry, anti-coating agent in colour photography, in dot etching and in amperometric biosensors.

The objective of this paper is to study oscillation of fourth-order neutral differential equation. By using Riccati substitution and comparison technique, new oscillation conditions are obtained which insure that all solutions of the studied equation are oscillatory. Our results complement some known results for neutral differential equations. An illustrative example is included.

This paper deals with the oscillation properties of higher-order nonlinear differential equations with distributed delay $$ \bigl[ b ( \ell ) \bigl( y^{ ( n-1 ) } ( \ell ) \bigr) ^{\gamma} \bigr] ^{\prime}+ \int_{c}^{d}q ( \ell, \xi ) y^{\gamma} \bigl( g ( \ell, \xi ) \bigr) \,d ( \xi ) =0, \quad \ell\geq\ell_{0}, $$ under the condition $$ \int_{\ell_{0}}^{\infty}\frac{1}{b^{\frac{1}{\gamma}} ( \ell ) }\,d\ell< \infty. $$ We obtain new oscillation criteria by employing a refinement of the generalized Riccati transformations and new comparison principles. We provide some examples to illustrate the main results.

In this paper, we present an intensive investigation of the finite volume method (FVM) compared to the finite difference methods (FDMs). In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate the previously mentioned methods. On the one hand, we simulate the results of the finite difference methods using the schemes of Lax–Friedrichs and Lax–Wendroff. On the other hand, we apply Godunov’s scheme to simulate the results of the finite volume method. Moreover, we show how starting with a variational formulation of the problem, the finite element technique provides piecewise formulations of functions defined by a collection of grid data points, while the finite difference technique begins with a differential formulation of the problem and continues to discretize the derivatives. Finally, some graphical and numerical comparisons are provided to illustrate and corroborate the differences between these two main methods.

The aim of this work is to study asymptotic properties of a class of fourth-order delay differential equations. Our results in this paper not only generalize some previous results, but also improve the earlier ones. Examples are considered to elucidate the main results.

The oscillation of a class of fourth-order nonlinear damped delay differential equations with distributed deviating arguments is the subject of this research. We propose a new explanation of the fourth-order equation oscillation in terms of the oscillation of a similar well-studied second-order linear differential equation without damping. The extended Riccati transformation, integral averaging approach, and comparison principles are used to provide some additional oscillatory criteria. An example demonstrates the efficacy of the acquired criteria.