computational methods

The main objective of this article is to solve stochastic delay differential equations via Haar wavelets‎. ‎We present fundamental concepts of stochastic process‎, ‎Haar‎, ‎block pulse functions‎, ‎and their operational matrix relevant to time-delayed Haar‎. ‎Analytic solutions of two examples are solved for the first time to approximate two kinds of single time-delayed stochastic differential equations with additive and multiplicative noise‎. ‎This orthogonal basis function not only simplifies the problem but also speeds up the computations and lessens the computational complexity of the stochastic delay differential equations to a lower triangular system of linear algebraic equations‎. ‎The equation can be solved via forward substitution‎, ‎such as lower-upper decomposition method‎. ‎Finally‎, ‎we examine the order of convergence and error analysis of two visual samples to validate the efficiency and effectiveness of the suggested procedure.

Three ordinary differential equations are used to represent mathematically the breakdown of a phenol and pcresolcombination in a constantly agitated bioreactor.The research offers a stability analysis of the model’sequilibrium locations. Three different kernels have also been used to examine the effects of the fractaldimension and the fractional order on the model with the fractal-fractional derivatives. We have developedextremely effective computational techniques for phenol, p-cresol, and biomass concentrations. Finally,computer simulations are used to confirm the correctness of the suggested strategy.

Hydrogen bonds between and within molecules are essential interactions that control how molecules behave in various chemical and biological systems. To better understand the complex nature of hydrogen bonding events, spectroscopic and computational methodologies were integrated in this abstract. Direct probing of the vibrational and electronic fingerprints linked to hydrogen bonds has been made possible by spectroscopic techniques, such as infrared and nuclear magnetic resonance spectroscopy, providing crucial details regarding bond strength, length, and dynamics. Molecular dynamics simulations and advanced computational techniques like density functional theory (DFT) have simultaneously produced a theoretical foundation for comprehending the energetics and geometry of hydrogen bonds. A thorough understanding of hydrogen bonding interactions in a variety of settings, including biomolecular systems, liquids, and solids, has been made possible by the synergistic interaction between experimental data and theoretical discoveries. The combined efforts of spectroscopic and computational research have revealed the relevance of hydrogen bonds in molecular recognition, reaction processes, and material properties, even though difficulties still exist in adequately simulating solvent effects and long-range interactions. This multidisciplinary approach continues to lead to discoveries as technology develops, providing a deeper understanding of hydrogen bonding and its ramifications across other scientific disciplines.

Polycyclic aromatic hydrocarbons (such as, anthracene, benzo[a]pyrene and so on) are non-polar, hydrophobic compounds, which are not ionized. They are only slightly soluble in water. They are very dangerous compounds in the environment. The single-walled carbon nanotube (SWNT) is used for removal and conversion of anthracene to low-risk products. In this study, electron transfer between anthracene and SWNT (8,8) is evaluated through density functional approach at the level of B3LYP/6-31G. The calculation of the electronic properties shows that SWNT is very sensitive to the presence of anthracene molecule. The HOMO/LUMO and gap energy (Eg) changes were considerable. According to the calculated thermodynamic parameters through the DFT method, it is expected that SWNT be a candidate in the elimination of anthracene as well as a gas sensor for its detection and conversion. Regarding the thermodynamic results, the absorption of pollutant on nano-surface of SWNT is exothermic and spontaneous. The results show that the pollutant can be reduced or eliminated from the environment by single-walled carbon nanotubes.